In this section, before we discuss our main results, let us denote, respectively

$$begin{aligned}& Delta ({tau }):= int _{0}^{tau } frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}mu )}{mu } ,mathrm{d} mu quad text{and}quad delta ({tau }):= int _{tau }^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}mu )}{mu } ,mathrm{d} mu , \& eta ({tau }):= int _{0}^{tau } frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}(m+1)}mu )}{mu } ,mathrm{d} mu quad text{and}quad Omega ({tau }):= int _{0}^{tau } frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )}mu )}{mu } ,mathrm{d} mu . end{aligned}$$

Generalized trapezium inequality

We now derive a new generalized fractional trapezium-type integral inequality using the class of harmonic convex functions. For brevity, we denote in the following ({Psi }(tau ):={frac{1}{tau }}).

Theorem 2.1

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]rightarrow mathbb{R}) be an harmonic convex function, then

$$begin{aligned} {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)& leq frac{1}{2eta (1)} biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] \ &leq frac{[{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})]}{2}, end{aligned}$$

where (min mathbb{N}).

Proof

Since ϒ is an harmonic convex function, then

$$begin{aligned} {Upsilon } biggl(frac{2xy}{x+y} biggr)leq frac{1}{2} bigl[{Upsilon }(x)+{ Upsilon }(y)bigr]. end{aligned}$$

This implies

$$begin{aligned} &2{Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) leq {Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr)+{Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr). end{aligned}$$

Multiplying both sides by (frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}) and integrating with respect to τ on ([0,1]), we have

$$begin{aligned} &2{Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }} ,mathrm{d} {tau } \ &quad leq biggl[ int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &qquad{} + int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

This implies

$$begin{aligned} &{2eta (1)} {Upsilon } biggl( frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) \ &quad leq int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &qquad{} + int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &quad = int _{frac{1}{{{b_{2}}}}}^{ frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}} frac{Phi (frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{ (frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{ Upsilon circ Psi }(x) ,mathrm{d}x+ int _{ frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}^{frac{1}{{b_{1}}}} frac{Phi (x-frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{ (x-frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{ Upsilon circ Psi }(x) ,mathrm{d}x \ &quad ={{}_{ (frac{1}{{{b_{2}}}} )^{+}}I_{Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr)+{_{ (frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl(frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr). end{aligned}$$

Now, we prove the second inequality, for this we have

$$begin{aligned} &{Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr)leq frac{m+{tau }}{m+1}{ Upsilon }({b_{1}})+ frac{1-{tau }}{m+1}{Upsilon }({{b_{2}}}). end{aligned}$$

(2.1)

$$begin{aligned} &{Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr)leq frac{m+{tau }}{m+1}{ Upsilon }({{b_{2}}})+ frac{1-{tau }}{m+1}{Upsilon }({b_{1}}). end{aligned}$$

(2.2)

Adding (2.1) and (2.2) and multiplying both sides by (frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}) and integrating with respect to τ on ([0,1]), we have

$$begin{aligned} & int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }\ &qquad {}+ int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &quad leq {bigl[{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}}) bigr]} int _{0}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{tau } )}{{tau }} ,mathrm{d} {tau }. end{aligned}$$

Using generalized fractional integrals, we obtain our second inequality. This completes the proof. □

Corollary 2.1

If we choose (Phi ({tau })={tau }) and (m=1) in Theorem 2.1, we have

$$begin{aligned} {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)& leq frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{ frac{1}{{{b_{2}}}}}^{frac{1}{{b_{1}}}}{Upsilon circ Psi }(x) ,mathrm{d}xleq frac{[{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})]}{2}. end{aligned}$$

Corollary 2.2

If we choose (Phi ({tau })=frac{{tau }^{{alpha }}}{Gamma (alpha )}) in Theorem 2.1, we obtain

$$begin{aligned} {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)leq{}& frac{({b_{1}{b_{2}}}(m+1))^{alpha }Gamma (alpha +1)}{2({{b_{2}}}-{b_{1}})^{alpha }}\ &{}times biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{J_{ (frac{1}{{b_{1}}} )^{-}}} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] \ leq {}&frac{[{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})]}{2}. end{aligned}$$

For (m=1), we obtain

$$begin{aligned} {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)leq {}& frac{2^{alpha -1}({b_{1}{b_{2}}})^{alpha }Gamma (alpha +1)}{({{b_{2}}}-{b_{1}})^{alpha }}\ &{}times biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} {Upsilon circ Psi } biggl( frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) +{J_{ (frac{1}{{b_{1}}} )^{-}}} {Upsilon circ Psi } biggl( frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) biggr] \ leq {}&frac{[{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})]}{2}. end{aligned}$$

Corollary 2.3

If we choose (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Theorem 2.1, we have

$$begin{aligned} {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) leq {}&frac{((m+1){b_{1}{b_{2}}})^{frac{alpha }{k}}Gamma _{k}(alpha +k)}{2({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} \ &{}timesbiggl[{_{k}J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} { Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] \ leq {}&frac{[{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})]}{2}. end{aligned}$$

For (m=1), we obtain

$$begin{aligned} {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) leq {}&frac{2^{frac{alpha }{k}-1}({b_{1}{b_{2}}})^{frac{alpha }{k}}Gamma _{k}(alpha +k)}{({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} \ &{}times biggl[{_{k}J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} { Upsilon circ Psi } biggl(frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) +{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} { Upsilon circ Psi } biggl(frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) biggr] \ leq{}& frac{[{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})]}{2}. end{aligned}$$

Auxiliary results

In this subsection, we derive three new fractional integral identities that will be used in the following.

Lemma 2.2

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]rightarrow mathbb{R}) be a differentiable function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}) and (min mathbb{N}), then

$$begin{aligned} &frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}\ &qquad {}- frac{1}{(m+1)eta (1)} biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] \ &quad =frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)} biggl[ int _{0}^{1} frac{eta ({tau })}{((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &qquad {} – int _{0}^{1} frac{eta ({tau })}{((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Proof

Consider the right-hand side

$$begin{aligned} I:={}&frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)} biggl[ int _{0}^{1} frac{eta ({tau })}{((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &{} – int _{0}^{1} frac{eta ({tau })}{((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } biggr] \ ={}&frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)}[I_{1}-I_{2}], end{aligned}$$

where

$$begin{aligned} I_{1}:={}& int _{0}^{1} frac{eta ({tau })}{((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ ={}& frac{eta (1){Upsilon }({{b_{2}}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\ &{}- frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)} int _{ frac{1}{{{b_{2}}}}}^{frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}} frac{Phi (frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x}{ Upsilon circ Psi }(x) ,mathrm{d}x \ ={}& frac{eta (1){Upsilon }({{b_{2}}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}- frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)}{_{ ( frac{1}{{{b_{2}}}} )^{+}}I_{Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr). end{aligned}$$

Similarly,

$$begin{aligned} I_{2}:={}& int _{0}^{1} frac{eta ({tau })}{((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ ={}&{-} frac{eta (1){Upsilon }({b_{1}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\ &{}+ frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)} int _{ frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}^{frac{1}{{b_{1}}}} frac{Phi (x-frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{x-frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}{ Upsilon circ Psi }(x) ,mathrm{d}x \ ={}&{-} frac{eta (1){Upsilon }({b_{1}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}+ frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)}{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr). end{aligned}$$

Substituting the values of (I_{1}) and (I_{2}) in I, we obtain our required result. □

Remark 2.1

If we choose (m=1) and (Phi ({tau })={tau }), we have

$$begin{aligned} &frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{2}- frac{{b_{1}{b_{2}}}}{({{b_{2}}}-{b_{1}})} int _{frac{1}{{{b_{2}}}}}^{ frac{1}{{b_{1}}}}{Upsilon circ Psi }(x) ,mathrm{d}x \ &quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} biggl[ int _{0}^{1} frac{{tau }}{((1+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}}{Upsilon }’ biggl(frac{2{b_{1}{b_{2}}}}{(1+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ & qquad {}- int _{0}^{1} frac{{tau }}{((1-{tau }){b_{1}}+(1+{tau }){{b_{2}}})^{2}}{Upsilon }’ biggl(frac{2{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(1+{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Corollary 2.4

If we take (m=1) and (Phi (tau )=frac{{tau }^{alpha }}{Gamma (alpha )}) in Lemma 2.2, we obtain

$$begin{aligned} &frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{2}\ &qquad {}- frac{2^{alpha -1}({b_{1}{b_{2}}})^{alpha }Gamma (alpha +1)}{({{b_{2}}}-{b_{1}})^{alpha }} biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} {Upsilon circ Psi } biggl(frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) +J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }{ Upsilon circ Psi } biggl(frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) biggr] \ &quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} biggl[ int _{0}^{1} frac{{tau }^{alpha }}{((1+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{2{b_{1}{b_{2}}}}{(1+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &qquad {} – int _{0}^{1} frac{{tau }^{alpha }}{((1-{tau }){b_{1}}+(1+{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{2{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Corollary 2.5

If we choose (m=1) and (Phi (tau )=frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Lemma 2.2, we obtain

$$begin{aligned} &frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{2}\ &qquad {}- frac{2^{frac{alpha }{k}-1}({b_{1}{b_{2}}})^{frac{alpha }{k}}Gamma _{k}(alpha +k)}{({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} biggl[{_{k}J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} { Upsilon circ Psi } biggl(frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) +{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} { Upsilon circ Psi } biggl(frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr) biggr] \ &quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} biggl[ int _{0}^{1} frac{{tau }^{frac{alpha }{k} }}{((1+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{2{b_{1}{b_{2}}}}{(1+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } \ &qquad {} – int _{0}^{1} frac{{tau }^{frac{alpha }{k}}}{((1-{tau }){b_{1}}+(1+{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{2{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Lemma 2.3

Let (Upsilon :[{b_{1}},{{b_{2}}}]rightarrow mathbb{R}) be a differentiable function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}) and (lambda ,mu in [0,infty )) with (lambda +mu neq0), then

$$begin{aligned} &frac{Omega (lambda )Upsilon ({{b_{2}}})+Omega (mu )Upsilon ({b_{1}})}{lambda +mu }\ &qquad {}- frac{1}{lambda +mu } biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }{ Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} +{_{ (frac{1}{{b_{1}}} )^{-}}I_{Phi }{ Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr] \ &quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ int _{0}^{lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) ,mathrm{d} {tau } \ &qquad {}- int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Proof

Consider the right-hand side

$$begin{aligned} I:={}&{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ int _{0}^{lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) ,mathrm{d} {tau } \ &{}- int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) ,mathrm{d} {tau } biggr] \ ={}&{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})[I_{3}-I_{4}], end{aligned}$$

(2.3)

where

$$begin{aligned} I_{3}:={}& int _{0}^{lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) ,mathrm{d} {tau } \ ={}& frac{Omega (lambda )Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )}\ &{}- frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )} int _{ frac{1}{{{b_{2}}}}}^{ frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )}} frac{Phi (frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )}-x )}{ (frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )}-x )}{ Upsilon circ Psi }(x) ,mathrm{d}x \ ={}& frac{Omega (lambda )Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )}- frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )}{_{ ( frac{1}{{{b_{2}}}} )^{+}}I_{Phi }{ Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)}. end{aligned}$$

Similarly,

$$begin{aligned} I_{4}:={}& int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) ,mathrm{d} {tau } \ ={}&{-} frac{Omega (mu )Upsilon ({b_{1}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )}\ &{}+ frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )} int _{ frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )}}^{ frac{1}{{b_{1}}}} frac{Phi (x-frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} )}{ (x-frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} )}{ Upsilon circ Psi }(x) ,mathrm{d}x \ ={}&{-} frac{Omega (mu )Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )}- frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(lambda +mu )}{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }{ Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)}. end{aligned}$$

Substituting the values of (I_{3}) and (I_{4}) in (2.3), we obtain our required result. □

Corollary 2.6

Choosing (Phi ({tau })={tau }) in Lemma 2.3, we have

$$begin{aligned} &frac{lambda Upsilon ({{b_{2}}})+mu Upsilon ({b_{1}})}{lambda +mu }- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{frac{1}{{{b_{2}}}}}^{ frac{1}{{b_{1}}}}{Upsilon circ Psi }(x) ,mathrm{d}x \ &quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ int _{0}^{lambda } frac{{tau }}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) ,mathrm{d} {tau } \ &qquad {}- int _{0}^{mu } frac{{tau }}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Corollary 2.7

Taking (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) in Lemma 2.3, we obtain

$$begin{aligned} &frac{lambda ^{alpha }Upsilon ({{b_{2}}})+mu ^{alpha }Upsilon ({b_{1}})}{lambda +mu }- frac{({b_{1}{b_{2}}})^{alpha }(lambda +mu )^{alpha -1}Gamma (alpha +1)}{({{b_{2}}}-{b_{1}})^{alpha }} biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }{ Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} \ &qquad {}+{J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr] \ &quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ int _{0}^{lambda } frac{{tau }^{alpha }}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) ,mathrm{d} {tau } \ &qquad {}- int _{0}^{mu } frac{{tau }^{alpha }}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Corollary 2.8

Choosing (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Lemma 2.3, we obtain

$$begin{aligned} &frac{lambda ^{frac{alpha }{k}}Upsilon ({{b_{2}}})+mu ^{frac{alpha }{k}}Upsilon ({b_{1}})}{lambda +mu }- frac{k({b_{1}{b_{2}}})^{frac{alpha }{k}}(lambda +mu )^{frac{alpha }{k}-1}Gamma _{k}(alpha +k)}{({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} biggl[{_{k}{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} { Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} \ &qquad {}+{{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} { Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr] \ &quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ int _{0}^{lambda } frac{{tau }^{frac{alpha }{k}}}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) ,mathrm{d} {tau } \ &qquad {}- int _{0}^{mu } frac{{tau }^{frac{alpha }{k}}}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} Upsilon ‘ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) ,mathrm{d} {tau } biggr]. end{aligned}$$

Lemma 2.4

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]subset (0,+infty )rightarrow mathbb{R}) be a differentiable mapping on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}), then

$$begin{aligned} &{Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{1}{2Delta (1)} biggl[_{frac{1}{{{b_{2}}}}^{+}}I_{Phi }{{ Upsilon }circ {Psi }} biggl( frac{1}{{b_{1}}} biggr)+ _{ frac{1}{{b_{1}}}^{-}}I_{Phi }{{Upsilon } circ {Psi }} biggl( frac{1}{{{b_{2}}}} biggr) biggr]\ &quad = frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2Delta (1)}sum_{j=1}^{4}M_{j}, end{aligned}$$

where

$$begin{aligned}& M_{1}:= int _{0}^{frac{1}{2}} frac{Delta ({tau })}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& M_{2}:= int _{0}^{frac{1}{2}} frac{ (-Delta ({tau }) )}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }, \& M_{3}:= int _{frac{1}{2}}^{1} frac{ (-delta ({tau }) )}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& M_{4}:= int _{frac{1}{2}}^{1} frac{delta ({tau })}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }. end{aligned}$$

Proof

Integrating by parts (M_{i}) for (i=1,2,3,4), and changing the variables, we have

$$begin{aligned}& begin{aligned} M_{1}={}&frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{Upsilon } biggl( frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) int _{0}^{ frac{1}{2}} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}mu )}{mu } ,mathrm{d} mu \ &{}-frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} int _{0}^{ frac{1}{2}} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, end{aligned} \& begin{aligned} M_{2}={}&frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{Upsilon } biggl( frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) int _{0}^{ frac{1}{2}} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}mu )}{mu } ,mathrm{d} mu \ &{}-frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} int _{0}^{ frac{1}{2}} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }, end{aligned} \& begin{aligned} M_{3}={}&frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{Upsilon } biggl( frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) int _{frac{1}{2}}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}mu )}{mu } ,mathrm{d} mu \ &{}-frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} int _{ frac{1}{2}}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, end{aligned} \& begin{aligned} M_{4}={}&frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{Upsilon } biggl( frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr) int _{frac{1}{2}}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}mu )}{mu } ,mathrm{d} mu \ &{}-frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} int _{ frac{1}{2}}^{1} frac{Phi (frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{tau } )}{{tau }}{ Upsilon } biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }. end{aligned} end{aligned}$$

Adding (M_{1}), (M_{2}), (M_{3}) and (M_{4}) and multiplying by the factor (frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2Delta (1)}), we obtain our required result. □

Corollary 2.9

Taking (Phi ({tau })={tau }) in Lemma 2.4, then

$$begin{aligned} &{Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{b_{1}}^{{b_{2}}} frac{{Upsilon }(x)}{x^{2}} ,mathrm{d}x= frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}sum_{j=1}^{4}L_{j}, end{aligned}$$

where

$$begin{aligned}& L_{1}:= int _{0}^{frac{1}{2}} frac{{tau }}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{Upsilon }’ biggl(frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{2}:= int _{0}^{frac{1}{2}} frac{{-tau }}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{Upsilon }’ biggl(frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{3}:= int _{frac{1}{2}}^{1} frac{-{tau }}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{Upsilon }’ biggl(frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{4}:= int _{frac{1}{2}}^{1} frac{{tau }}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }. end{aligned}$$

Corollary 2.10

Choosing (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) in Lemma 2.4, then

$$begin{aligned} &{Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{Gamma (alpha +1)}{2} biggl( frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} biggr)^{alpha } biggl[J_{ frac{1}{{{b_{2}}}}^{+}}^{alpha }{{ Upsilon }circ {Psi }} biggl( frac{1}{{b_{1}}} biggr) +J_{frac{1}{{b_{1}}}^{-}}^{alpha }{{ Upsilon }circ {Psi }} biggl( frac{1}{{{b_{2}}}} biggr) biggr] \ &quad =frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}sum_{j=5}^{8}L_{j}, end{aligned}$$

where

$$begin{aligned}& L_{5}:= int _{0}^{frac{1}{2}} frac{{tau }^{alpha }}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{Upsilon }’ biggl(frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{6}:= int _{0}^{frac{1}{2}} frac{(-{tau })^{alpha }}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{7}:= int _{frac{1}{2}}^{1} frac{(-{tau })^{alpha }}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{8}:= int _{frac{1}{2}}^{1} frac{{tau }^{alpha }}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{Upsilon }’ biggl(frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }. end{aligned}$$

Corollary 2.11

Taking (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Lemma 2.4, then

$$begin{aligned} &{Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{Gamma _{k}(alpha +k)}{2} biggl( frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} biggr)^{frac{alpha }{k}} biggl[ _{k}J_{frac{1}{{{b_{2}}}}^{+}}^{alpha }{{ Upsilon }circ { Psi }} biggl(frac{1}{{b_{1}}} biggr) + _{k}J_{frac{1}{{b_{1}}}^{-}}^{ alpha }{{Upsilon }circ {Psi }} biggl(frac{1}{{{b_{2}}}} biggr) biggr] \ &quad =frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}sum_{j=9}^{12}L_{j}, end{aligned}$$

where

$$begin{aligned}& L_{9}:= int _{0}^{frac{1}{2}} frac{{tau }^{frac{alpha }{k}}}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{10}:= int _{0}^{frac{1}{2}} frac{(-{tau })^{frac{alpha }{k}}}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{11}:= int _{frac{1}{2}}^{1} frac{(-{tau })^{frac{alpha }{k}}}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{12}:= int _{frac{1}{2}}^{1} frac{{tau }^{frac{alpha }{k}}}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }. end{aligned}$$

Corollary 2.12

Choosing (Phi ({tau })=frac{{tau }}{alpha }exp (-A{tau } )) in Lemma 2.4with (A=frac{1-alpha }{alpha }) and (alpha in (0,1]), then

$$begin{aligned} &{Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{1-alpha }{2(1-exp (-A))} biggl( frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} biggr)^{alpha } biggl[I_{ frac{1}{{{b_{2}}}}^{+}}^{alpha }{{ Upsilon }circ {Psi }} biggl( frac{1}{{b_{1}}} biggr) +I_{frac{1}{{b_{1}}}^{-}}^{alpha }{{ Upsilon }circ {Psi }} biggl( frac{1}{{{b_{2}}}} biggr) biggr] \ &quad =frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2(1-exp (-A))}sum_{j=13}^{16}L_{j}, end{aligned}$$

where

$$begin{aligned}& L_{13}:= int _{0}^{frac{1}{2}} frac{[exp (-A{tau })-1]}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{14}:= int _{0}^{frac{1}{2}} frac{[1-exp (-A{tau })]}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{15}:= int _{frac{1}{2}}^{1} frac{[exp (-A(1-{tau }))-exp (-A{tau })]}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{{tau b_{1}}+(1-{tau }){{b_{2}}}} biggr) ,mathrm{d} {tau }, \& L_{16}:= int _{frac{1}{2}}^{1} frac{[exp (-A{tau })-exp (-A(1-{tau }))]}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}}{ Upsilon }’ biggl( frac{{b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+{tau {b_{2}}}} biggr) ,mathrm{d} {tau }. end{aligned}$$

Further results

Now, utilizing auxiliary results obtained in the previous subsection, we derive some further generalized fractional trapezium-like inequalities using the class of harmonic convex functions.

Theorem 2.5

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]rightarrow mathbb{R}) be a continuous function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}) and (|{Upsilon }’|^{q}) be an harmonic convex function with (frac{1}{p}+frac{1}{q}=1), then

$$begin{aligned} & bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}\ &qquad {}- frac{1}{(m+1)eta (1)} biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)} biggl[pi _{1}^{ frac{1}{p}} biggl( int _{0}^{1}eta ^{q}({tau }) biggl( frac{1-{tau }}{m+1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+frac{m+{tau }}{m+1} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} {tau } biggr)^{ frac{1}{q}} \ &qquad {}+ pi _{2}^{frac{1}{p}} biggl( int _{0}^{1}eta ^{q}({tau }) biggl(frac{m+{tau }}{m+1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+ frac{1-{tau }}{m+1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} { tau } biggr)^{frac{1}{q}} biggr], end{aligned}$$

where

$$begin{aligned} &pi _{1} := frac{(m{b_{1}}+{{b_{2}}})^{1-2p}}{({{b_{2}}}-{b_{1}})(1-2p)} biggl[1- biggl( frac{(m+1){b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)^{1-2p} biggr], \ &pi _{2} := frac{({b_{1}}+m{{b_{2}}})^{1-2p}}{({{b_{2}}}-{b_{1}})(1-2p)} biggl[ biggl( frac{(m+1){{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)^{1-2p}-1 biggr]. end{aligned}$$

Proof

Using Lemma 2.2, the modulus property, Hölder’s inequality and the harmonic convexity of (|{Upsilon }’|^{q}), we have

$$begin{aligned} & bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}\ &qquad {}- frac{1}{(m+1)eta (1)} biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)} biggl[ int _{0}^{1} frac{eta ({tau })}{((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) biggrvert ,mathrm{d} {tau } \ &qquad {} + int _{0}^{1} frac{eta ({tau })}{((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}})^{2}} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) biggrvert ,mathrm{d} {tau } biggr] \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)} biggl[ biggl( int _{0}^{1}bigl((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}bigr)^{-2p} ,mathrm{d} { tau } biggr)^{frac{1}{p}}\ &qquad {}times biggl( int _{0}^{1}{eta ^{p}({tau })} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) biggrvert ^{q} ,mathrm{d} {tau } biggr)^{frac{1}{q}} \ & qquad {}+ biggl( int _{0}^{1}bigl((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}bigr)^{-2p} ,mathrm{d} {tau } biggr)^{frac{1}{p}}\ &qquad {}times biggl( int _{0}^{1}{eta ^{q}({ tau })} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) biggrvert ^{q} ,mathrm{d} {tau } biggr)^{frac{1}{q}} biggr] \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)} biggl[ biggl( int _{0}^{1}bigl((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}bigr)^{-2p} ,mathrm{d} { tau } biggr)^{frac{1}{p}} \ &qquad {}timesbiggl( int _{0}^{1}eta ^{q}({tau }) biggl(frac{1-{tau }}{m+1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+ frac{m+{tau }}{m+1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} { tau } biggr)^{frac{1}{q}} \ &qquad {} + biggl( int _{0}^{1}bigl((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}bigr)^{-2p} ,mathrm{d} {tau } biggr)^{frac{1}{p}}\ &qquad {}times biggl( int _{0}^{1}eta ^{q}({ tau }) biggl(frac{m+{tau }}{m+1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+ frac{1-{tau }}{m+1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} { tau } biggr)^{frac{1}{q}} biggr]. end{aligned}$$

After simple calculations, we obtain our required result. □

Corollary 2.13

Choosing (Phi ({tau })={tau }) in Theorem 2.5, we have

$$begin{aligned} & bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{frac{1}{{{b_{2}}}}}^{ frac{1}{{b_{1}}}}{Upsilon circ Psi }(x) ,mathrm{d}x biggrvert \ &quad leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\ &qquad {}times biggl[pi _{1}^{frac{1}{p}} biggl(frac{1}{(m+1)(q+1)(q+2)} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+ frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr)^{frac{1}{q}} \ &qquad {}+ pi _{2}^{frac{1}{p}} biggl(frac{1}{(m+1)(q+1)(q+2)} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q}+frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} biggr)^{frac{1}{q}} biggr], end{aligned}$$

Corollary 2.14

Taking (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) in Theorem 2.5, we obtain

$$begin{aligned} &bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}- frac{(m+1)^{alpha -1}({b_{1}{b_{2}}})^{alpha }Gamma (alpha +1)}{({{b_{2}}}-{b_{1}})^{alpha }} biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} biggr) \ &qquad {}+J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }{ Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr]biggrvert \ &quad leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} biggl[pi _{1}^{frac{1}{p}} biggl(frac{1}{(m+1)({alpha }q+1)({alpha }q+2)} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}\ &qquad {}+ frac{m({alpha }q+2)+({alpha }q+1)}{(m+1)({alpha }q+1)({alpha }q+2)} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr)^{ frac{1}{q}} \ &qquad {}+ pi _{2}^{frac{1}{p}} biggl( frac{1}{(m+1)({alpha } q+1)({alpha }q+2)} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}\ &qquad {}+ frac{m({alpha }q+2)+({alpha }q+1)}{(m+1)({alpha }q+1)({alpha }q+2)} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} biggr)^{frac{1}{q}} biggr], end{aligned}$$

where (pi _{1}) and (pi _{2}) are already defined.

Corollary 2.15

Choosing (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Theorem 2.5, we obtain

$$begin{aligned} &bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}- frac{(m+1)^{frac{alpha }{k}-1}({b_{1}{b_{2}}})^{frac{alpha }{k}}Gamma _{k}(alpha +k)}{({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} biggl[{_{k}J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} { Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} biggr) \ &qquad {} +{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr]biggrvert \ &quad leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} biggl[pi _{1}^{frac{1}{p}} biggl(frac{k}{(m+1)({alpha }q+k)({alpha }q+2k)} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}\ &qquad {}+ frac{km({alpha }q+2k)+k({alpha }q+k)}{(m+1)({alpha }q+k)({alpha }q+2k)} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr)^{frac{1}{q}} \ &qquad {}+ pi _{2}^{frac{1}{p}} biggl( frac{k}{(m+1)({alpha } q+k)({alpha }q+2k)} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}\ &qquad {}+ frac{mk({alpha }q+2k)+(k{alpha }q+k)}{(m+1)({alpha }q+k)({alpha }q+2k)} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} biggr)^{frac{1}{q}} biggr]. end{aligned}$$

Theorem 2.6

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]rightarrow mathbb{R}) be a continuous function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}) and (|{Upsilon }’|^{q}) be an harmonic convex function with (qgeq 1), then

$$begin{aligned} & bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}\ &qquad {}- frac{1}{(m+1)eta (1)} biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)}biggl[ biggl( int _{0}^{1}eta ({tau }) bigl((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}bigr)^{-2} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {}times biggl( int _{0}^{1}{eta ({tau })}bigl((m+{tau }){b_{1}}+(1-{ tau }){{b_{2}}}bigr)^{-2} biggl(frac{1-{tau }}{m+1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+ frac{m+{tau }}{m+1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} { tau } biggr)^{frac{1}{q}} \ &qquad {}+ biggl( int _{0}^{1}eta ({tau }) bigl((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}bigr)^{-2} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {}times biggl( int _{0}^{1}{eta ({tau })}bigl((1-{tau }){b_{1}}+(m+{ tau }){{b_{2}}}bigr)^{-2}\ &qquad {}times biggl(frac{m+{tau }}{m+1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+ frac{1-{tau }}{m+1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} { tau } biggr)^{frac{1}{q}}biggr]. end{aligned}$$

Proof

Using Lemma 2.2, the modulus property, the power mean inequality and the convexity of (|{Upsilon }’|^{q}), we have

$$begin{aligned} & bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}\ &qquad {}- frac{1}{(m+1)eta (1)} biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) +{_{ ( frac{1}{{b_{1}}} )^{-}}I_{Phi }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)}biggl[ int _{0}^{1} frac{eta ({tau })}{((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}})^{2}} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) biggrvert ,mathrm{d} {tau } \ &qquad {}+ int _{0}^{1} frac{eta ({tau })}{((1-{tau }){b_{1}}+(m+{tau }){{b_{2}}})^{2}} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) biggrvert ,mathrm{d} {tau }biggr] \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)}biggl[ biggl( int _{0}^{1}eta ({tau }) bigl((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}bigr)^{-2} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {}times biggl( int _{0}^{1}{eta ({tau })}bigl((m+{ tau }){b_{1}}+(1-{tau }){{b_{2}}}bigr)^{-2} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}} biggr) biggrvert ^{q} ,mathrm{d} { tau } biggr)^{frac{1}{q}} \ &qquad {}+ biggl( int _{0}^{1}eta ({tau }) bigl((1-{tau }){b_{1}}+(m+{ tau }){{b_{2}}}bigr)^{-2} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {} times biggl( int _{0}^{1}{eta ({tau })}bigl((1-{ tau }){b_{1}}+(m+{tau }){{b_{2}}}bigr)^{-2} bigglvert {Upsilon }’ biggl( frac{(m+1){b_{1}{b_{2}}}}{(1-{tau }){b_{1}}+(m+{tau }){{b_{2}}}} biggr) biggrvert ^{q} ,mathrm{d} {tau } biggr)^{frac{1}{q}}biggr] \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{eta (1)}biggl[ biggl( int _{0}^{1}eta ({tau }) bigl((m+{tau }){b_{1}}+(1-{tau }){{b_{2}}}bigr)^{-2} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {} times biggl( int _{0}^{1}{eta ({tau })}bigl((m+{ tau }){b_{1}}+(1-{tau }){{b_{2}}}bigr)^{-2} biggl(frac{1-{tau }}{m+1} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q}+frac{m+{tau }}{m+1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} {tau } biggr)^{frac{1}{q}} \ &qquad {} + biggl( int _{0}^{1}eta ({tau }) bigl((1-{tau }){b_{1}}+(m+{ tau }){{b_{2}}}bigr)^{-2} biggr)^{1-frac{1}{q}} \ &qquad {}times biggl( int _{0}^{1}{eta ({tau })}bigl((1-{ tau }){b_{1}}+(m+{tau }){{b_{2}}}bigr)^{-2}\ &qquad {}times biggl(frac{m+{tau }}{m+1} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q}+frac{1-{tau }}{m+1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr) ,mathrm{d} {tau } biggr)^{frac{1}{q}}biggr]. end{aligned}$$

After simple calculations, we obtain our required result. □

Corollary 2.16

If we take (Phi ({tau })={tau }) in Theorem 2.6, we have

$$begin{aligned} & bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{frac{1}{{{b_{2}}}}}^{ frac{1}{{b_{1}}}}{Upsilon circ Psi }(x) ,mathrm{d}x biggrvert \ &quad leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} bigl[pi _{3}^{1- frac{1}{q}} bigl(pi _{4} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+pi _{5} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+pi _{6}^{1- frac{1}{q}} bigl(pi _{7} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+pi _{8} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr], end{aligned}$$

where

$$begin{aligned} &pi _{3} :=frac{(m{b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} biggl(2,2,3, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, \ &pi _{4} :=frac{(m{b_{1}}+{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} biggl(2,2,4, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, \ &pi _{5} :=frac{m(m{b_{1}}+{{b_{2}}})^{-2}}{2(m+1)}{_{2}F_{1} biggl(2,2,3, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}+ frac{(m{b_{1}}+{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} biggl(2,3,4, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, \ &pi _{6} :=frac{({b_{1}}+m{{b_{2}}})^{-2}}{2}{_{2}F_{1} biggl(2,2,3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}, \ &pi _{7} :=frac{m({b_{1}}+m{{b_{2}}})^{-2}}{2(m+1)}{_{2}F_{1} biggl(2,2,3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}+ frac{({b_{1}}+m{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} biggl(2,3,4, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}, \ &pi _{8} :=frac{({b_{1}}+m{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} biggl(2,2,4, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}. end{aligned}$$

Corollary 2.17

If we choose (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) in Theorem 2.6, we obtain

$$begin{aligned} &bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}- frac{(m+1)^{alpha -1}({b_{1}{b_{2}}})^{alpha }Gamma (alpha +1)}{({{b_{2}}}-{b_{1}})^{alpha }} biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} {Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} biggr) \ &qquad {}+{J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} { Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr]biggrvert \ &quad leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} bigl[pi _{9}^{1- frac{1}{q}} bigl(pi _{10} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+pi _{11} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+pi _{12}^{1- frac{1}{q}} bigl(pi _{13} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+pi _{14} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr], end{aligned}$$

where

$$begin{aligned} &pi _{9} :=frac{(m{b_{1}}+{{b_{2}}})^{-2}}{alpha +1}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, \ &pi _{10} := frac{(m{b_{1}}+{{b_{2}}})^{-2}}{(alpha +2)(alpha +1)(m+1)}{_{2}F_{1} biggl(2,alpha +1,alpha +3, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, \ & begin{aligned} pi _{11} :={}&frac{m(m{b_{1}}+{{b_{2}}})^{-2}}{(alpha +1)(m+1)}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)} \ &{} + frac{(m{b_{1}}+{{b_{2}}})^{-2}}{(alpha +2)(alpha +1)(m+1)}{_{2}F_{1} biggl(2,alpha +2, alpha +3, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, end{aligned} \ &pi _{12} :=frac{({b_{1}}+m{{b_{2}}})^{-2}}{alpha +1}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}, \ & begin{aligned} pi _{13} :={}&frac{m({b_{1}}+m{{b_{2}}})^{-2}}{(alpha +1)(m+1)}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)} \ & {}+ frac{({b_{1}}+m{{b_{2}}})^{-2}}{(alpha +2)(alpha +1)(m+1)} {_{2}F_{1} biggl(2,alpha +2,alpha +3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}, end{aligned} \ &pi _{14} := frac{({b_{1}}+m{{b_{2}}})^{-2}}{(alpha +2)(alpha +1)(m+1)}{_{2}F_{1} biggl(2,alpha +1,alpha +3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}. end{aligned}$$

Corollary 2.18

If we take (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Theorem 2.6, we obtain

$$begin{aligned} &bigglvert frac{{Upsilon }({b_{1}})+{Upsilon }({{b_{2}}})}{m+1}- frac{(m+1)^{frac{alpha }{k}-1}({b_{1}{b_{2}}})^{frac{alpha }{k}}Gamma _{k}(alpha +k)}{({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} biggl[{_{k}J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }} { Upsilon circ Psi } biggl( frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} biggr) \ & qquad {}+{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} {Upsilon circ Psi } biggl( frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} biggr) biggr]biggrvert \ &quad leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} bigl[pi _{15}^{1- frac{1}{q}} bigl(pi _{16} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+pi _{17} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+pi _{18}^{1- frac{1}{q}} bigl(pi _{19} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+pi _{20} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr], end{aligned}$$

where

$$begin{aligned} &pi _{15} :=frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{alpha +k}{_{2}F_{1,k} biggl(2k,alpha +k,alpha +2k, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, \ &pi _{16} := frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{(alpha +2k)(alpha +k)(m+1)}{_{2}F_{1,k} biggl(2k,alpha +k,alpha +3k, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, \ & begin{aligned} pi _{17} :={}&frac{km(m{b_{1}}+{{b_{2}}})^{-2}}{(alpha +k)(m+1)}{_{2}F_{1,k} biggl(2k,alpha +k,alpha +2k, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)} \ &{} + frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{(alpha +2k)(alpha +k)(m+1)}{_{2}F_{1,k} biggl(2k,alpha +2k,alpha +3k, frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} biggr)}, end{aligned} \ &pi _{18} :=frac{k({b_{1}}+m{{b_{2}}})^{-2}}{alpha +k}{_{2}F_{1,k} biggl(2k,alpha +k,alpha +2k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}, \ & begin{aligned} pi _{19} :={}&frac{km({b_{1}}+m{{b_{2}}})^{-2}}{(alpha +k)(m+1)}{_{2}F_{1,k} biggl(2k,alpha +k,alpha +2k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)} \ &{} + frac{k({b_{1}}+m{{b_{2}}})^{-2}}{(alpha +2k)(alpha +k)(m+1)} {_{2}F_{1,k} biggl(2k,alpha +2k,alpha +3k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}, end{aligned} \ &pi _{20} := frac{k({b_{1}}+m{{b_{2}}})^{-2}}{(alpha +2k)(alpha +k)(m+1)}{_{2}F_{1,k} biggl(2k,alpha +k,alpha +3k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} biggr)}. end{aligned}$$

Theorem 2.7

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]rightarrow mathbb{R}) be a continuous function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}) and (|{Upsilon }’|^{q}) be an harmonic convex function with (frac{1}{p}+frac{1}{q}=1) and (lambda ,mu in [0,infty )) with (lambda +mu neq0), then

$$begin{aligned} & bigglvert frac{Omega (lambda ){Upsilon }({{b_{2}}})+Omega (mu ){Upsilon }({b_{1}})}{lambda +mu }\ &qquad{}- frac{1}{lambda +mu } biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} +{_{ (frac{1}{{b_{1}}} )^{-}}I_{Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr] biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( int _{0}^{ lambda }{Omega ^{p}({tau })} ,mathrm{d} {tau } biggr)^{frac{1}{p}} bigl(sigma _{1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{2} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad {}+ biggl( int _{0}^{mu }{Omega ^{p}({tau })} ,mathrm{d} {tau } biggr)^{frac{1}{p}} bigl(sigma _{3} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{4} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}} biggr], end{aligned}$$

where

$$begin{aligned}& begin{aligned} sigma _{1}& := int _{0}^{lambda }frac{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{-2q}(mu +{tau })}{lambda +mu } ,mathrm{d} { tau } \ &= frac{mu (lambda {{b_{2}}}+mu {b_{1}})^{1-2q}-(lambda +mu )((lambda +mu ){b_{1}})^{1-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})(1-2q)}- frac{((lambda +mu ){b_{1}})^{2-2q}-(lambda {{b_{2}}}+mu {b_{1}})^{2-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, end{aligned}\& begin{aligned} sigma _{2}& := int _{0}^{lambda }frac{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{-2q}(lambda -{tau })}{lambda +mu } ,mathrm{d} { tau } \ &= frac{lambda (lambda {{b_{2}}}+mu {b_{1}})^{1-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})(1-2q)}+ frac{((lambda +mu ){b_{1}})^{2-2q}-(lambda {{b_{2}}}+mu {b_{1}})^{2-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, end{aligned}\& begin{aligned} sigma _{3}& := int _{0}^{mu }frac{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{-2q}(mu -{tau })}{lambda +mu } ,mathrm{d} { tau } \ &= frac{((lambda +mu ){{b_{2}}})^{2-2q}-(lambda {{b_{2}}}+mu {b_{1}})^{2-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}- frac{mu (lambda {{b_{2}}}+mu {b_{1}})^{1-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})(1-2q)}, end{aligned}\& begin{aligned} sigma _{4}& := int _{0}^{mu }frac{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{-2q}(lambda +{tau })}{lambda +mu } ,mathrm{d} { tau } \ &= frac{(lambda +mu )((lambda +mu ){{b_{2}}})^{1-2q}-lambda (lambda {{b_{2}}}+mu {b_{1}})^{1-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})(1-2q)}- frac{((lambda +mu ){{b_{2}}})^{2-2q}-(lambda {{b_{2}}}+mu {b_{1}})^{2-2q}}{(lambda +mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}. end{aligned} end{aligned}$$

Proof

Using Lemma 2.3, the modulus property, Hölder’s inequality and the harmonic convexity of (|{Upsilon }’|^{q}), we have

$$begin{aligned} & bigglvert frac{Omega (lambda ){Upsilon }({{b_{2}}})+Omega (mu ){Upsilon }({b_{1}})}{lambda +mu }\ &qquad {}- frac{1}{lambda +mu } biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} +{_{ (frac{1}{{b_{1}}} )^{-}}I_{Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr] biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( int _{0}^{ lambda }{Omega ^{p}({tau })} ,mathrm{d} {tau } biggr)^{frac{1}{p}}\ &qquad{}times biggl( int _{0}^{lambda } frac{1}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2p}} bigglvert {Upsilon }’ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) biggrvert ^{q} ,mathrm{d} {tau } biggr)^{frac{1}{q}} \ &qquad {}+ biggl( int _{0}^{mu }{Omega ^{p}({tau })} ,mathrm{d} {tau } biggr)^{frac{1}{p}}\ &qquad{}times biggl( int _{0}^{mu } frac{1}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2p}} bigglvert {Upsilon }’ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) biggrvert ^{q} ,mathrm{d} {tau } biggr)^{frac{1}{q}} biggr] \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( int _{0}^{ lambda }{Omega ^{p}({tau })} ,mathrm{d} {tau } biggr)^{frac{1}{p}} biggl( int _{0}^{lambda }{bigl((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}bigr)^{-2q}}\ &qquad{}times biggl(frac{mu +{tau }}{lambda +mu } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} +frac{lambda -{tau }}{lambda +mu } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} biggr) ,mathrm{d} {tau } biggr)^{frac{1}{q}} \ &qquad {}+ biggl( int _{0}^{mu }{Omega ^{p}({tau })} ,mathrm{d} {tau } biggr)^{frac{1}{p}} biggl( int _{0}^{mu }{bigl((lambda +{tau }){{b_{2}}}+( mu -{tau }){b_{1}}bigr)^{-2q}}\ &qquad{}times biggl(frac{mu -{tau }}{lambda +mu } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} + frac{lambda +{tau }}{lambda +mu } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} biggr) ,mathrm{d} {tau } biggr)^{frac{1}{q}} biggr]. end{aligned}$$

After simple calculations, we obtain our required result. □

Corollary 2.19

Choosing (Phi ({tau })={tau }) in Theorem 2.7, we have

$$begin{aligned} & bigglvert frac{lambda {Upsilon }({{b_{2}}})+mu {Upsilon }({b_{1}})}{lambda +mu }- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{frac{1}{{{b_{2}}}}}^{ frac{1}{{b_{1}}}}{Upsilon circ Psi }(x) ,mathrm{d}x biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( frac{lambda ^{p+1}}{p+1} biggr)^{frac{1}{p}} bigl(sigma _{1} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{2} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ biggl(frac{mu ^{p+1}}{p+1} biggr)^{ frac{1}{p}} \ &qquad {}times bigl(sigma _{3} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{4} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}} biggr], end{aligned}$$

where (sigma _{1}), (sigma _{2}), (sigma _{3}) and (sigma _{4}) are already defined in Theorem 2.7.

Corollary 2.20

Taking (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) in Theorem 2.7, we obtain

$$begin{aligned} &bigglvert frac{lambda ^{alpha }{Upsilon }({{b_{2}}})+mu ^{alpha }{Upsilon }({b_{1}})}{lambda +mu }- frac{({b_{1}{b_{2}}})^{alpha }(lambda +mu )^{alpha -1}Gamma (alpha +1)}{({{b_{2}}}-{b_{1}})^{alpha }} biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} \ &qquad {}+{{J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} { Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr]biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( frac{lambda ^{{alpha }p+1}}{{alpha }p+1} biggr)^{frac{1}{p}} bigl(sigma _{1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{2} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ biggl(frac{mu ^{{alpha }p+1}}{{alpha }p+1} biggr)^{frac{1}{p}} \ &qquad {}times bigl(sigma _{3} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{4} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}} biggr], end{aligned}$$

where (sigma _{1}), (sigma _{2}), (sigma _{3}) and (sigma _{4}) are already defined in Theorem 2.7.

Corollary 2.21

Choosing (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Theorem 2.7, we obtain

$$begin{aligned} &bigglvert frac{lambda ^{frac{alpha }{k}}{Upsilon }({{b_{2}}})+mu ^{frac{alpha }{k}}{Upsilon }({b_{1}})}{lambda +mu }- frac{k({b_{1}{b_{2}}})^{frac{alpha }{k}}(lambda +mu )^{frac{alpha }{k}-1}Gamma _{k}(alpha +k)}{({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} biggl[{_{k}{J_{ ( frac{1}{{{b_{2}}}} )^{+}}^{alpha }} {Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} \ &qquad {}+{{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{ alpha }} {Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr]biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( frac{klambda ^{frac{{alpha }p+k}{k}}}{{alpha }p+k} biggr)^{ frac{1}{p}} bigl(sigma _{1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{2} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ biggl( frac{kmu ^{frac{{alpha }p+k}{k}}}{{{alpha }p}+k} biggr)^{ frac{1}{p}} \ &qquad {}times bigl(sigma _{3} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{4} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}} biggr], end{aligned}$$

where (sigma _{1}), (sigma _{2}), (sigma _{3}) and (sigma _{4}) are already defined in Theorem 2.7.

Theorem 2.8

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]rightarrow mathbb{R}) be a function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}) and (|{Upsilon }’|^{q}) be an harmonic convex function with (qgeq 1) and (lambda ,mu in [0,infty )) with (lambda +mu neq0), then

$$begin{aligned} & bigglvert frac{Omega (lambda ){Upsilon }({{b_{2}}})+Omega (mu ){Upsilon }({b_{1}})}{lambda +mu }\ &qquad{}- frac{1}{lambda +mu } biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} +{_{ (frac{1}{{b_{1}}} )^{-}}I_{Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr] biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( int _{0}^{ lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}}\ &qquad{}timesbiggl( int _{0}^{lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} biggl(frac{mu +{tau }}{lambda +mu } biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q}+ frac{lambda -{tau }}{lambda +mu } biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} biggr) ,mathrm{d} {tau }biggr)^{frac{1}{q}}\ &qquad{}+ biggl( int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {}times biggl( int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} biggl( frac{mu -{tau }}{lambda +mu } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+ frac{lambda +{tau }}{lambda +mu } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} biggr) ,mathrm{d} {tau } biggr)^{frac{1}{q}}biggr]. end{aligned}$$

Proof

Using Lemma 2.3, the modulus property, the power mean inequality and the harmonic convexity of of (|{Upsilon }’|^{q}), we have

$$begin{aligned} & bigglvert frac{Omega (lambda ){Upsilon }({{b_{2}}})+Omega (mu ){Upsilon }({b_{1}})}{lambda +mu }\ &qquad{}- frac{1}{lambda +mu } biggl[{_{ (frac{1}{{{b_{2}}}} )^{+}}I_{ Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} +{_{ (frac{1}{{b_{1}}} )^{-}}I_{Phi }{Upsilon circ Psi } biggl( frac{lambda {{b_{2}}}+mu {b_{1}}}{{b_{1}{b_{2}}}(lambda +mu )} biggr)} biggr] biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ int _{0}^{lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} bigglvert {Upsilon }’ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) biggrvert ,mathrm{d} {tau } \ &qquad {}+ int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} bigglvert {Upsilon }’ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) biggrvert ,mathrm{d} {tau }biggr] \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( int _{0}^{ lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}}\ &qquad{}timesbiggl( int _{0}^{lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} \ &qquad {}times bigglvert {Upsilon }’ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}}} biggr) biggrvert ,mathrm{d} {tau }biggr)^{frac{1}{q}}\ &qquad{}+ biggl( int _{0}^{ mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {}times biggl( int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} bigglvert {Upsilon }’ biggl( frac{{b_{1}{b_{2}}}(lambda +mu )}{(lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}}} biggr) biggrvert ,mathrm{d} {tau } biggr)^{frac{1}{q}}biggr] \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) biggl[ biggl( int _{0}^{ lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}}\ &qquad{}timesbiggl( int _{0}^{lambda } frac{Omega ({tau })}{((lambda -{tau }){{b_{2}}}+(mu +{tau }){b_{1}})^{2}} \ &qquad {} times biggl(frac{mu +{tau }}{lambda +mu } biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q}+ frac{lambda -{tau }}{lambda +mu } biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} biggr) ,mathrm{d} {tau }biggr)^{frac{1}{q}}\ &qquad{}+ biggl( int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} \ &qquad {}times biggl( int _{0}^{mu } frac{Omega ({tau })}{((lambda +{tau }){{b_{2}}}+(mu -{tau }){b_{1}})^{2}} biggl( frac{mu -{tau }}{lambda +mu } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+ frac{lambda +{tau }}{lambda +mu } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} biggr) ,mathrm{d} {tau } biggr)^{frac{1}{q}}biggr]. end{aligned}$$

This completes the proof. □

Corollary 2.22

Choosing (Phi ({tau })={tau }) and (lambda =mu =1) in Theorem 2.8, we have

$$begin{aligned} & bigglvert frac{ {Upsilon }({{b_{2}}})+ {Upsilon }({b_{1}})}{2}- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{frac{1}{{{b_{2}}}}}^{ frac{1}{{b_{1}}}}{Upsilon circ Psi }(x) ,mathrm{d}x biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) bigl[sigma _{5}^{1- frac{1}{q}} bigl(sigma _{6} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{7} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+ sigma _{8}^{1- frac{1}{q}} bigl(sigma _{9} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{10} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr], end{aligned}$$

where

$$begin{aligned} &sigma _{5} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} biggl(2,2,3, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{6} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{4}{_{2}F_{1} biggl(2,2,3, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}+ frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} biggl(2,3,4, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{7} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} biggl(2,2,4, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{8} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} biggl(2,2,3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{9} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} biggl(2,2,4, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{10} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{4}{_{2}F_{1} biggl(2,2,3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}+ frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} biggl(2,3,4, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}. end{aligned}$$

Corollary 2.23

Taking (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) and (lambda =mu =1) in Theorem 2.8, we obtain

$$begin{aligned} &bigglvert frac{{Upsilon }({{b_{2}}})+{Upsilon }({b_{1}})}{2}- frac{({b_{1}{b_{2}}})^{alpha }2^{alpha -1}Gamma (alpha +1)}{({{b_{2}}}-{b_{1}})^{alpha }} biggl[{J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }{Upsilon circ Psi } biggl( frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr)} \ &qquad {}+{{J_{ (frac{1}{{b_{1}}} )^{-}}^{alpha }} { Upsilon circ Psi } biggl( frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr)} biggr]biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) bigl[sigma _{11}^{1- frac{1}{q}} bigl(sigma _{12} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{13} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+ sigma _{14}^{1- frac{1}{q}} bigl(sigma _{15} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{16} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr], end{aligned}$$

where

$$begin{aligned} &sigma _{11} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{alpha +1}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ & begin{aligned} sigma _{12} :={}&frac{({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +1)}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)} \ &{}+frac{({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +1)(alpha +2)} {_{2}F_{1} biggl(2,alpha +2, alpha +3, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, end{aligned} \ &sigma _{13} := frac{({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +1)(alpha +2)}{_{2}F_{1} biggl(2,alpha +1,alpha +3, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{14} :=frac{({b_{1}}+{{b_{2}}})^{-2}}{alpha +1}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{15} := frac{({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +1)(alpha +2)}{_{2}F_{1} biggl(2,alpha +1,alpha +3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &begin{aligned} sigma _{16}:={}&frac{({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +1)}{_{2}F_{1} biggl(2,alpha +1,alpha +2, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)} \ &{}+frac{({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +1)(alpha +2)}{_{2}F_{1} biggl(2,alpha +2, alpha +3, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}. end{aligned} end{aligned}$$

Corollary 2.24

Choosing (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) and (lambda =mu =1) in Theorem 2.8, we obtain

$$begin{aligned} &bigglvert frac{{Upsilon }({{b_{2}}})+{Upsilon }({b_{1}})}{2}- frac{({b_{1}{b_{2}}})^{frac{alpha }{k}}2^{frac{alpha }{k}-1}Gamma _{k}(alpha +k)}{({{b_{2}}}-{b_{1}})^{frac{alpha }{k}}} biggl[{_{k}J_{ (frac{1}{{{b_{2}}}} )^{+}}^{alpha }{ Upsilon circ Psi } biggl(frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr)} \ &qquad {}+{{_{k}J_{ (frac{1}{{b_{1}}} )^{-}}^{ alpha }} {Upsilon circ Psi } biggl( frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} biggr)} biggr]biggrvert \ &quad leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) bigl[sigma _{16}^{ frac{1}{p}} bigl(sigma _{17} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{18} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+ sigma _{19}^{ frac{1}{p}} bigl(sigma _{20} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q}+sigma _{21} biglvert { Upsilon }'({b_{1}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr], end{aligned}$$

where

$$begin{aligned} &sigma _{16} :=frac{k({b_{1}}+{{b_{2}}})^{-2}}{alpha +k}{_{2}F_{1} biggl(2,alpha +k,alpha +2k, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ & begin{aligned} sigma _{17} :={}&frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(alpha k)}{_{2}F_{1} biggl(2,alpha +k,alpha +2k, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)} \ &{}+frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +k)(alpha +2k)} {_{2}F_{1} biggl(2,alpha +2k, alpha +3k, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, end{aligned} \ &sigma _{18} := frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +k)(alpha +2k)}{_{2}F_{1} biggl(2,alpha +k,alpha +3k, frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{19} :=frac{k({b_{1}}+{{b_{2}}})^{-2}}{alpha +k}{_{2}F_{1} biggl(2,alpha +k,alpha +2k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ &sigma _{20} := frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +k)(alpha +2k)}{_{2}F_{1} biggl(2,alpha +k,alpha +3k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}, \ & begin{aligned} sigma _{21}:={}&frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +k)}{_{2}F_{1} biggl(2,alpha +k,alpha +2k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)} \ &{}+frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(alpha +k)(alpha +2k)}{_{2}F_{1} biggl(2,alpha +2k, alpha +3k, frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)}. end{aligned} end{aligned}$$

Theorem 2.9

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]subset (0,+infty )rightarrow mathbb{R}) be a differentiable function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}). If (vert {Upsilon }’ vert ^{q}) is an harmonic convex function with (q>1) and (frac{1}{p}+frac{1}{q}=1), then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{1}{2Delta (1)} biggl[_{frac{1}{{{b_{2}}}}^{+}}I_{ Phi }{{Upsilon }circ {Psi }} biggl(frac{1}{{b_{1}}} biggr)+ _{ frac{1}{{b_{1}}}^{-}}I_{Phi }{{ Upsilon }circ {Psi }} biggl( frac{1}{{{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2Delta (1)} biggl( biggl( int _{0}^{frac{1}{2}} biglvert Delta ({tau }) bigrvert ^{p} ,mathrm{d} {tau } biggr)^{frac{1}{p}}+ biggl( int _{frac{1}{2}}^{1} biglvert delta ({tau }) bigrvert ^{p} ,mathrm{d} {tau } biggr)^{frac{1}{p}} biggr) \ &qquad {}times bigl( bigl(N_{1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{2} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ bigl(N_{3} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{4} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr), end{aligned}$$

where

$$begin{aligned}& begin{aligned} N_{1}&:= int _{0}^{frac{1}{2}} frac{(1-{tau })}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2q}} ,mathrm{d} { tau } \ &= frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, end{aligned}\& begin{aligned} N_{2}&:= int _{0}^{frac{1}{2}} frac{{tau }}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2q}} ,mathrm{d} { tau } \ &={{b_{2}}} frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, end{aligned}\& begin{aligned} N_{3}&:= int _{0}^{frac{1}{2}} frac{{tau }}{((1-{tau }){tau b_{1}}+{tau {b_{2}}})^{2q}} ,mathrm{d} { tau } \ &= frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, end{aligned}\& begin{aligned} N_{4}&:= int _{0}^{frac{1}{2}} frac{1-{tau }}{((1-{tau }){tau b_{1}}+{tau {b_{2}}})^{2q}} ,mathrm{d} {tau } \ &={{b_{2}}} frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, end{aligned}\& begin{aligned} N_{5}&:= int _{frac{1}{2}}^{1} frac{1-{tau }}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2q}} ,mathrm{d} { tau } \ &= frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, end{aligned}\& begin{aligned} N_{6}&:= int _{frac{1}{2}}^{1} frac{{tau }}{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2q}} ,mathrm{d} { tau } \ &={{b_{2}}} frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, end{aligned}\& begin{aligned} N_{7}&:= int _{frac{1}{2}}^{1} frac{{tau }}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2q}} ,mathrm{d} { tau } \ &= frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, end{aligned}\& begin{aligned} N_{8}&:= int _{frac{1}{2}}^{1} frac{1-{tau }}{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2q}} ,mathrm{d} { tau } \ &={{b_{2}}} frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}. end{aligned} end{aligned}$$

Also, it is easy to verify that (N_{1}=N_{7}), (N_{2}=N_{8}), (N_{3}=N_{5}) and (N_{4}=N_{6}).

Proof

By using Lemma 2.4, the property of modulus, Hölder’s inequality and the harmonic convexity of (vert {Upsilon }’ vert ^{q}), we obtain the desired result. We omit here the proof. □

Corollary 2.25

Taking (Phi ({tau })={tau }) in Theorem 2.9, then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{b_{1}}^{{b_{2}}} frac{{Upsilon }(x)}{x^{2}} ,mathrm{d}x biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} biggl( frac{1}{2^{p+1}(p+1)} biggr)^{frac{1}{p}} \ &qquad {}times bigl( bigl(N_{1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{2} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ bigl(N_{3} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{4} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr). end{aligned}$$

Corollary 2.26

Choosing (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) in Theorem 2.9, then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{Gamma (alpha +1)}{2} biggl( frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} biggr)^{alpha } biggl[J_{ frac{1}{{{b_{2}}}}^{+}}^{alpha }{{Upsilon }circ {Psi }} biggl( frac{1}{{b_{1}}} biggr) +J_{frac{1}{{b_{1}}}^{-}}^{alpha }{{ Upsilon } circ {Psi }} biggl(frac{1}{{{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} biggl( frac{ (1+2^{alpha p-1}(alpha p-1) )}{2^{alpha p}(alpha p+1)} biggr)^{frac{1}{p}} \ &qquad {}times bigl( bigl(N_{1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{2} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ bigl(N_{3} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{4} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr). end{aligned}$$

Corollary 2.27

Taking (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Theorem 2.9, then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{Gamma _{k}(alpha +k)}{2} biggl( frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} biggr)^{frac{alpha }{k}} biggl[ _{k}J_{frac{1}{{{b_{2}}}}^{+}}^{alpha }{{Upsilon }circ { Psi }} biggl(frac{1}{{b_{1}}} biggr) + _{k}J_{frac{1}{{b_{1}}}^{-}}^{ alpha }{{ Upsilon }circ {Psi }} biggl(frac{1}{{{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} biggl( frac{ (k+2^{frac{alpha p}{k}-1}(alpha p-k) )}{2^{frac{alpha p}{k}}(alpha p+k)} biggr)^{frac{1}{p}} \ &qquad {}times bigl( bigl(N_{1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{2} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ bigl(N_{3} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+N_{4} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr). end{aligned}$$

Theorem 2.10

Let ({Upsilon }:[{b_{1}},{{b_{2}}}]subset (0,+infty )rightarrow mathbb{R}) be a differentiable function on (({b_{1}},{{b_{2}}})) with ({b_{1}}<{{b_{2}}}). If (vert {Upsilon }’ vert ^{q}) is an harmonic convex function with (qgeq 1), then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{1}{2Delta (1)} biggl[_{frac{1}{{{b_{2}}}}^{+}}I_{ Phi }{{Upsilon }circ {Psi }} biggl(frac{1}{{b_{1}}} biggr)+ _{ frac{1}{{b_{1}}}^{-}}I_{Phi }{{ Upsilon }circ {Psi }} biggl( frac{1}{{{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2Delta (1)} \ &qquad {}times biggl[ biggl( int _{0}^{frac{1}{2}} biglvert Delta ({tau }) bigrvert ,mathrm{d} {tau } biggr)^{1-frac{1}{q}} biggl{ biggl( int _{0}^{ frac{1}{2}} frac{(1-{tau }) vert Delta ({tau }) vert }{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}\ &qquad{}+ int _{0}^{ frac{1}{2}} frac{{tau } vert Delta ({tau }) vert }{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr)^{ frac{1}{q}} \ &qquad {}+ biggl( int _{0}^{frac{1}{2}} frac{{tau } vert Delta ({tau }) vert }{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+ int _{0}^{ frac{1}{2}} frac{(1-{tau }) vert Delta ({tau }) vert }{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr)^{ frac{1}{q}}biggr} \ &qquad {}+ biggl( int _{frac{1}{2}}^{1} biglvert delta ({tau }) bigrvert ,mathrm{d} {tau } biggr)^{1-frac{1}{q}}biggl{ biggl( int _{ frac{1}{2}}^{1} frac{(1-{tau }) vert delta ({tau }) vert }{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}\ &qquad{} + int _{ frac{1}{2}}^{1} frac{{tau } vert delta ({tau }) vert }{({tau b_{1}}+(1-{tau }){{b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr)^{ frac{1}{q}} \ &qquad {} + biggl( int _{frac{1}{2}}^{1} frac{{tau } vert delta ({tau }) vert }{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}\ &qquad{}+ int _{ frac{1}{2}}^{1} frac{(1-{tau }) vert delta ({tau }) vert }{((1-{tau }){b_{1}}+{tau {b_{2}}})^{2}} ,mathrm{d} {tau } biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} biggr)^{ frac{1}{q}}biggr} biggr]. end{aligned}$$

Proof

By using Lemma 2.4, the property of modulus, the power mean inequality and the convexity of (vert {Upsilon }’ vert ^{q}) we obtain the desired result. We omit here the proof. □

Corollary 2.28

Taking (Phi ({tau })={tau }) in Theorem 2.10, then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} int _{b_{1}}^{{b_{2}}} frac{{Upsilon }(x)}{x^{2}} ,mathrm{d}x biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} biggl(frac{1}{8} biggr)^{1- frac{1}{q}} \ &qquad {}times bigl[ bigl(M_{1} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{2} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ bigl(M_{3} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{4} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} \ &qquad {}+ bigl(M_{2} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} +M_{1} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}+ bigl(M_{4} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{3} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr], end{aligned}$$

where

$$begin{aligned}& M_{1}:=frac{{{b_{2}}}^{-2}}{8}{_{2}F_{1}} biggl(2,2,3, frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr)- frac{{{b_{2}}}^{-2}}{24}{_{2}F_{1}} biggl(2,3,4, frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr), \& M_{2}:=frac{{{b_{2}}}^{-2}}{24}{_{2}F_{1}} biggl(2,3,4, frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr),qquad M_{3}:= frac{{b_{1}}^{-2}}{24}{_{2}F_{1}} biggl(2,3,4, frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr), \& M_{4}:=frac{{b_{1}}^{-2}}{8}{_{2}F_{1}} biggl(2,2,3, frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr)-frac{{b_{1}}^{-2}}{24}{_{2}F_{1}} biggl(2,3,4, frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr). end{aligned}$$

Corollary 2.29

Choosing (Phi ({tau })=frac{{tau }^{alpha }}{Gamma (alpha )}) in Theorem 2.10, then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{Gamma (alpha +1)}{2} biggl( frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} biggr)^{alpha } biggl[J_{ frac{1}{{{b_{2}}}}^{+}}^{alpha }{{Upsilon }circ {Psi }} biggl( frac{1}{{b_{1}}} biggr) +J_{frac{1}{{b_{1}}}^{-}}^{alpha }{{ Upsilon } circ {Psi }} biggl(frac{1}{{{b_{2}}}} biggr) biggr] biggrvert \ &quad leq frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \ &qquad {}times biggl[ biggl(frac{1}{2^{alpha +1}(alpha +1)} biggr)^{1- frac{1}{q}} bigl{ bigl(M_{5} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{6} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+ bigl(M_{7} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{8} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr} \ &qquad {}+ biggl( frac{2^{alpha }(alpha -1)+1}{2^{alpha +1}(alpha +1)} biggr)^{1- frac{1}{q}} bigl{ bigl(M_{6} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} +M_{5} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}}\ &qquad{}+ bigl(M_{8} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{7} biglvert { Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr} biggr], end{aligned}$$

where

$$begin{aligned}& begin{aligned} M_{5}:={}&frac{{{b_{2}}}^{-2}}{2^{alpha +1}(alpha +1)}{_{2}F_{1}} biggl(2,alpha +1,alpha +2, frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr)\ &{}-frac{{{b_{2}}}^{-2}}{2^{alpha +2}(alpha +2)}{_{2}F_{1}} biggl(2,alpha +2,alpha +3, frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr), end{aligned} \& M_{6}:=frac{{{b_{2}}}^{-2}}{2^{alpha +2}(alpha +2)}{_{2}F_{1}} biggl(2,alpha +2,alpha +3, frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr), \& M_{7}:=frac{{b_{1}}^{-2}}{2^{alpha +2}(alpha +2)}{_{2}F_{1}} biggl(2, alpha +2,alpha +3, frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr), \& begin{aligned} M_{8}:={}&frac{{b_{1}}^{-2}}{2^{alpha +1}(alpha +1)}{_{2}F_{1}} biggl(2, alpha +1,alpha +2, frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr)\ &{}- frac{{b_{1}}^{-2}}{2^{alpha +2}(alpha +2)}{_{2}F_{1}} biggl(2, alpha +2, alpha +3, frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr). end{aligned} end{aligned}$$

Corollary 2.30

Taking (Phi ({tau })= frac{{tau }^{frac{alpha }{k}}}{kGamma _{k}(alpha )}) in Theorem 2.10, then

$$begin{aligned} & bigglvert {Upsilon } biggl(frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} biggr)- frac{Gamma _{k}(alpha +k)}{2} biggl( frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} biggr)^{frac{alpha }{k}} biggl[_{k}J_{frac{1}{{{b_{2}}}}^{+}}^{alpha }{{Upsilon }circ {Psi }} biggl(frac{1}{{b_{1}}} biggr) + _{k}J_{frac{1}{{b_{1}}}^{-}}^{ alpha }{{ Upsilon }circ {Psi }} biggl(frac{1}{{{b_{2}}}} biggr) biggr] biggrvert \ &qquad {}times biggl[ biggl(frac{k}{2^{frac{alpha }{k}+1}(alpha +k)} biggr)^{1-frac{1}{q}} bigl{ bigl(M_{9} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{10} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{ frac{1}{q}}\ &qquad{}+ bigl(M_{11} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{12} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr} \ &qquad {}+ biggl( frac{2^{frac{alpha }{k}}(alpha -k)+k}{2^{frac{alpha }{k}+1}(alpha +k)} biggr)^{1-frac{1}{q}} bigl{ bigl(M_{10} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q} +M_{9} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{ frac{1}{q}}\ &qquad{}+ bigl(M_{12} biglvert {Upsilon }'({b_{1}}) bigrvert ^{q}+M_{11} biglvert {Upsilon }'({{b_{2}}}) bigrvert ^{q} bigr)^{frac{1}{q}} bigr} biggr], end{aligned}$$

where

$$begin{aligned}& begin{aligned} M_{9}:={}&frac{k{{b_{2}}}^{-2}}{2^{frac{alpha }{k}+1}(alpha +k)}{_{2}F_{1}} biggl(2,alpha +k,alpha +2k, frac{1}{k} biggl( frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr) biggr) \ &{}-frac{k{{b_{2}}}^{-2}}{2^{frac{alpha }{k}+2}(alpha +2k)}{_{2}F_{1}} biggl(2,alpha +2k, alpha +3k, frac{1}{k} biggl( frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr) biggr), end{aligned}\& M_{10}:=frac{k{{b_{2}}}^{-2}}{2^{frac{alpha }{k}+2}(alpha +2k)}{_{2}F_{1}} biggl(2,alpha +2k,alpha +3k, frac{1}{k} biggl( frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} biggr) biggr), \& M_{11}:=frac{k{b_{1}}^{-2}}{2^{frac{alpha }{k}+2}(alpha +2k)}{_{2}F_{1}} biggl(2,alpha +2k,alpha +3k, frac{1}{k} biggl( frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr) biggr),\& begin{aligned} M_{12}:={}&frac{k{b_{1}}^{-2}}{2^{frac{alpha }{k}+1}(alpha +k)}{_{2}F_{1}} biggl(2,alpha +k,alpha +2k, frac{1}{k} biggl( frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr) biggr) \ &{}-frac{k{b_{1}}^{-2}}{2^{frac{alpha }{k}+2}(alpha +2k)}{_{2}F_{1}} biggl(2,alpha +2k, alpha +3k, frac{1}{k} biggl( frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} biggr) biggr). end{aligned} end{aligned}$$

Remark

For other suitable choices of function Φ, several new interesting inequalities can be found from our results. We omit here their proofs and the details are left to the interested reader.

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