We now discuss our main results of the paper.
2.1 Post quantum Montgomery identity
We now derive a significant result of this paper. Next results of this paper depend on this lemma.
Lemma 2.1
Let \({\Psi }:[e,e+\xi (f,e)]\to \mathbb{R}\) be a \((\mathrm{p},\mathrm{q})\)-differentiable function such that \({}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }\) is \((\mathrm{p},\mathrm{q})\)-integrable on \([e,e+\xi (f,e)]\), then
$$\begin{aligned} {\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda }=\xi (f,e) \int _{0}^{1}K_{\mathrm{q}}({ \lambda }) \,{}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }, \end{aligned}$$
where
$$ K_{\mathrm{q}}({\lambda })= \textstyle\begin{cases} \mathrm{q}{\lambda } & \textit{for } {\lambda }\in [0, \frac{x-e}{\xi (f,e)}], \\ \mathrm{q}{\lambda }-1 & \textit{for } {\lambda }\in ( \frac{x-e}{\xi (f,e)},1]. \end{cases} $$
Proof
It suffices to show that
$$\begin{aligned} &\xi (f,e) \int _{0}^{1}K_{\mathrm{q}}({\lambda }) \,{}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \\ &\quad =\xi (f,e) \biggl[ \int _{0}^{\frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda } \,{}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \\ &\qquad {}+ \int _{\frac{x-e}{\xi (f,e)}}^{1}(\mathrm{q} {\lambda }-1) \,{}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \biggr] \\ &\quad =\xi (f,e) \biggl[ \int _{0}^{\frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda } \,{}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \\ &\qquad {}+ \int _{0}^{1}(\mathrm{q} {\lambda }-1) \,{}_{e}D_{\mathrm{p}, \mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \\ &\qquad {}- \int _{0}^{\frac{x-e}{\xi (f,e)}}(\mathrm{q} {\lambda }-1) \,{}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \biggr] \\ &\quad =\xi (f,e) \biggl[ \int _{0}^{1}(\mathrm{q} {\lambda }-1) \,{}_{e}D_{\mathrm{p}, \mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \\ &\qquad {}+ \int _{0}^{\frac{x-e}{\xi (f,e)}}\,{}_{e}D_{\mathrm{p}, \mathrm{q}}{ \Psi }\bigl(e+{\lambda }\xi (f,e)\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{ \lambda } \biggr] \\ &\quad =\xi (f,e) \biggl[ \int _{0}^{1}\mathrm{q} {\lambda } \,{}_{e}D_{\mathrm{p}, \mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \\ &\qquad {}- \int _{0}^{1}\,{}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }\bigl(e+{ \lambda }\xi (f,e)\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda } \\ &\qquad {}+ \int _{0}^{\frac{x-e}{\xi (f,e)}}\,{}_{e}D_{\mathrm{p}, \mathrm{q}}{ \Psi }\bigl(e+{\lambda }\xi (f,e)\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{ \lambda } \biggr] \\ &\quad =\frac{1}{\mathrm{p}-\mathrm{q}} \biggl[ \mathrm{q} \biggl[ \int _{0}^{1}{\Psi }\bigl(e+\mathrm{q}\lambda \xi (f,e) \bigr)\,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } - \int _{0}^{1}{\Psi }\bigl(e+ \mathrm{q} {\lambda } \xi (f,e)\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda }\biggr] \\ &\qquad {}- \biggl[ \int _{0}^{1}\frac{{\Psi }(e+\mathrm{q}\lambda \xi (f,e))}{{\lambda }}\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } - \int _{0}^{1} \frac{{\Psi }(e+\mathrm{q}{\lambda }\xi (f,e))}{{\lambda }}\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr] \\ &\qquad {}+ \biggl[ \int _{0}^{\frac{x-e}{\xi (f,e)}} \frac{{\Psi }(e+\mathrm{q}\lambda \xi (f,e))}{{\lambda }}\,{}_{0} \mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } - \int _{0}^{ \frac{x-e}{\xi (f,e)}} \frac{{\Psi }(e+\mathrm{q}{\lambda }\xi (f,e))}{{\lambda }}\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr] \biggr] \\ &\quad =\frac{1}{\mathrm{p}-\mathrm{q}} \Biggl[ \mathrm{q}(\mathrm{p}-\mathrm{q}) \Biggl[ \sum_{n=0}^{\infty }\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{ \Psi } \biggl(e+\frac{\mathrm{q}^{n}}{\mathrm{p}^{n}}\xi (f,e) \biggr) \\ &\qquad {}-\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{ \Psi } \biggl(e+\frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}-(\mathrm{p}-\mathrm{q}) \Biggl[ \sum_{n=0}^{\infty }{ \Psi } \biggl(e+ \frac{\mathrm{q}^{n}}{\mathrm{p}^{n}}\xi (f,e) \biggr) \\ &\qquad {}-\sum_{n=0}^{\infty }{\Psi } \biggl(e+ \frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}+(\mathrm{p}-\mathrm{q}) \biggl(\frac{x-e}{\xi (f,e)} \biggr) \Biggl[ \sum_{n=0}^{\infty } \frac{{\Psi } (e+\frac{\mathrm{q}^{n}}{\mathrm{p}^{n}} (\frac{x-e}{\xi (f,e)} )\xi (f,e) )}{\frac{x-e}{\xi (f,e)}} \\ &\qquad {}-\sum_{n=0}^{\infty } \frac{{\Psi } (e+\frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+1}} (\frac{x-e}{\xi (f,e)} )\xi (f,e) )}{\frac{x-e}{\xi (f,e)}} \Biggr] \Biggr] \\ &\quad = \Biggl[ \mathrm{q} \Biggl[ \sum _{n=0}^{\infty }\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{ \Psi } \biggl(e+ \frac{\mathrm{q}^{n}}{\mathrm{p}^{n}}\xi (f,e) \biggr) \\ &\qquad {}-\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{ \Psi } \biggl(e+\frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}- \Biggl[ \sum_{n=0}^{\infty }{\Psi } \biggl(e+ \frac{\mathrm{q}^{n}}{\mathrm{p}^{n}}\xi (f,e) \biggr) \\ &\qquad {}-\sum_{n=0}^{\infty }{\Psi } \biggl(e+ \frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}+ \Biggl[ \sum_{n=0}^{\infty }{\Psi } \biggl(e+ \frac{\mathrm{q}^{n}}{\mathrm{p}^{n}} \biggl(\frac{x-e}{\xi (f,e)} \biggr)\xi (f,e) \biggr) \\ &\qquad {}-\sum_{n=0}^{\infty }{\Psi } \biggl(e+ \frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+1}} \biggl(\frac{x-e}{\xi (f,e)} \biggr)\xi (f,e) \biggr) \Biggr] \Biggr] \\ &\quad = \Biggl[ \mathrm{q} \Biggl[ \sum _{n=0}^{\infty }\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{ \Psi } \biggl(e+ \mathrm{p}\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \\ &\qquad {}-\frac{\mathrm{p}}{\mathrm{q}}\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+2}}{\Psi } \biggl(e+\mathrm{p} \frac{\mathrm{q}^{n+1}}{\mathrm{p}^{n+2}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}- \bigl[ {\Psi }\bigl(e+\xi (f,e)\bigr)-{\Psi }(e) \bigr] \\ &\qquad {}+ \biggl[ {\Psi } \biggl(e+ \biggl(\frac{x-e}{\xi (f,e)} \biggr)\xi (f,e) \biggr)-{ \Psi }(e) \biggr] \Biggr] \\ &\quad = \Biggl[ \mathrm{q} \Biggl[ \sum _{n=0}^{\infty }\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{ \Psi } \biggl(e+ \mathrm{p}\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \\ &\qquad {}-\frac{\mathrm{p}}{\mathrm{q}}\sum_{n=1}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{\Psi } \biggl(e+\mathrm{p} \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}-{\Psi }\bigl(e+\xi (f,e)\bigr)+{\Psi } \biggl(e+ \biggl( \frac{x-e}{\xi (f,e)} \biggr)\xi (f,e) \biggr) \Biggr] \\ &\quad = \Biggl[ \mathrm{q} \Biggl[ \frac{{\Psi }(e+\xi (f,e))}{\mathrm{q}}-\biggl( \frac{\mathrm{p}}{\mathrm{q}}-1\biggr) \sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{ \Psi } \biggl(e+\mathrm{p}\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}-{\Psi }\bigl(e+\xi (f,e)\bigr)+{\Psi } \biggl(e+ \biggl( \frac{x-e}{\xi (f,e)} \biggr)\xi (f,e) \biggr) \Biggr] \\ &\quad = \Biggl[ \mathrm{q} \Biggl[ \frac{{\Psi }(e+\xi (f,e))}{\mathrm{q}}-\biggl( \frac{\mathrm{p}-\mathrm{q}}{\mathrm{q}}\biggr)\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{\Psi } \biggl(e+\mathrm{p} \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \Biggr] \\ &\qquad {}-{\Psi }\bigl(e+\xi (f,e)\bigr)+{\Psi } \biggl(e+ \biggl( \frac{x-e}{\xi (f,e)} \biggr)\xi (f,e) \biggr) \Biggr] \\ &\quad ={\Psi }(x)-(\mathrm{p}-\mathrm{q})\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}{\Psi } \biggl(e+\mathrm{p} \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}}\xi (f,e) \biggr) \\ &\quad ={\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda }. \end{aligned}$$
This completes the proof. □
Remark 2.2
If we take \(x= \frac{(\mathrm{p}+\mathrm{q})e+\mathrm{p}\xi (f,e)}{\mathrm{p}+\mathrm{q}}\) in Lemma 2.1, then we have the following new equality:
$$\begin{aligned} &{\Psi } \biggl( \frac{(\mathrm{p}+\mathrm{q})e+\mathrm{p}\xi (f,e)}{\mathrm{p}+\mathrm{q}} \biggr) -\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \\ &\quad = \xi (f,e) \biggl[ \int _{0}^{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}} \mathrm{q} {\lambda } \,{}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }\bigl(+{\lambda } \xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \\ &\qquad {}+ \int _{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}^{1}( \mathrm{q} {\lambda }-1) \,{}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{ \lambda }\xi (f,e)\bigr) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr]. \end{aligned}$$
2.2 Estimation of bounds
We now discuss some results depending upon Lemma 2.1.
Theorem 2.3
Let \({\Psi }:[e,e+\xi (f,e)]\) be a function such that \({}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }\) is \((\mathrm{p},\mathrm{q})\)-integrable on \([e,e+\xi (f,e)]\). If \(|{}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }|^{r}\), \(r> 1\) is preinvex on \([e,e+\xi (f,e)]\), then
$$\begin{aligned} & \biggl\vert {\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr\vert \\ &\quad \leq \xi (f,e) \bigl[ L_{1}^{1-\frac{1}{r}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}L_{2}+ \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{ \Psi }(f) \bigr\vert ^{r}L_{3} \bigr]^{\frac{1}{r}} \\ &\qquad {}+L_{4}^{1-\frac{1}{r}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}L_{5}+ \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }(f) \bigr\vert ^{r}L_{6} \bigr]^{\frac{1}{r}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &L_{1}= \int _{0}^{\frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda } \,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{q}}{\mathrm{p}+\mathrm{q}} \biggl(\frac{x-e}{\xi (f,e)} \biggr)^{2}, \\ &L_{2}= \int _{0}^{\frac{x-e}{\xi (f,e)}}\bigl(\mathrm{q} {\lambda }- \mathrm{q} {\lambda }^{2}\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda }=L_{1}-L_{3}, \\ &L_{3}= \int _{0}^{\frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda }^{2} \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{q}}{\mathrm{p}+pq+\mathrm{q}^{2}} \biggl( \frac{x-e}{\xi (f,e)} \biggr)^{3}, \\ &L_{4}= \int _{\frac{x-e}{\xi (f,e)}}^{1}(1-\mathrm{q} {\lambda }) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}- \frac{\mathrm{q}}{\mathrm{p}+\mathrm{q}} \biggl(\frac{x-e}{\xi (f,e)} \biggr) \biggl(1-\frac{\mathrm{q}}{\mathrm{p}+\mathrm{q}} \frac{x-e}{\xi (f,e)} \biggr), \\ &L_{5}= \int _{\frac{x-e}{\xi (f,e)}}^{1}\bigl(1-\mathrm{q} {\lambda }-{ \lambda }+\mathrm{q} {\lambda }^{2}\bigr)\,{}_{0} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda }=L_{4}-L_{6}, \\ &\begin{aligned} L_{6}&= \int _{\frac{x-e}{\xi (f,e)}}^{1}\bigl({\lambda }-\mathrm{q} { \lambda }^{2}\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \\ &= \frac{\mathrm{p}^{2}}{(\mathrm{p}+\mathrm{q})(\mathrm{p}^{2}+pq+\mathrm{q}^{2})}- \frac{1}{\mathrm{p}+\mathrm{q}} \biggl(\frac{x-e}{\xi (f,e)} \biggr)^{2}+ \frac{\mathrm{q}}{\mathrm{p}^{2}+pq+\mathrm{q}^{2}} \biggl( \frac{x-e}{\xi (f,e)} \biggr)^{3}. \end{aligned} \end{aligned}$$
Proof
Using Lemma 2.1, the power mean integral inequality, and the preinvexity of \(|{}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }|^{r}\), we obtain
$$\begin{aligned} & \biggl\vert {\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr\vert \\ &\quad \leq \xi (f,e) \biggl[ \int _{0}^{\frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda } \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \bigr\vert {}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \\ &\qquad {}+ \int _{\frac{x-e}{\xi (f,e)}}^{1}(1-\mathrm{q} {\lambda }) \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \bigr\vert {}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \biggr] \\ &\quad \leq \xi (f,e) \biggl[ \biggl( \int _{0}^{\frac{x-e}{\xi (f,e)}} \mathrm{q} {\lambda } \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{1-\frac{1}{r}} \biggl( \int _{0}^{ \frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda } \bigl\vert {}_{e}D_{\mathrm{p}, \mathrm{q}}{\Psi }\bigl(e+{\lambda }\xi (f,e)\bigr) \bigr\vert ^{r}\,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \biggr)^{\frac{1}{r}} \\ &\qquad {}+ \biggl( \int _{\frac{x-e}{\xi (f,e)}}^{1}(1- \mathrm{q} {\lambda }) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{1-\frac{1}{r}} \biggl( \int _{\frac{x-e}{\xi (f,e)}}^{1}(1- \mathrm{q} {\lambda }) \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }\bigl(e+{\lambda } \xi (f,e)\bigr) \bigr\vert ^{r}\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{\frac{1}{r}} \biggr] \\ &\quad \leq \xi (f,e) \biggl[ \biggl( \int _{0}^{\frac{x-e}{\xi (f,e)}} \mathrm{q} {\lambda } \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{1-\frac{1}{r}} \\ &\qquad{} \times \biggl( \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }(e) \bigr\vert ^{r} \int _{0}^{\frac{x-e}{\xi (f,e)}}\bigl(\mathrm{q} { \lambda }- \mathrm{q} {\lambda }^{2}\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{ \lambda }+ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(f) \bigr\vert ^{r} \int _{0}^{\frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda }^{2} \,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{\frac{1}{r}} \\ &\qquad{} + \biggl( \int _{\frac{x-e}{\xi (f,e)}}^{1}(1- \mathrm{q} {\lambda }) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{1-\frac{1}{r}} \\ &\qquad {}\times \biggl( \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }(e) \bigr\vert ^{r} \int _{\frac{x-e}{\xi (f,e)}}^{1}\bigl(1-\mathrm{q} {\lambda }-{ \lambda }+\mathrm{q} {\lambda }^{2}\bigr)\,{}_{0} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda }\\ &\qquad {}+ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }(f) \bigr\vert ^{r} \int _{\frac{x-e}{\xi (f,e)}}^{1}\bigl({\lambda }-\mathrm{q} { \lambda }^{2}\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{ \frac{1}{r}} \biggr], \end{aligned}$$
The proof is accomplished. □
Corollary 2.4
In Theorem 2.3, the following quantum estimates hold under the following conditions:
I. \(r=1\)
$$\begin{aligned} & \biggl\vert {\Psi }(x)-\frac{1}{\xi (f,e)} \int _{e}^{e+\xi (f,e)}{ \Psi }({\lambda }) \,{}_{e}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr\vert \\ &\quad \leq \xi (f,e) \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }(e) \bigr\vert [L_{2}+L_{5}] + \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(f) \bigr\vert [L_{3}+L_{6}] \bigr]. \end{aligned}$$
II. \(x= \frac{(\mathrm{p}+\mathrm{q})e+\mathrm{p}\xi (f,e)}{\mathrm{p}+\mathrm{q}}\), then we have a new inequality:
$$\begin{aligned} & \biggl\vert {\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr\vert \\ &\quad \leq \xi (f,e) \bigl[ L_{7}^{1-\frac{1}{r}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}L_{8} + \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{ \Psi }(f) \bigr\vert ^{r}L_{9} \bigr]^{\frac{1}{r}} \\ &\qquad {}+L_{10}^{1-\frac{1}{r}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}L_{11} + \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }(f) \bigr\vert ^{r}L_{12} \bigr]^{\frac{1}{r}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &L_{7}= \int _{0}^{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}} \mathrm{q} {\lambda } \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{q}}{\mathrm{p}+\mathrm{q}} \biggl( \frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}} \biggr)^{2}, \\ &L_{8}= \int _{0}^{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}\bigl( \mathrm{q} {\lambda }- \mathrm{q} {\lambda }^{2}\bigr)\,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{ \lambda }=L_{7}-L_{9}, \\ &L_{9}= \int _{0}^{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}} \mathrm{q} {\lambda }^{2} \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda }=\frac{\mathrm{q}}{\mathrm{p}+pq+\mathrm{q}^{2}} \biggl( \frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}} \biggr)^{3}, \\ &L_{10}= \int _{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}^{1}(1- \mathrm{q} {\lambda }) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{q}}{\mathrm{p}+\mathrm{q}} \biggl(1- \frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}} \biggr) \biggl(1- \frac{\mathrm{q}}{\mathrm{p}+\mathrm{q}} \frac{x-e}{\xi (f,e)} \biggr), \\ &L_{11}= \int _{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}^{1}\bigl(1- \mathrm{q} {\lambda }-{ \lambda }+\mathrm{q} {\lambda }^{2}\bigr)\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }=L_{10}-L_{12}, \\ &L_{12}= \int _{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}^{1}\bigl({ \lambda }-\mathrm{q} {\lambda }^{2}\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda }= \frac{1}{\mathrm{p}+\mathrm{q}}- \frac{\mathrm{p}^{2}}{(\mathrm{p}+\mathrm{q})^{3}}+ \frac{1}{\mathrm{p}^{2}+\mathrm{q}^{2}+pq} \biggl(1- \biggl( \frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}} \biggr)^{3} \biggr). \end{aligned}$$
Theorem 2.5
Let \({\Psi }:[e,e+\xi (f,e)]\) be a function such that \({}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }\) is \((\mathrm{p},\mathrm{q})\)-integrable on \([e,e+\xi (f,e)]\). If \(|{}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }|^{r}, r>1\), \(s^{-1}+r^{-1}=1\) is preinvex on \([e,e+\xi (f,e)]\), then
$$\begin{aligned} & \biggl\vert {\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr\vert \\ &\quad \leq \mathrm{q}\xi (f,e) \bigl[ K_{1}^{\frac{1}{s}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}K_{2}+ \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{ \Psi }(f) \bigr\vert ^{r}K_{3} \bigr]^{\frac{1}{r}} \\ &\qquad {}+K_{4}^{\frac{1}{s}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}K_{5}+ \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }(f) \bigr\vert ^{r}K_{6} \bigr]^{\frac{1}{r}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &K_{1}= \int _{0}^{\frac{x-e}{\xi (f,e)}}{\lambda }^{s} \,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \biggl( \frac{x-e}{\xi (f,e)} \biggr)^{s+1} \frac{\mathrm{p}-\mathrm{q}}{\mathrm{p}^{s+1}-\mathrm{q}^{s+1}}, \\ &K_{2}= \int _{0}^{\frac{x-e}{\xi (f,e)}}(1-{\lambda })\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }=\frac{x-e}{\xi (f,e)}- \frac{1}{\mathrm{p}+\mathrm{q}} \biggl( \frac{x-e}{\xi (f,e)} \biggr)^{2}, \\ &K_{3}= \int _{0}^{\frac{x-e}{\xi (f,e)}}{\lambda }\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{1}{\mathrm{p}+\mathrm{q}} \biggl(\frac{x-e}{\xi (f,e)} \biggr)^{2}, \\ & \begin{aligned} K_{4}&= \int _{\frac{x-e}{\xi (f,e)}}^{1} \biggl( \frac{1}{\mathrm{q}}-{\lambda } \biggr)^{s}\,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \\ &= (\mathrm{p}-\mathrm{q}) \Biggl[ \sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggl(\frac{1}{\mathrm{q}}-\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggr)^{s} \\ &\quad {}-\frac{x-e}{\xi (f,e)}\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggl(\frac{1}{\mathrm{q}}- \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggl( \frac{x-e}{\xi (f,e)} \biggr) \biggr)^{s} \Biggr], \end{aligned} \\ &K_{5}= \int _{\frac{x-e}{\xi (f,e)}}^{1}(1-{\lambda })\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{p}+\mathrm{q}-1}{\mathrm{p}+\mathrm{q}}- \frac{x-e}{\xi (f,e)}+ \frac{1}{\mathrm{p}+\mathrm{q}} \biggl( \frac{x-e}{\xi (f,e)} \biggr)^{2}, \\ &K_{6}= \int _{\frac{x-e}{\xi (f,e)}}^{1}{\lambda }\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{1}{\mathrm{p}+\mathrm{q}} \biggl(1- \biggl( \frac{x-e}{\xi (f,e)} \biggr)^{2} \biggr). \end{aligned}$$
Proof
Using Lemma 2.1, Holder’s inequality, and the preinvexity of \(|{}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }|^{r}\), we obtain
$$\begin{aligned} & \biggl\vert {\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr\vert \\ &\quad \leq \xi (f,e) \biggl[ \int _{0}^{\frac{x-e}{\xi (f,e)}}\mathrm{q} {\lambda } \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+\xi (f,e)\bigr) \bigr\vert {}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \\ &\qquad {}+ \int _{\frac{x-e}{\xi (f,e)}}^{1}(1-\mathrm{q} {\lambda }) \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }\bigl(e+\xi (f,e)\bigr) \bigr\vert {}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \biggr] \\ &\quad \leq \xi (f,e) \biggl[ \biggl( \int _{0}^{\frac{x-e}{\xi (f,e)}}( \mathrm{q} {\lambda })^{s}\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda } \biggr)^{\frac{1}{s}} \biggl( \int _{0}^{ \frac{x-e}{\xi (f,e)}} \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }\bigl(e+\xi (f,e)\bigr) \bigr\vert ^{r} \,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{ \frac{1}{r}} \\ &\qquad {}+ \biggl( \int _{\frac{x-e}{\xi (f,e)}}^{1}(1- \mathrm{q} {\lambda })^{s}\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda } \biggr)^{\frac{1}{s}} \biggl( \int _{ \frac{x-e}{\xi (f,e)}}^{1} \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }\bigl(e+ \xi (f,e)\bigr) \bigr\vert ^{r}\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{\frac{1}{r}} \biggr] \\ &\quad \leq \mathrm{q}\xi (f,e) \biggl[ \biggl( \int _{0}^{ \frac{x-e}{\xi (f,e)}}{\lambda }^{s} \,{}_{0}\mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr)^{\frac{1}{s}}\\ &\qquad {}\times \biggl( \bigl\vert {}_{e}D_{\mathrm{p}, \mathrm{q}}{\Psi }(e) \bigr\vert ^{r} \int _{0}^{\frac{x-e}{\xi (f,e)}}(1-{ \lambda })\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }+ \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{ \Psi }(f) \bigr\vert ^{r} \int _{0}^{ \frac{x-e}{\xi (f,e)}}{\lambda }\,{}_{0} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr)^{\frac{1}{r}} \\ &\qquad{} + \biggl( \int _{\frac{x-e}{\xi (f,e)}}^{1} \biggl( \frac{1}{\mathrm{q}}-{\lambda } \biggr)^{s}\,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda } \biggr)^{\frac{1}{s}} \\ &\qquad {}\times \biggl( \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{ \Psi }(e) \bigr\vert ^{r} \int _{\frac{x-e}{\xi (f,e)}}^{1}(1-{\lambda })\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }+ \bigl\vert {}_{e}D_{\mathrm{p}, \mathrm{q}}{ \Psi }(f) \bigr\vert ^{r} \int _{\frac{x-e}{\xi (f,e)}}^{1}{ \lambda }\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \biggr)^{ \frac{1}{r}} \biggr]. \end{aligned}$$
The proof is accomplished. □
Remark 2.6
If we take \(x=\frac{e+\mathrm{q}(e+\xi (f,e))}{1+\mathrm{q}}\) in Theorem 2.5, then we have a new inequality:
$$\begin{aligned} & \biggl\vert {\Psi }(x)-\frac{1}{\mathrm{p}\xi (f,e)} \int _{e}^{e+ \mathrm{p}\xi (f,e)}{\Psi }({\lambda })\,{}_{e} \mathrm{d}_{\mathrm{p}, \mathrm{q}}{\lambda } \biggr\vert \\ &\quad \leq \mathrm{q}\xi (f,e) \bigl[ K_{7}^{\frac{1}{s}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}K_{8}+ \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{ \Psi }(f) \bigr\vert ^{r}K_{9} \bigr]^{\frac{1}{r}} \\ &\qquad {}+K_{10}^{\frac{1}{s}} \bigl[ \bigl\vert {}_{e}D_{\mathrm{p},\mathrm{q}}{\Psi }(e) \bigr\vert ^{r}K_{11}+ \bigl\vert {}_{e}D_{ \mathrm{p},\mathrm{q}}{\Psi }(f) \bigr\vert ^{r}K_{12} \bigr]^{\frac{1}{r}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &K_{7}= \int _{0}^{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}{ \lambda }^{s} \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \biggl( \frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}} \biggr)^{s+1} \frac{\mathrm{p}-\mathrm{q}}{\mathrm{p}^{s+1}-\mathrm{q}^{s+1}}, \\ &K_{8}= \int _{0}^{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}(1-{ \lambda })\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}- \frac{\mathrm{p}^{2}}{(\mathrm{p}+\mathrm{q})^{3}}, \\ &K_{9}= \int _{0}^{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}{ \lambda }\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{p}^{2}}{(\mathrm{p}+\mathrm{q})^{3}}, \\ & \begin{aligned} K_{10}&= \int _{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}^{1} \biggl(\frac{1}{\mathrm{q}}-{\lambda } \biggr)^{s}\,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda }, \\ &= (\mathrm{p}-\mathrm{q}) \Biggl[ \sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggl(\frac{1}{\mathrm{q}}-\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggr)^{s} \\ &\quad {}-\frac{1}{\mathrm{p}+\mathrm{q}}\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n}} \biggl(\frac{1}{\mathrm{q}}- \frac{\mathrm{q}^{n}}{\mathrm{p}^{n}} \biggl( \frac{1}{\mathrm{p}+\mathrm{q}} \biggr) \biggr)^{s} \Biggr], \end{aligned} \\ &K_{11}= \int _{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}^{1}(1-{ \lambda })\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{q}-1}{\mathrm{p}+\mathrm{q}}- \frac{\mathrm{p}^{2}}{(\mathrm{p}+\mathrm{q})^{3}}, \\ &K_{12}= \int _{\frac{\mathrm{p}}{\mathrm{p}+\mathrm{q}}}^{1}{ \lambda }\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{1}{\mathrm{p}+\mathrm{q}}- \frac{\mathrm{p}^{2}}{(\mathrm{p}+\mathrm{q})^{3}}. \end{aligned}$$
2.3 Applications
We now discuss some applications of the results obtained in the previous section. First of all we recall some previously known concepts. For arbitrary real numbers, consider the following means:
$$\begin{aligned}& \text{Arithmetic mean}:\quad A(e,f)=\frac{e+f}{2}, \\& \text{Generalized logarithmic mean}:\quad L_{\mathrm{q}}(e,f)= \biggl[ \frac{f^{\mathrm{q}+1}-e^{\mathrm{q}+1}}{(\mathrm{q}+1)(f-e)} \biggr]^{\frac{1}{\mathrm{q}}}, \end{aligned}$$
where \(\mathrm{q}\in \mathbb{R}\setminus \{-1,0\}\), \(e,f\in \mathbb{R}\) with \(e\neq f\).
Proposition 2.7
Let \(0< e< f\), \(n\in \mathbb{N}\), \(0<\mathrm{q}<\mathrm{p}<1\), then
$$\begin{aligned} & \biggl\vert A^{n}(e,f)- \frac{(n+1)(\mathrm{p}-\mathrm{q})}{\mathrm{p}(\mathrm{p}^{n+1}-\mathrm{q}^{n+1})}L_{n}^{n} \bigl(e,(1- \mathrm{p})e+pf\bigr) \biggr\vert \\ &\quad \leq (f-e) \biggl[ H_{1}^{1-\frac{1}{r}} \biggl[ \bigl\vert ne^{n-1} \bigr\vert ^{r}H_{2} + \biggl\vert \frac{(pf+(1-\mathrm{p})e)^{n}-(qf+(1-\mathrm{q})e)^{n}}{(f-e)(1-\mathrm{q})} \biggr\vert ^{r}H_{3} \biggr]^{\frac{1}{r}} \\ &\qquad {}+H_{4}^{1-\frac{1}{r}} \biggl[ \bigl\vert ne^{n-1} \bigr\vert ^{r}H_{5} + \biggl\vert \frac{(pf+(1-\mathrm{p})e)^{n}-(qf+(1-\mathrm{q})e)^{n}}{(f-e)(1-\mathrm{q})} \biggr\vert ^{r}H_{6} \biggr]^{\frac{1}{r}} \biggr], \end{aligned}$$
where
$$\begin{aligned} &H_{1}= \int _{0}^{\frac{1}{2}}\mathrm{q} {\lambda } \,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{q}}{4(\mathrm{p}+\mathrm{q})}, \\ &H_{2}= \int _{0}^{\frac{1}{2}}\bigl(\mathrm{q} {\lambda }-\mathrm{q} { \lambda }^{2}\bigr)\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda }= \frac{\mathrm{q}(\mathrm{p}+2pq+2\mathrm{q}^{2}-\mathrm{q})}{8(\mathrm{p}+\mathrm{q})(\mathrm{p}^{2}+pq+\mathrm{q}^{2})}, \\ &H_{3}= \int _{0}^{\frac{1}{2}}\mathrm{q} {\lambda }^{2} \,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{\mathrm{q}}{8(\mathrm{p}+pq+\mathrm{q}^{2})}, \\ &H_{4}= \int _{\frac{1}{2}}^{1}(1-\mathrm{q} {\lambda }) \,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{2\mathrm{p}-\mathrm{q}}{4(\mathrm{p}+\mathrm{q})}, \\ &H_{5}= \int _{\frac{1}{2}}^{1}\bigl(1-\mathrm{q} {\lambda }-{\lambda }+ \mathrm{q} {\lambda }^{2}\bigr)\,{}_{0} \mathrm{d}_{\mathrm{p},\mathrm{q}}{ \lambda }= \frac{4\mathrm{p}^{3}+2pq^{2}+2\mathrm{p}^{2}\mathrm{q}-2\mathrm{q}^{3}-6\mathrm{p}^{2}+pq+\mathrm{q}^{2}}{8(\mathrm{p}+\mathrm{q})(\mathrm{p}^{2}+pq+\mathrm{q}^{2})}, \\ &H_{6}= \int _{\frac{1}{2}}^{1}\bigl({\lambda }-\mathrm{q} {\lambda }^{2}\bigr) \,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda }= \frac{6\mathrm{p}^{2}-pq-\mathrm{q}^{2}}{8(\mathrm{p}+\mathrm{q})(\mathrm{p}^{2}+pq+\mathrm{q}^{2})}. \end{aligned}$$
Proof
The proof directly follows from Theorem 2.3 applied for \({\Psi }(x)=x^{n}\), \(\xi (f,e)=f-e\) and considering \(x=\frac{e+f}{2}\). □
Proposition 2.8
Let \(0< e< f\), \(n\in \mathbb{N}\), \(0<\mathrm{q}<1\), then
$$\begin{aligned} & \biggl\vert A^{n}(e,f)- \frac{(n+1)(\mathrm{p}-\mathrm{q})}{\mathrm{p}(\mathrm{p}^{n+1}-\mathrm{q}^{n+1})}L_{n}^{n} \bigl(e,(1- \mathrm{p})e+pf\bigr) \biggr\vert \\ &\quad \leq \mathrm{q}(f-e) \biggl[ M_{1}^{\frac{1}{s}} \biggl[ \bigl\vert ne^{n-1} \bigr\vert ^{r}M_{2}+ \biggl\vert \frac{(pf+(1-\mathrm{p})e)^{n}-(qf+(1-\mathrm{q})e)^{n}}{(f-e)(1-\mathrm{q})} \biggr\vert ^{r}M_{3} \biggr]^{\frac{1}{r}} \\ &\qquad {}+M_{4}^{\frac{1}{s}} \biggl[ \bigl\vert ne^{n-1} \bigr\vert ^{r}M_{5}+ \biggl\vert \frac{(pf+(1-\mathrm{p})e)^{n}-(qf+(1-\mathrm{q})e)^{n}}{(f-e)(1-\mathrm{q})} \biggr\vert ^{r}M_{6} \biggr]^{\frac{1}{r}} \biggr], \end{aligned}$$
where
$$\begin{aligned} &M_{1}= \int _{0}^{\frac{1}{2}}{\lambda }^{s} \,{}_{0}\mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda }=\frac{1}{2^{s+1}} \frac{\mathrm{p}-\mathrm{q}}{\mathrm{p}^{s+1}-\mathrm{q}^{s+1}}, \\ &M_{2}= \int _{0}^{\frac{1}{2}}(1-{\lambda })\,{}_{0} \mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda }= \frac{2\mathrm{p}+2\mathrm{q}-1}{4(\mathrm{p}+\mathrm{q})}, \\ &M_{3}= \int _{0}^{\frac{1}{2}}{\lambda }\,{}_{0} \mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda }=\frac{1}{4(\mathrm{p}+\mathrm{q})}, \\ & \begin{aligned} M_{4}&= \int _{\frac{1}{2}}^{1} \biggl(\frac{1}{\mathrm{q}}-{ \lambda } \biggr)^{s}\,{}_{0}\mathrm{d}_{\mathrm{p},\mathrm{q}}{\lambda } \\ &= (\mathrm{p}-\mathrm{q}) \Biggl[ \sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggl(\frac{1}{\mathrm{q}}-\frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggr)^{s} \\ &\quad {}-\frac{1}{2}\sum_{n=0}^{\infty } \frac{\mathrm{q}^{n}}{\mathrm{p}^{n+1}} \biggl(\frac{1}{\mathrm{q}}- \frac{\mathrm{q}^{n}}{2\mathrm{p}^{n+1}} \biggr)^{s} \Biggr], \end{aligned} \\ &M_{5}= \int _{\frac{1}{2}}^{1}(1-{\lambda })\,{}_{0} \mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda }= \frac{2\mathrm{p}+2\mathrm{q}-3}{4(\mathrm{p}+\mathrm{q})}, \\ &M_{6}= \int _{\frac{1}{2}}^{1}{\lambda }\,{}_{0} \mathrm{d}_{ \mathrm{p},\mathrm{q}}{\lambda }=\frac{3}{4(\mathrm{p}+\mathrm{q})}. \end{aligned}$$
Proof
The proof directly follows from Theorem 2.5 applied for \({\Psi }(x)=x^{n}\), \(\xi (f,e)=f-e\) and considering \(x=\frac{e+f}{2}\). □