1 Introduction

Let \(\psi:[b_{1},b_{2}]\to \mathbb{R}\) be a convex function. Then the following double inequality holds:

$$ \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)\leq \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x) \,dx\leq \frac{\psi (b_{1})+\psi (b_{2})}{2}. $$
(1.1)

It is known in the literature as the Hermite–Hadamard inequality. This inequality has instigated pletora of papers. Results concerning generalization, refinement, and extension of (1.1) are also found; see [1–9, 12, 14, 15, 17–20, 23, 27–30] and the references therein.

In the early 16th century, the concept of q-calculus was introduced. Ever since, integral inequalities of the trapeziod, Ostrowski, Cauchy–Bunyakovsky–Schwarz, Grüss, Hölder, Grüss–C̆ebys̆ev, and other types have been established in the q-calculus sense. In 2014, Tariboon and Ntouyas [33] obtained the following q-calculus version of (1.1).

Theorem 1.1

Let\(\psi :[b_{1},b_{2}]\to \mathbb{R}\)be a convex continuous function on\((b_{1},b_{2})\), and let\(0< q<1\). Then

$$ \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)\leq \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x) _{b_{1}} \,d_{q}x\leq \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}. $$
(1.2)

In 2016, Kunt and Isçan [21] observed, via a counterexample, that the left-hand side of inequality (1.2) is not necessarily true. Subsequently, Alp et al. [11] proved the following correct version of (1.2).

Theorem 1.2

([11])

Let\(\psi :[b_{1},b_{2}]\to \mathbb{R}\)be a convex differentiable function on\((b_{1},b_{2})\), and let\(0< q<1\). Then

$$ \psi \biggl(\frac{qb_{1}+b_{2}}{1+q} \biggr)\leq \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x) _{b_{1}} \,d_{q}x \leq \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}. $$
(1.3)

Remark 1.3

It is important to note that the inequality in Theorem 1.2 was first established by Marinković et al. [24, Theorem 5.3].

The aim of this work is to recast inequality (1.3) in Theorem 1.2 via another approach different from that presented in [11]. Specifically, we do this using a Green function. In the process, we establish some identities that are also used to obtain more results in this direction.

We organize this paper as follows. Section 2 contains a brief introduction of the quantum calculus. Our main results are then framed and proved in Sect. 3.

2 Preliminaries

Quantum calculus is known as the calculus without limits. In this section, we present a quick overview of the theory of q-calculus. The interested reader is invited to the book [16] for an in-depth study of this subject. We begin with these basic definitions.

Definition 2.1

([32])

Let \(\psi :[b_{1},b_{2}]\to \mathbb{R}\) be a continuous function, and let \(w\in [b_{1},b_{2}]\). Then the expression

$$ {_{b_{1}}}D_{q}\psi (w)= \frac{\psi (w)-\psi (qw+(1-q)b_{1} )}{(1-q)(w-b_{1})},\quad w \neq b_{1}, {_{b_{1}}}D_{q}\psi (b_{1})= \lim_{w\to b_{1}}{_{b_{1}}}D_{q} \psi (w) $$
(2.1)

is called the q-derivative on \([b_{1},b_{2}]\) of the function at w.

We call ψq-differentiable on \([b_{1},b_{2}]\) if \({_{b_{1}}}D_{q}\psi (w)\) exists for all \(w\in [b_{1},b_{2}]\).

Definition 2.2

([32])

Let \(\psi :[b_{1},b_{2}]\to \mathbb{R}\) be a continuous function. Then the q-integral on \([b_{1},b_{2}]\) is defined as

$$ \int _{b_{1}}^{w}\psi (x) _{b_{1}} \,d_{q}x=(1-q) (w-b_{1})\sum_{k=0}^{ \infty }q^{k} \psi \bigl(q^{k}w+\bigl(1-q^{k}\bigr)b_{1} \bigr) $$
(2.2)

for \(w\in [b_{1},b_{2}]\). Moreover, if \(c\in (b_{1},w)\), then the q-integral on \([b_{1},b_{2}]\) is defined as

$$ \int _{c}^{w}\psi (x) _{b_{1}} \,d_{q}x= \int _{b_{1}}^{w}\psi (x) _{b_{1}} \,d_{q}x- \int _{b_{1}}^{c}\psi (x) _{b_{1}} \,d_{q}x. $$
(2.3)

Remark 2.3

In light of Definitions 2.1 and 2.2, we make the following remarks:

  1. 1.

    By taking \(b_{1}=0\) expression (2.1) boils down to the well-known q-derivative \(D_{q}\psi (w)\) of the function \(\psi (w)\) defined by

    $$ D_{q}\psi (w)=\frac{\psi (w)-\psi (qw)}{(1-q)w}. $$
  2. 2.

    Also, if \(b_{1}=0\), then (2.2) reduces to the classical q-integral of a function \(\psi :[0,\infty )\to \mathbb{R}\) defined by

    $$ \int _{0}^{w}\psi (x) _{0} \,d_{q}x=(1-q)w\sum_{k=0}^{\infty }q^{k} \psi \bigl(q^{k}w\bigr). $$

Some known results in continuous calculus have been extended to the q-calculus framework as follows.

Theorem 2.4

([32])

Let\(\psi :[b_{1},b_{2}]\to \mathbb{R}\)be a continuous function. Then we have

$$ \int _{\delta }^{w}{_{b_{1}}}D_{q}\psi (w) _{b_{1}}\,d_{q}x=\psi (w)- \psi (\delta ) \quad\textit{for } \delta \in (b_{1},w). $$

Theorem 2.5

([13])

Let\(\psi ,\phi :[b_{1},b_{2}]\to \mathbb{R}\)be two continuous functions and suppose\(\psi (x)\leq \phi (x)\)for all\(x\in [b_{1},b_{2}]\). Then

$$ \int _{b_{1}}^{w}\psi (x) _{b_{1}} \,d_{q}x\leq \int _{b_{1}}^{w}\phi (x) _{b_{1}} \,d_{q}x. $$

Theorem 2.6

([32])

Let\(\psi :[b_{1},b_{2}]\to \mathbb{R}\)be a continuous function. Then

$$\begin{aligned} &{}_{b_{1}} D_{q} \int _{b_{1}}^{w}\psi (x) _{b_{1}} \,d_{q}x=\psi (w); \\ &\int _{c}^{w}{_{b_{1}} D_{q}}\psi (x) _{b_{1}} \,d_{q}x=\psi (w)-\psi (c),\quad \textit{for } c\in (b_{1},w). \end{aligned}$$

Theorem 2.7

([32])

Let\(\psi ,\phi :[b_{1},b_{2}]\to \mathbb{R}\)be continuous functions, and let\(\alpha \in \mathbb{R}\). Then, for\(w\in [b_{1},b_{2}]\)and\(c\in (b_{1},w)\), we have

$$\begin{aligned} &\int _{b_{1}}^{w} \bigl[\psi (x)+\phi (x) \bigr] _{b_{1}} \,d_{q}x= \int _{b_{1}}^{w}\psi (x) _{b_{1}} \,d_{q}x+ \int _{b_{1}}^{w}\phi (x) _{b_{1}} \,d_{q}x; \\ &\int _{b_{1}}^{w}\alpha \psi (x) _{b_{1}} \,d_{q}x=\alpha \int _{b_{1}}^{w} \psi (x) _{b_{1}} \,d_{q}x; \\ &\int _{c}^{w}\psi (x){_{b_{1}} D_{q}}\phi (x) _{b_{1}} \,d_{q}x\\ &\quad =\psi (w) \phi (w)- \psi (c)\phi (c)- \int _{c}^{w}\phi \bigl(qx+(1-q)b_{1} \bigr){_{b_{1}} D_{q}} \psi (x) _{b_{1}} \,d_{q}x. \end{aligned}$$

3 Main results

We will prove our fundamental results with the help of the following lemma.

Lemma 3.1

([10, 25])

Let G be the Green function defined on \([b_{1},b_{2}]\times [b_{1},b_{2}]\) by

$$ G(x,u)= \textstyle\begin{cases} b_{1}-u , & b_{1}\leq u\leq x; \\ b_{1}-x, & x\leq u\leq b_{2}. \end{cases} $$

Then any \(\psi \in C^{2}([b_{1},b_{2}])\) can be expressed as

$$ \psi (x)= \psi (b_{1})+(x-b_{1})\psi '(b_{2})+ \int _{b_{1}}^{b_{2}} G(x, \mu )\psi ''( \mu ) \,d\mu. $$
(3.1)

We now state and justify our main results.

Theorem 3.2

Let\(\psi :[b_{1},b_{2}]\to \mathbb{R}\)be a convex twice differentiable function on\((b_{1},b_{2})\). If\(0< q<1\), then

$$ \psi \biggl(\frac{qb_{1}+b_{2}}{q+1} \biggr)\leq \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)_{b_{1}} \,d_{q}x\leq \frac{q\psi (b_{1})+\psi (b_{2})}{q+1}. $$
(3.2)

Proof

If we set \(x=\frac{qb_{1}+b_{2}}{q+1}\) in (3.1), then we get

$$\begin{aligned} \psi \biggl(\frac{qb_{1}+b_{2}}{q+1} \biggr)&=\psi (b_{1})+ \biggl( \frac{qb_{1}+b_{2}}{q+1}-b_{1} \biggr) \psi '(b_{2})+ \int _{b_{1}}^{b_{2}}G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) \psi ''(u)\,du \\ &=\psi (b_{1})+\frac{b_{2}-b_{1}}{q+1} \psi '(b_{2})+ \int _{b_{1}}^{b_{2}}G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) \psi ''(u)\,du. \end{aligned}$$
(3.3)

By computing we obtain that

$$\begin{aligned} &\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)_{b_{1}} \,d_{q}x \\ &\quad=\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}} \biggl\{ \psi (b_{1})+(x-b_{1}) \psi '(b_{2})+ \int _{b_{1}}^{b_{2}}G(x,u) \psi ''(u) \,du \biggr\} ~_{b_{1}}\,d_{q}x \\ &\quad=\psi (b_{1})+\frac{b_{2}-b_{1}}{q+1}\psi '(b_{2})+ \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}} \int _{b_{1}}^{b_{2}}G(x,u) \psi ''(u) \,du _{b_{1}}\,d_{q}x. \end{aligned}$$
(3.4)

Subtracting (3.4) from (3.3), we get:

$$\begin{aligned} &\psi \biggl(\frac{qb_{1}+b_{2}}{q+1} \biggr)-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}} \psi (x)_{b_{1}} \,d_{q}x \\ &\quad=\psi (b_{1})+\frac{b_{2}-b_{1}}{q+1}\psi '(b_{2})+ \int _{b_{1}}^{b_{2}}G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) \psi ''(u)\,du \\ &\qquad{} -\psi (b_{1})-\frac{b_{2}-b_{1}}{q+1}\psi '(b_{2})- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}} \int _{b_{1}}^{b_{2}}G(x,u) \psi ''(u) \,du _{b_{1}}\,d_{q}x \\ &\quad= \int _{b_{1}}^{b_{2}}G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) \psi ''(u)\,du -\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}} \int _{b_{1}}^{b_{2}}G(x,u) \psi ''(u) \,du _{b_{1}}\,d_{q}x \\ &\quad= \int _{b_{1}}^{b_{2}} \biggl\{ G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) - \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}G(x,u)~ _{b_{1}} \,d_{q}x \biggr\} \psi ''(u)\,du \\ &\quad= \int _{b_{1}}^{b_{2}} \biggl[G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du. \end{aligned}$$
(3.5)

Next, we consider the function

$$\begin{aligned} f(u)=G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr)+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} . \end{aligned}$$
(3.6)

For this, the following cases are possible.

Case 1. If \(b_{1}\leq u\leq \frac{qb_{1}+b_{2}}{q+1}\), then

$$\begin{aligned} f(u)&=b_{1}-u+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} . \end{aligned}$$

Therefore

$$\begin{aligned} &f'(u)=-1+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{2(u-b_{1})}{q+1}+(b_{2}-u)-(u-b_{1}) \biggr\} ; \\ &f''(u)=\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{2}{q+1}-2 \biggr\} = \frac{-2q}{(1+q)(b_{2}-b_{1})}< 0. \end{aligned}$$

This implies that \(f'\) is decreasing and \(f'(b_{1})=0\), which shows that \(f'(u)\leq 0\). Thus f is also decreasing, and \(f(b_{1})=0\), that is, \(f(u)\leq 0\) for all \(u\in [b_{1},\frac{qb_{1}+b_{2}}{q+1} ]\).

Case 2. If \(\frac{qb_{1}+b_{2}}{q+1}\leq u\leq b_{2} \), then

$$\begin{aligned} &f(u)=\frac{b_{1}-b_{2}}{q+1}+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} ; \\ &f'(u)=\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{u-b_{1}}{q+1} (1-q )+(b_{2}-u) \biggr\} > 0. \end{aligned}$$

Hence f is increasing and \(f(b_{2})=0\). So, \(f(u)\leq 0\) for all \(u\in [\frac{qb_{1}+b_{2}}{q+1},b_{2} ]\).

Now, using (3.5) and the fact that \(\psi ''(u)\geq 0\) for all \(u\in [b_{1},b_{2}]\), since ψ is convex, we obtain the first inequality:

$$ \psi \biggl(\frac{qb_{1}+b_{2}}{q+1} \biggr)\leq \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)_{b_{1}} \,d_{q}x. $$

For the right-hand side inequality, we recall that

$$\begin{aligned} &\psi (x)=\psi (b_{1})+(x-b_{1})\psi '(b_{2})+ \int _{b_{1}}^{b_{2}}G(x,u) \psi ''(u) \,du; \\ &\psi (b_{2})=\psi (b_{1})+(b_{2}-b_{1}) \psi '(b_{2})+ \int _{b_{1}}^{b_{2}}G(b_{2},u) \psi ''(u)\,du; \\ &\frac{q\psi (b_{1})+\psi (b_{2})}{q+1}=\psi (b_{1})+ \frac{b_{2}-b_{1}}{q+1}\psi '(b_{2})+\frac{1}{q+1} \int _{b_{1}}^{b_{2}} G(b_{2},u)\psi ''(u)\,du. \end{aligned}$$
(3.7)

Subtracting (3.4) from (3.7), we get

$$\begin{aligned} &\frac{q\psi (b_{1})+\psi (b_{2})}{q+1}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}} \psi (x)_{b_{1}} \,d_{q}x \\ &\quad= \int _{b_{1}}^{b_{2}} \biggl[\frac{G(b_{2},u)}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du. \end{aligned}$$
(3.8)

Let

$$ F(u)=\frac{G(b_{2},u)}{q+1}+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} . $$

Then

$$\begin{aligned} &F(u)=\frac{b_{1}-u}{q+1}+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} ; \\ &F'(u)=\frac{-1}{q+1}+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{2(u-b_{1})}{q+1}+(b_{2}-u)-(u-b_{1}) \biggr\} ; \\ &F''(u)=\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{2}{q+1}-2 \biggr\} = \frac{-2q}{(1+q)(b_{2}-b_{1})}< 0. \end{aligned}$$

Here we also observe two cases.

Case 3. If \(b_{1}\leq u\leq \frac{b_{1}+b_{2}}{2}\), then \(F''(u)<0\). Therefore \(F'\) is decreasing, and also \(F' (\frac{b_{1}+b_{2}}{2} )=0\), which shows that \(F'(u)\geq 0\). Moreover, F is increasing, and \(F(b_{1})=0\). Hence \(F(u)\geq 0\) for all \(u\in [b_{1},\frac{b_{1}+b_{2}}{2} ]\).

Case 4. Also, if \(\frac{b_{1}+b_{2}}{2}\leq u\leq b_{2}\), then \(F''(u)<0\). So, \(F'\) is decreasing, and \(F' (\frac{b_{1}+b_{2}}{2} )=0\), which implies that \(F'(u)\leq 0\). Hence F is decreasing, and \(F(b_{2})=0\), and then \(F(u)\geq 0\) for all \(u\in [\frac{b_{1}+b_{2}}{2},b_{2} ]\).

Combining these two cases, we conclude that \(F(u)\geq 0\) for all \(u\in [b_{1},b_{2}]\). Applying (3.8) and the convexity of ψ, we establish the right-hand side of the desired inequality. The proof is complete. □

Next, we prove new quantum Hermite–Hadamard inequalities for the class of monotone and convex functions.

Theorem 3.3

Let\(\psi \in C^{2}([b_{1},b_{2}])\)and\(0< q<1\). Then:

  1. (i).

    If\(|\psi ''|\)is an increasing function, then

    $$\begin{aligned} \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q}x \biggr\vert \leq \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{q(b_{2}-b_{1})^{2}}{6(1+q)} \biggr]. \end{aligned}$$
  2. (ii).

    If\(|\psi ''|\)is a decreasing function, then

    $$\begin{aligned} \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q}x \biggr\vert \leq \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[\frac{q(b_{2}-b_{1})^{2}}{6(1+q)} \biggr]. \end{aligned}$$
  3. (iii).

    If\(|\psi ''|\)is a convex function, then

    $$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q}x \biggr\vert \\ &\quad \leq \max \bigl\{ \bigl\vert \psi ''(b_{1}) \bigr\vert , \bigl\vert \psi ''(b_{2}) \bigr\vert \bigr\} \biggl[\frac{q(b_{2}-b_{1})^{2}}{6(1+q)} \biggr]. \end{aligned}$$

Proof

To prove (i), by (3.8) we get:

$$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q}x \biggr\vert \\ &\quad= \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[\frac{G(b_{2},u)}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du \biggr\vert \\ &\quad= \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[\frac{b_{1}-u}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-b_{1}b_{2}+(b_{1}+b_{2})u-u^{2} \biggr\} \biggr]\psi ''(u)\,du \biggr\vert \\ &\quad\leq \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{b_{1}}{1+q} \int _{b_{1}}^{b_{2}}1\,du- \frac{1}{1+q} \int _{b_{1}}^{b_{2}}u \,du +\frac{1}{b_{2}-b_{1}} \biggl\{ \int _{b_{1}}^{b_{2}}\frac{(u-b_{1})^{2}}{1+q}\,du \\ &\qquad{} -b_{1}b_{2} \int _{b_{1}}^{b_{2}}1\,du+(b_{1}+b_{2}) \int _{b_{1}}^{b_{2}}u \,du- \int _{b_{1}}^{b_{2}}u^{2} \,du \biggr\} \biggr] \\ &\quad= \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{b_{1}(b_{2}-b_{1})}{1+q}- \frac{u^{2}}{2(1+q)}\bigg|_{b_{1}}^{b_{2}} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{3}}{3(1+q)}\bigg|_{b_{1}}^{b_{2}}-b_{1}b_{2}(b_{2}-b_{1}) \\ &\qquad{} +(b_{1}+b_{2})\frac{u^{2}}{2}\bigg|_{b_{1}}^{b_{2}}- \frac{u^{3}}{3}\bigg|_{b_{1}}^{b_{2}} \biggr\} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{b_{1}(b_{2}-b_{1})}{1+q}- \frac{b_{2}^{2}-b_{1}^{2}}{2(1+q)} +\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{3}}{3(1+q)}-b_{1}b_{2}(b_{2}-b_{1}) \\ &\qquad{} +(b_{1}+b_{2})\frac{b_{2}^{2}-b_{1}^{2}}{2}- \frac{b_{2}^{3}-b_{1}^{3}}{3} \biggr\} \biggr] \\ &\quad= \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{q(b_{2}-b_{1})^{2}}{6(1+q)} \biggr], \end{aligned}$$

which proves the inequality in (i).

Part (ii) can be proved in a similar fashion. For part (iii), using (3.8) and the fact that \(|\psi ''|\) is bounded above, on the interval \([b_{1},b_{2}]\), by \(\max \{ |\psi ''(b_{1}) |, |\psi ''(b_{2}) | \}\) as a convex function, we obtain:

$$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q}x \biggr\vert \\ &\quad \leq \max \bigl\{ \bigl\vert \psi ''(b_{1}) \bigr\vert , \bigl\vert \psi ''(b_{2}) \bigr\vert \bigr\} \biggl[\frac{q(b_{2}-b_{1})^{2}}{6(1+q)} \biggr]. \end{aligned}$$

 □

Theorem 3.4

Let\(\psi \in C^{2}([b_{1},b_{2}])\), and let\(\vert \psi '' \vert \)be a concave function. Then, for\(0< q<1\),

$$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad\leq (b_{2}-b_{1})^{2} \biggl[ \frac{1}{2(q+1)} \biggl\vert \psi '' \biggl( \frac{b_{1}+2b_{2}}{3} \biggr) \biggr\vert \\ &\qquad{} +\frac{1}{3(q+1)} \biggl\vert \psi '' \biggl( \frac{b_{1}+3b_{2}}{4} \biggr) \biggr\vert +\frac{1}{6} \biggl\vert \psi '' \biggl(\frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert \biggr]. \end{aligned}$$

Proof

Employing identity (3.8), we have:

$$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad= \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[\frac{G(b_{2},u)}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du \biggr\vert \\ &\quad= \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[\frac{b_{1}-u}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du \biggr\vert . \end{aligned}$$

Suppose \(u=(1-t)b_{1}+tb_{2}\) with \(t\in [0,1]\). Then

$$\begin{aligned} ={}& \biggl\vert \int _{0}^{1} \biggl[\frac{b_{1}-(1-t)b_{1}-tb_{2}}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{((1-t)b_{1}+tb_{2}-b_{1})^{2}}{q+1} \\ &{} + \bigl(b_{2}-(1-t)b_{1}-tb_{2} \bigr) (-tb_{1}+tb_{2} ) \biggr\} \biggr]\psi '' \bigl((1-t)b_{1}+tb_{2}\bigr) (b_{2}-b_{1}) \,dt \biggr\vert \\ ={}& \biggl\vert \int _{0}^{1} \biggl[\frac{-t(b_{2}-b_{1})}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{2}(b_{2}-b_{1})^{2}}{q+1} \\ &{} +t(1-t) (b_{2}-b_{1}) (b_{2}-b_{1} ) \biggr\} \biggr]\psi ''\bigl((1-t)b_{1}+tb_{2} \bigr) (b_{2}-b_{1})\,dt \biggr\vert \\ \leq{}& (b_{2}-b_{1})^{2} [\frac{1}{q+1} \biggl\vert \int _{0}^{1}t\psi '' \bigl((1-t)b_{1}+tb_{2} \bigr)\,dt \biggr\vert \\ &{} +\frac{1}{q+1} \biggl\vert \int _{0}^{1}t^{2}\psi '' \bigl((1-t)b_{1}+tb_{2} \bigr) \,dt \biggr\vert + \biggl\vert \int _{0}^{1}t(1-t)\psi '' \bigl((1-t)b_{1}+tb_{2} \bigr)\,dt \biggr\vert ]. \end{aligned}$$
(3.9)

Now, using the Jensen integral inequality, we get the following estimates:

$$\begin{aligned} & \biggl\vert \int _{0}^{1}t\psi '' \bigl((1-t)b_{1}+tb_{2} \bigr)\,dt \biggr\vert \\ &\quad\leq \int _{0}^{1}t\,dt \biggl\vert \psi '' \biggl( \frac{\int _{0}^{1}t ((1-t)b_{1}+tb_{2} )\,dt}{\int _{0}^{1}t\,dt} \biggr) \biggr\vert \\ &\quad=\frac{1}{2} \biggl\vert \psi '' \biggl( \frac{b_{1}\int _{o}^{1}(t-t^{2})\,dt+b_{2}\int _{0}^{1}t^{2}\,dt}{\frac{1}{2}} \biggr) \biggr\vert \\ &\quad =\frac{1}{2} \biggl\vert \psi '' \biggl( \frac{b_{1}+2b_{2}}{3} \biggr) \biggr\vert , \end{aligned}$$
(3.10)
$$\begin{aligned} & \biggl\vert \int _{0}^{1}t^{2}\psi '' \bigl((1-t)b_{1}+tb_{2} \bigr) \,dt \biggr\vert \\ &\quad\leq \int _{0}^{1}t^{2}\,dt \biggl\vert \psi '' \biggl( \frac{\int _{0}^{1}t^{2} ((1-t)b_{1}+tb_{2} )\,dt}{\int _{0}^{1}t^{2}\,dt} \biggr) \biggr\vert \\ &\quad=\frac{1}{3} \biggl\vert \psi '' \biggl( \frac{b_{1}\int _{0}^{1}(t^{2}-t^{3})\,dt+b_{2}\int _{0}^{1}t^{3}\,dt}{\frac{1}{3}} \biggr) \biggr\vert \\ &\quad =\frac{1}{3} \biggl\vert \psi '' \biggl( \frac{b_{1}+3b_{2}}{4} \biggr) \biggr\vert , \end{aligned}$$
(3.11)

and

$$\begin{aligned} & \biggl\vert \int _{0}^{1}\bigl(t-t^{2}\bigr)\psi '' \bigl((1-t)b_{1}+tb_{2} \bigr) \,dt \biggr\vert \\ &\quad \leq \int _{0}^{1}\bigl(t-t^{2}\bigr)\,dt \biggl\vert \psi '' \biggl( \frac{\int _{0}^{1}(t-t^{2}) ((1-t)b_{1}+tb_{2} )\,dt}{\int _{0}^{1}(t-t^{2})\,dt} \biggr) \biggr\vert \\ &\quad =\frac{1}{6} \biggl\vert \psi '' \biggl( \frac{b_{1}\int _{0}^{1}(t-t^{2})(1-t)\,dt+b_{2}\int _{0}^{1}(t^{2}-t^{3})\,dt}{\frac{1}{6}} \biggr) \biggr\vert \\ &\quad =\frac{1}{6} \biggl\vert \psi '' \biggl( \frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert . \end{aligned}$$
(3.12)

Putting (3.10), (3.11), and (3.12) into (3.9), we get

$$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad\leq (b_{2}-b_{1})^{2} \biggl[ \frac{1}{2(q+1)} \biggl\vert \psi '' \biggl( \frac{b_{1}+2b_{2}}{3} \biggr) \biggr\vert \\ &\qquad{} +\frac{1}{3(q+1)} \biggl\vert \psi '' \biggl( \frac{b_{1}+3b_{2}}{4} \biggr) \biggr\vert +\frac{1}{6} \biggl\vert \psi '' \biggl(\frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert \biggr]. \end{aligned}$$

 □

Theorem 3.5

Suppose\(\psi \in C^{2}([b_{1},b_{2}])\)and\(q\in (0,1)\). Then:

  1. (i).

    If\(|\psi ''|\)is an increasing function, then

    $$\begin{aligned} & \biggl\vert \psi '' \biggl(\frac{q b_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad \leq \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr] \\ &\qquad{}-\frac{ \vert \psi ''(b_{2}) \vert }{6(b_{2}-b_{1})} \bigl[2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2} \bigr]. \end{aligned}$$
  2. (ii).

    If\(|\psi ''|\)is a decreasing function, then

    $$\begin{aligned} & \biggl\vert \psi '' \biggl(\frac{q b_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad \leq \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr] \\ &\qquad{}-\frac{ \vert \psi ''(b_{2}) \vert }{6(b_{2}-b_{1})} \bigl[2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2} \bigr]. \end{aligned}$$
  3. (iii).

    If\(|\psi ''|\)is a convex function, then

    $$\begin{aligned} &\biggl\vert \psi '' \biggl(\frac{q b_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad \leq \max \bigl\{ \bigl\vert \psi ''(b_{2}) \bigr\vert , \bigl\vert \psi ''(b_{1}) \bigr\vert \bigr\} \biggl[\frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1} \frac{qb_{1}+b_{2}}{1+q}+\frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}} \\ &\qquad{}-\frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr]-\min \bigl\{ \bigl\vert \psi ''(b_{1}) \bigr\vert , \bigl\vert \psi ''(b_{2}) \bigr\vert \bigr\} \biggl[ \frac{2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2}}{6(b_{2}-b_{1})} \biggr]. \end{aligned}$$

Proof

To prove item (i), we use (3.5) and the fact that \(|\psi ''|\) is an increasing function to obtain:

$$\begin{aligned} & \biggl\vert \psi '' \biggl(\frac{q b_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad = \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) +\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du \biggr\vert \\ &\quad\leq \bigl\vert \psi ''(b_{2}) \bigr\vert \int _{b_{1}}^{b_{2}} \biggl\vert G \biggl( \frac{qb_{1}+b_{2}}{q+1},u \biggr) +\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr\vert \,du \\ &\quad \leq \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \int _{b_{1}}^{ \frac{qb_{1}+b_{2}}{1+q}} \biggl\vert b_{1}-u + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-u^{2}+(b_{1}+b_{2})u-b_{1}b_{2} \biggr\} \biggr\vert \,du \\ &\qquad{}+ \int _{\frac{qb_{1}+b_{2}}{1+q}}^{b_{2}} \biggl\vert \frac{b_{1}-b_{2}}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-u^{2}+(b_{1}+b_{2})u-b_{1}b_{2} \biggr\} \biggr\vert \,du \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \int _{b_{1}}^{ \frac{qb_{1}+b_{2}}{1+q}} \biggl\{ u-b_{1} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-u^{2}+(b_{1}+b_{2})u-b_{1}b_{2} \biggr\} \biggr\} \,du \\ &\qquad{}+ \int _{\frac{qb_{1}+b_{2}}{1+q}}^{b_{2}} \biggl\{ \frac{b_{2}-b_{1}}{q+1} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-u^{2}+(b_{1}+b_{2})u-b_{1}b_{2} \biggr\} \biggr\} \,du \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \biggl(\frac{u^{2}}{2}-b_{1}u \biggr) \bigg|_{b_{1}}^{\frac{qb_{1}+b_{2}}{1+q}}- \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{3}}{3(q+1)}\bigg|_{b_{1}}^{ \frac{qb_{1}+b_{2}}{1+q}} - \frac{u^{3}}{3}\bigg|_{b_{1}}^{ \frac{qb_{1}+b_{2}}{1+q}} \\ &\qquad{}+(b_{1}+b_{2})\frac{u^{2}}{2}\bigg|_{b_{1}}^{ \frac{qb_{1}+b_{2}}{1+q}}-b_{2}b_{1}u\bigg|_{b_{1}}^{ \frac{qb_{1}+b_{2}}{1+q}} \biggr\} +\frac{b_{2}-b_{1}}{1+q} \biggl(b_{2}- \frac{qb_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \\ &\qquad{}\times \biggl\{ \frac{(u-b_{1})^{3}}{3(q+1)}|^{b_{2}}_{ \frac{qb_{1}+b_{2}}{1+q}} - \frac{u^{3}}{3}|^{b_{2}}_{ \frac{qb_{1}+b_{2}}{1+q}}+(b_{1}+b_{2}) \frac{u^{2}}{2}|^{b_{2}}_{ \frac{qb_{1}+b_{2}}{1+q}} -b_{2}b_{1}u|^{b_{2}}_{ \frac{qb_{1}+b_{2}}{1+q}} \biggr\} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \biggl( \frac{(\frac{qb_{1}+b_{2}}{1+q})^{2}}{2}- b_{1}\biggl( \frac{qb_{1}+b_{2}}{1+q} \biggr)-\frac{b_{1}^{2}}{2}+b_{1}^{2} \biggr)- \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{ (\frac{qb_{1}+b_{2}}{1+q}-b_{1} )^{3}}{3(q+1)} \\ &\qquad{}-\frac{ (qb_{1}+b_{2} )^{3}}{3(1+q)^{3}} +\frac{b_{1}^{3}}{3}+(b_{1}+b_{2}) \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}-(b_{1}+b_{2}) \frac{b_{1}^{2}}{2}-b_{2}b_{1} \biggl(\frac{qb_{1}+b_{2}}{1+q}-b_{1}\biggr) \biggr\} \\ &\qquad{}+\frac{b_{2}-b_{1}}{1+q} \biggl(b_{2}-\frac{qb_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{3}}{3(q+1)} - \frac{ (\frac{qb_{1}+b_{2}}{1+q}-b_{1} )^{3}}{3(q+1)}- \frac{b_{2}^{3}}{3} \\ &\qquad{}+\frac{(qb_{1}+b_{2})^{3}}{3(1+q)^{3}}+(b_{1}+b_{2}) \frac{b_{2}^{2}}{2}-(b_{1}+b_{2}) \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}-b_{2}b_{1} \biggl(b_{2}- \frac{qb_{1}+b_{2}}{1+q} \biggr) \biggr\} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \biggl( \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\biggl(\frac{qb_{1}+b_{2}}{1+q} \biggr)+ \frac{b_{1}^{2}}{2} \biggr)-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{ (b_{2}-b_{1} )^{3}}{3(q+1)^{4}} \\ &\qquad{}-\frac{ (qb_{1}+b_{2} )^{3}}{3(1+q)^{3}} +\frac{b_{1}^{3}}{3}+(b_{1}+b_{2}) \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}-(b_{1}+b_{2}) \frac{b_{1}^{2}}{2} -b_{2}b_{1}\biggl(\frac{b_{2}-b_{1}}{1+q}\biggr) \biggr\} \\ &\qquad{}+\frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{3}}{3(q+1)} - \frac{ (b_{2}-b_{1} )^{3}}{3(q+1)^{4}}-\frac{b_{2}^{3}}{3}+ \frac{(qb_{1}+b_{2})^{3}}{3(1+q)^{3}} \\ &\qquad{}+(b_{1}+b_{2})\frac{b_{2}^{2}}{2}-(b_{1}+b_{2}) \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}-b_{2}b_{1}\frac{q(b_{2}-b_{1})}{1+q} \biggr\} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\biggl(\frac{qb_{1}+b_{2}}{1+q}\biggr)+ \frac{b_{1}^{2}}{2}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{b_{1}^{3}}{3}- \frac{(b_{1}+b_{2})b_{1}^{2}}{2} \\ &\qquad{}-b_{2}b_{1}\biggl(\frac{b_{2}-b_{1}}{1+q}\biggr) \biggr\} + \frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{1}{b_{2}-b_{1}} \biggl\{ - \frac{b_{2}^{3}}{3}+ \frac{(b_{1}+b_{2})b_{2}^{2}}{2}+ \frac{(b_{2}-b_{1})^{3}}{3(q+1)} \\ &\qquad{}-b_{2}b_{1}\frac{q(b_{2}-b_{1})}{1+q} \biggr\} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}-b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{b_{1}^{2}}{2} +\frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}- \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{b_{1}^{3}}{3}- \frac{b_{2}^{3}}{3} \\ &\qquad{}-\frac{(b_{1}+b_{2})b_{1}^{2}}{2}-b_{2}b_{1}\biggl( \frac{b_{2}-b_{1}}{1+q} \biggr)+\frac{(b_{1}+b_{2})b_{2}^{2}}{2}+ \frac{(b_{2}-b_{1})^{3}}{3(q+1)}-b_{2}b_{1} \frac{q(b_{2}-b_{1})}{1+q} \biggr\} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{b_{1}^{2}}{2}+\frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}- \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{b_{1}^{3}}{3} \\ &\qquad{}-\frac{(b_{1}+b_{2})b_{1}^{2}}{2}-\frac{b_{2}^{3}}{3}+ \frac{(b_{1}+b_{2})b_{2}^{2}}{2}+\frac{(b_{2}-b_{1})^{3}}{3(q+1)} -b_{2}b_{1}(b_{2}-b_{1}) \biggr\} \biggr] \\ &\quad=\frac{ \vert \psi ''(b_{2}) \vert }{b_{2}-b_{1}} \biggl[ \frac{(qb_{1}+b_{2})^{2}(b_{2}-b_{1})}{2(1+q)^{2}}- b_{1} \frac{(qb_{1}+b_{2})(b_{2}-b_{1})}{1+q}+ \frac{b_{1}^{2}(b_{2}-b_{1})}{2}-\frac{b_{1}^{3}}{3} \\ &\qquad{}+\frac{q(b_{2}-b_{1})^{3}}{(1+q)^{2}}+ \frac{(b_{1}+b_{2})b_{1}^{2}}{2}+\frac{b_{2}^{3}}{3}- \frac{(b_{1}+b_{2})b_{2}^{2}}{2} -\frac{(b_{2}-b_{1})^{3}}{3(q+1)}+b_{2}^{2}b_{1}-b_{2}b_{1}^{2} \biggr] \\ &\quad= \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr] \\ &\qquad{}+\frac{ \vert \psi ''(b_{2}) \vert }{b_{2}-b_{1}} \\ &\qquad{}\times \biggl[ \frac{3b_{1}^{2}b_{2}-3b_{1}^{3}-2b_{1}^{3}+3b_{1}^{3}+3b_{1}^{2}b_{2}+2b_{2}^{3} -3b_{1}b_{2}^{2}-3b_{2}^{3}+6b_{2}^{2}b_{1}-6b_{2}b_{1}^{2}}{6} \biggr] \\ &\quad= \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr] \\ &\qquad{}-\frac{ \vert \psi ''(b_{2}) \vert }{6(b_{2}-b_{1})} \bigl[2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2} \bigr]. \end{aligned}$$

Part (ii) can be proved in a similar way. For part (iii), using (3.5) and the fact that \(|\psi ''|\) is bounded above, on the interval \([b_{1},b_{2}]\), by \(\max \{ |\psi ''(b_{1}) |, |\psi ''(b_{2}) | \}\) as a convex function, we obtain:

$$\begin{aligned} & \biggl\vert \psi '' \biggl(\frac{q b_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad \leq \max \biggl\{ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr] \\ &\qquad{}-\frac{ \vert \psi ''(b_{2}) \vert }{6(b_{2}-b_{1})} \bigl[2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2} \bigr], \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}-b_{1}\frac{qb_{1}+b_{2}}{1+q} \\ &\qquad{}+\frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}- \frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr]- \frac{ \vert \psi ''(b_{1}) \vert }{6(b_{2}-b_{1})} \bigl[2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2} \bigr] \biggr\} \\ &\quad=\max \biggl\{ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+ \frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}-\frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr], \\ & \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[\frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1}\frac{qb_{1}+b_{2}}{1+q}+\frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}- \frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr] \biggr\} \\ &\qquad{}+\max \biggl\{ - \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2}}{6(b_{2}-b_{1})} \biggr], \\ &\qquad{}- \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2}}{6(b_{2}-b_{1})} \biggr] \biggr\} \\ &\quad =\max \bigl\{ \bigl\vert \psi ''(b_{2}) \bigr\vert , \bigl\vert \psi ''(b_{1}) \bigr\vert \bigr\} \biggl[\frac{(qb_{1}+b_{2})^{2}}{2(1+q)^{2}}- b_{1} \frac{qb_{1}+b_{2}}{1+q}+\frac{q(b_{2}-b_{1})^{2}}{(1+q)^{2}}- \frac{(b_{2}-b_{1})^{2}}{3(q+1)} \biggr] \\ &\qquad{}-\min \bigl\{ \bigl\vert \psi ''(b_{1}) \bigr\vert , \bigl\vert \psi ''(b_{2}) \bigr\vert \bigr\} \biggl[ \frac{2b_{1}^{3}+b_{2}^{3}-3b_{1}b_{2}^{2}}{6(b_{2}-b_{1})} \biggr], \end{aligned}$$

which gives the inequality in item (iii). □

Theorem 3.6

Let\(\psi \in C^{2}([b_{1},b_{2}])\), and let\(|\psi '' |\)be a convex function. Then for\(q\in (0,1)\), the following inequality holds:

$$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q}x \biggr\vert \\ &\quad \leq \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{4b_{1}b_{2}+q(b_{1}+b_{2})^{2}}{12(q+1)} \biggr] + \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{q(b_{2}-b_{1})^{2}-2(q+1)b_{1}b_{2}}{12(q+1)} \biggr]. \end{aligned}$$

Proof

Employing (3.8), we get:

$$\begin{aligned} & \biggl\vert \frac{q\psi (b_{1})+\psi (b_{2})}{1+q}-\frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q}x \biggr\vert \\ &\quad = \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[\frac{G(b_{2},u)}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du \biggr\vert \\ &\quad = \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[\frac{b_{1}-u}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du \biggr\vert . \\ &\quad \leq \int _{b_{1}}^{b_{2}} \biggl[\frac{b_{1}-u}{q+1} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-u^{2}+(b_{1}+b_{2})u-b_{1}b_{2} \biggr\} \biggr] \bigl\vert \psi ''(u) \bigr\vert \,du. \end{aligned}$$

Putting \(u=(1-t)b_{1}+tb_{2}\) with \(t\in [0,1]\), we get

$$\begin{aligned} ={}& \int _{0}^{1} \biggl[\frac{b_{1}-(1-t)b_{1}-tb_{2}}{q+1}+ \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{((1-t)b_{1}+tb_{2}-b_{1})^{2}}{q+1}-\bigl((1-t)b_{1}+tb_{2} \bigr)^{2} \\ &{}+(b_{1}+b_{2}) \bigl((1-t)b_{1}+tb_{2} \bigr)-b_{1}b_{2} \biggr\} \biggr] \bigl\vert \psi ''\bigl((1-t)b_{1}+tb_{2}\bigr) \bigr\vert (b_{2}-b_{1})\,dt \\ \leq{} & \int _{0}^{1} \biggl[\frac{t(b_{1}-b_{2})}{q+1}+ \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{2}(b_{2}-b_{1})^{2}}{q+1} -(1-t)^{2}b_{1}^{2}-t^{2}b_{2}^{2}-2(1-t)tb_{1}b_{2} \\ &{}+(b_{1}+b_{2})b_{1}(1-t)+(b_{1}+b_{2})b_{2}t-b_{1}b_{2} \biggr\} \biggr] \bigl[(1-t) \bigl\vert \psi ''(b_{1}) \bigr\vert +t \bigl\vert \psi ''(b_{2}) \bigr\vert \bigr](b_{2}-b_{1})\,dt \\ ={}&(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \int _{0}^{1} \biggl[ \frac{t(1-t)(b_{1}-b_{2})}{q+1}+ \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{2}(1-t)(b_{2}-b_{1})^{2}}{q+1} \\ &{}-(1-t)^{3}b_{1}^{2}-t^{2}(1-t)b_{2}^{2}-2(1-t)^{2}tb_{1}b_{2}+(b_{1}+b_{2})b_{1}(1-t)^{2} \\ &{}+(b_{1}+b_{2})b_{2}t(1-t)-b_{1}b_{2}(1-t) \biggr\} \biggr]\,dt+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \int _{0}^{1} \biggl[ \frac{t^{2}(b_{1}-b_{2})}{q+1} \\ &{}+\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{3}(b_{2}-b_{1})^{2}}{q+1}-t(1-t)^{2}b_{1}^{2}+t^{3}b_{2}^{2}+2(1-t)t^{2}b_{1}b_{2} \\ &{}+(b_{1}+b_{2})b_{1}t(1-t)+(b_{1}+b_{2})b_{2}t^{2}-b_{1}b_{2}t \biggr\} \biggr]\,dt \\ ={}& \bigl\vert \psi ''(b_{1}) \bigr\vert \int _{0}^{1} \biggl[ \frac{(t^{2}-t)(b_{2}-b_{1})^{2}}{q+1}+ \frac{(t^{2}-t^{3})(b_{2}-b_{1})^{2}}{q+1}-(1-t)^{3}b_{1}^{2}- \bigl(t^{2}-t^{3}\bigr)b_{2}^{2} \\ &{}-2(1-t)^{2}tb_{1}b_{2}+(b_{1}+b_{2})b_{1}(1-t)^{2}+(b_{1}+b_{2})b_{2} \bigl(t-t^{2}\bigr)-b_{1}b_{2}(1-t) \biggr]\,dt \\ &{}+ \bigl\vert \psi ''(b_{2}) \bigr\vert \int _{0}^{1} \biggl[ \frac{-t^{2}(b_{2}-b_{1})^{2}}{q+1}+ \frac{t^{3}(b_{2}-b_{1})^{2}}{q+1} -t(1-t)^{2}b_{1}^{2}-t^{3}b_{2}^{2}-2 \bigl(t^{2}-t^{3}\bigr)b_{1}b_{2} \\ &{}+(b_{1}+b_{2})b_{1}\bigl(t-t^{2} \bigr)+(b_{1}+b_{2})b_{2}t^{2}-b_{1}b_{2}t \} \biggr]\,dt \\ ={}& \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{ (\frac{t^{3}}{3}-\frac{t^{2}}{2} )(b_{2}-b_{1})^{2}}{q+1} +\frac{(\frac{t^{3}}{3}-\frac{t^{4}}{4})(b_{2}-b_{1})^{2}}{q+1}+ \frac{(1-t)^{4}}{4}b_{1}^{2}- \biggl(\frac{t^{3}}{3} -\frac{t^{4}}{4} \biggr)b_{2}^{2} \\ &{}+b_{1}b_{2} \biggl(t^{2}+\frac{t^{4}}{2}- \frac{4t^{3}}{3} \biggr)-(b_{1}+b_{2})b_{1} \frac{(1-t)^{3}}{3}+(b_{1}+b_{2})b_{2} \biggl( \frac{t^{2}}{2}- \frac{t^{3}}{3} \biggr) \\ &{}-b_{1}b_{2} \biggl(t-\frac{t^{2}}{2} \biggr) \biggr]_{0}^{1}+ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{-t^{3}(b_{2}-b_{1})^{2}}{3(q+1)}+\frac{t^{4}(b_{2}-b_{1})^{2}}{4(q+1)}- \biggl( \frac{t^{2}}{2}+ \frac{t^{4}}{4}-\frac{2t^{3}}{3} \biggr)b_{1}^{2} \\ &{}-\frac{t^{4}}{4}b_{2}^{2}-2 \biggl(\frac{t^{3}}{3}- \frac{t^{4}}{4} \biggr)b_{1}b_{2}+(b_{1}+b_{2})b_{1} \biggl(\frac{t^{2}}{2} - \frac{t^{3}}{3} \biggr)+(b_{1}+b_{2})b_{2} \frac{t^{3}}{3}-b_{1}b_{2} \frac{t^{2}}{2} \biggr]_{0}^{1} \\ ={}& \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[\frac{-(b_{2}-b_{1})^{2}}{6(q+1)} + \frac{(b_{2}-b_{1})^{2}}{12(q+1)}-\frac{b_{1}^{2}}{4}- \frac{b_{2}^{2}}{12}+b_{1}b_{2} \biggl(1+\frac{1}{2}- \frac{4}{3} \biggr) \\ &{}+\frac{(b_{1}+b_{2})b_{1}}{3}+\frac{(b_{1}+b_{2})b_{2}}{6}- \frac{b_{1}b_{2}}{2} \biggr] + \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[- \frac{(b_{2}-b_{1})^{2}}{3(q+1)}+\frac{(b_{2}-b_{1})^{2}}{4(q+1)} \\ &{}- \biggl(\frac{1}{2}+\frac{1}{4}-\frac{2}{3} \biggr)b_{1}^{2}- \frac{b_{2}^{2}}{4}-\frac{b_{1}b_{2}}{3} + \frac{(b_{1}+b_{2})b_{1}}{6}+\frac{(b_{1}+b_{2})b_{2}}{3}- \frac{b_{1}b_{2}}{2} \biggr] \\ ={}& \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{4b_{1}b_{2}+q(b_{1}+b_{2})^{2}}{12(q+1)} \biggr] + \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{q(b_{2}-b_{1})^{2}-2(q+1)b_{1}b_{2}}{12(q+1)} \biggr]. \end{aligned}$$

 □

We now present our last result.

Theorem 3.7

Let\(\psi \in ([b_{1},b_{2}] )\)be such that\(|\psi '' |\)is a convex function. Then for any\(q\in (0,1)\), the following inequality holds:

$$\begin{aligned} & \biggl\vert \psi '' \biggl(\frac{q b_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad\leq \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{(b_{2}-b_{1})^{2} (5q^{2}+4q+1 )-4(b_{1}+b_{2})b_{1}}{12(1+q)^{3}} +\frac{3b_{1}^{2}-b_{2}^{2}+6b_{1}b_{2}}{12} \biggr] \\ &\qquad {}+ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(b_{2}-b_{1})^{2} (3(1+q)^{2}+10 )}{12(1+q)^{3}}-\frac{(b_{2}-b_{1})^{2}}{12} \biggr]. \end{aligned}$$

Proof

Using (3.5), we get:

$$\begin{aligned} & \biggl\vert \psi '' \biggl(\frac{q b_{1}+b_{2}}{1+q} \biggr)- \frac{1}{b_{2}-b_{1}} \int _{b_{1}}^{b_{2}}\psi (x)~_{b_{1}} \,d_{q} x \biggr\vert \\ &\quad= \biggl\vert \int _{b_{1}}^{b_{2}} \biggl[G \biggl(\frac{qb_{1}+b_{2}}{q+1},u \biggr) +\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr]\psi ''(u)\,du \biggr\vert \\ &\quad \leq \int _{b_{1}}^{\frac{qb_{1}+b_{2}}{1+q}} \biggl\vert b_{1}-u + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr\vert \bigl\vert \psi ''(u) \bigr\vert \,du \\ &\qquad{}+ \int _{\frac{qb_{1}+b_{2}}{1+q}}^{b_{2}} \biggl\vert \frac{b_{1}-b_{2}}{1+q} + \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}+(b_{2}-u) (u-b_{1}) \biggr\} \biggr\vert \bigl\vert \psi ''(u) \bigr\vert \,du \\ &\quad = \int _{b_{1}}^{\frac{qb_{1}+b_{2}}{1+q}} \biggl[u-b_{1} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-u^{2}+(b_{1}+b_{2})u-b_{1}b_{2} \biggr\} \biggr] \bigl\vert \psi ''(u) \bigr\vert \,du \\ &\qquad{}+ \int _{\frac{qb_{1}+b_{2}}{1+q}}^{b_{2}} \biggl[ \frac{b_{2}-b_{1}}{1+q} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(u-b_{1})^{2}}{q+1}-u^{2}+(b_{1}+b_{2})u-b_{1}b_{2} \biggr\} \biggr] \bigl\vert \psi ''(u) \bigr\vert \,du \\ &\quad =(b_{2}-b_{1}) \int _{0}^{\frac{1}{1+q}} \biggl[(1-t)b_{1}+tb_{2}-b_{1} -\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{((1-t)b_{1}+tb_{2}-b_{1})^{2}}{q+1} \\ &\qquad{}-\bigl((1-t)b_{1}+tb_{2}\bigr)^{2}+(b_{1}+b_{2}) \bigl((1-t)b_{1}+tb_{2}\bigr)-b_{1}b_{2} \biggr\} \biggr] \bigl\vert \psi '' \bigl((1-t)b_{1}+tb_{2}\bigr) \bigr\vert \,dt \\ &\qquad{}+ \int _{\frac{1}{1+q}}^{1} \biggl[\frac{b_{2}-b_{1}}{1+q} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{((1-t)b_{1}+tb_{2}-b_{1})^{2}}{q+1}-\bigl((1-t)b_{1}+tb_{2} \bigr)^{2} \\ &\qquad{}+(b_{1}+b_{2}) \bigl((1-t)b_{1}+tb_{2} \bigr)-b_{1}b_{2} \biggr\} \biggr] \bigl\vert \psi ''\bigl((1-t)b_{1}+tb_{2}\bigr) \bigr\vert (b_{2}-b_{1})\,dt \\ &\quad \leq (b_{2}-b_{1}) \int _{0}^{\frac{1}{1+q}} \biggl[t(b_{2}-b_{1}) - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{2}(b_{2}-b_{1})^{2}}{q+1}-(1-t)^{2}b_{1}^{2}-t^{2}b_{2}^{2} \\ &\qquad{}-2t(1-t)b_{1}b_{2}+(b_{1}+b_{2})b_{1}(1-t)+t(b_{1}+b_{2})b_{2}-b_{1}b_{2} \biggr\} \biggr] \bigl[(1-t) \bigl\vert \psi ''(b_{1}) \bigr\vert \\ &\qquad{}+t \bigl\vert \psi ''(b_{2}) \bigr\vert \bigr]\,dt+ \int _{\frac{1}{1+q}}^{1} \biggl[ \frac{b_{2}-b_{1}}{1+q} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{2}(b_{2}-b_{1})^{2}}{q+1}-(1-t)^{2}b_{1}^{2} \\ &\qquad{}-t^{2}b_{2}^{2}-2t(1-t)b_{1}b_{2}+(b_{1}+b_{2})b_{1}(1-t)+t(b_{1}+b_{2})b_{2}-b_{1}b_{2} \biggr\} \biggr] \\ &\qquad{}\times \bigl[(1-t) \bigl\vert \psi ''(b_{1}) \bigr\vert +t \bigl\vert \psi ''(b_{2}) \bigr\vert \bigr](b_{2}-b_{1})\,dt \\ &\quad =(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \int _{0}^{\frac{1}{1+q}} \biggl[t(1-t) (b_{2}-b_{1}) -\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{2}(1-t)(b_{2}-b_{1})^{2}}{q+1} \\ &\qquad{}-(1-t)^{3}b_{1}^{2}-t^{2}(1-t)b_{2}^{2}-2t(1-t)^{2}b_{1}b_{2}+(b_{1}+b_{2})b_{1}(1-t)^{2} \\ &\qquad{}+t(1-t) (b_{1}+b_{2})b_{2}-b_{1}b_{2}(1-t) \biggr\} \biggr]+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \int _{0}^{\frac{1}{1+q}} \biggl[t^{2}(b_{2}-b_{1}) \\ &\qquad{}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{3}(b_{2} -b_{1})^{2}}{q+1}-t(1-t)^{2}b_{1}^{2}-t^{3}b_{2}^{2}-2t^{2}(1-t) b_{1}b_{2}\\ &\qquad{}+(b_{1}+b_{2})b_{1}t(1-t) \end{aligned}$$
$$\begin{aligned} &\qquad{}+t^{2}(b_{1}+b_{2})b_{2}-tb_{1}b_{2} \biggr\} \biggr]\,dt+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \int _{\frac{1}{1+q}}^{1} \biggl[ \frac{b_{2}-b_{1}}{1+q}(1-t) \\ &\qquad{}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{2}(1-t)(b_{2}-b_{1})^{2}}{q+1}-(1-t)^{3}b_{1}^{2}-t^{2}(1-t)b_{2}^{2}-2t(1-t)^{2}b_{1}b_{2} \\ &\qquad{}+(b_{1}+b_{2})b_{1}(1-t)^{2}+t(1-t) (b_{1}+b_{2})b_{2}-b_{1}b_{2}(1-t) \biggr\} \biggr]\,dt+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \\ &\qquad{}\times \int _{\frac{1}{1+q}}^{1} \biggl[\frac{b_{2}-b_{1}}{1+q}t - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{3}(b_{2}-b_{1})^{2}}{q+1}-t(1-t)^{2}b_{1}^{2}-t^{3}b_{2}^{2}-2t^{2}(1-t)b_{1}b_{2} \\ &\qquad{}+(b_{1}+b_{2})b_{1}t(1-t)+t^{2}(b_{1}+b_{2})b_{2}-tb_{1}b_{2} \biggr\} \biggr]\,dt \\ &\quad =(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \int _{0}^{\frac{1}{1+q}} \biggl[\bigl(t-t^{2}\bigr) (b_{2}-b_{1}) -\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(t^{2}-t^{3})(b_{2}-b_{1})^{2}}{q+1} \\ &\qquad{}-(1-t)^{3}b_{1}^{2}-\bigl(t^{2}-t^{3} \bigr)b_{2}^{2}-2\bigl(t+t^{3}-2t^{2} \bigr)b_{1}b_{2}+(b_{1}+b_{2})b_{1}(1-t)^{2} \\ &\qquad{}+\bigl(t-t^{2}\bigr) (b_{1}+b_{2})b_{2}-b_{1}b_{2}(1-t) \biggr\} \biggr]+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \int _{0}^{\frac{1}{1+q}} \biggl[t^{2}(b_{2}-b_{1}) \\ &\qquad{}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{3}(b_{2}-b_{1})^{2}}{q+1}-\bigl(t+t^{3}-2t^{2} \bigr)b_{1}^{2}-t^{3}b_{2}^{2}-2 \bigl(t^{2}-t^{3}\bigr)b_{1}b_{2} \\ &\qquad{}+(b_{1} +b_{2})b_{1}\bigl(t-t^{2}\bigr) \\ &\qquad{}+t^{2}(b_{1}+b_{2})b_{2}-tb_{1}b_{2} \biggr\} \biggr]\,dt+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \int _{\frac{1}{1+q}}^{1} \biggl[ \frac{b_{2}-b_{1}}{1+q}(1-t) \\ &\qquad{}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(t^{2}-t^{3})(b_{2}-b_{1})^{2}}{q+1}-(1-t)^{3}b_{1}^{2}- \bigl(t^{2}-t^{3}\bigr)b_{2}^{2}-2 \bigl(t+t^{3}-t^{2}\bigr)b_{1}b_{2} \\ &\qquad{}+(b_{1}+b_{2})b_{1}(1-t)^{2}+ \bigl(t-t^{2}\bigr) (b_{1}+b_{2})b_{2}-b_{1}b_{2}(1-t) \biggr\} \biggr]\,dt+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \\ &\qquad{}\times \int _{\frac{1}{1+q}}^{1} \biggl[\frac{b_{2}-b_{1}}{1+q}t - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{3}(b_{2}-b_{1})^{2}}{q+1}-\bigl(t+t^{3}-2t^{2} \bigr)b_{1}^{2}-t^{3}b_{2}^{2}-2 \bigl(t^{2}-t^{3}\bigr)b_{1}b_{2} \\ &\qquad{}+(b_{1}+b_{2})b_{1}\bigl(t-t^{2} \bigr)+t^{2}(b_{1}+b_{2})b_{2}-tb_{1}b_{2} \biggr\} \biggr]\,dt \\ &\quad =(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \biggl(\frac{t^{2}}{2}- \frac{t^{3}}{3} \biggr) (b_{2}-b_{1}) -\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(\frac{t^{3}}{3}-\frac{t^{4}}{4})(b_{2}-b_{1})^{2}}{q+1} \\ &\qquad{}+\frac{(1-t)^{4}}{4}b_{1}^{2}- \biggl(\frac{t^{3}}{3}- \frac{t^{4}}{4} \biggr)b_{2}^{2}-2 \biggl( \frac{t^{2}}{2}+\frac{t^{4}}{4} -2 \frac{t^{3}}{3} \biggr)b_{1}b_{2}-(b_{1}+b_{2})b_{1} \frac{(1-t)^{3}}{3} \\ &\qquad{}+ \biggl(\frac{t^{2}}{2}-\frac{t^{3}}{3} \biggr) (b_{1}+b_{2})b_{2}-b_{1}b_{2} \biggl(t-\frac{t^{2}}{2} \biggr) \biggr\} \biggr]_{0}^{\frac{1}{1+q}}+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \\ &\qquad{}\times \biggl[\frac{t^{3}}{3}(b_{2}-b_{1})- \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{4}(b_{2}-b_{1})^{2}}{4(q+1)}- \biggl(\frac{t^{2}}{2} + \frac{t^{4}}{4}-\frac{2t^{3}}{3} \biggr)b_{1}^{2}-2 \biggl(\frac{t^{3}}{3} - \frac{t^{4}}{4} \biggr)b_{1}b_{2} \\ &\qquad{}-\frac{t^{4}}{4}b_{2}^{2}+(b_{1}+b_{2})b_{1} \biggl(\frac{t^{2}}{2}- \frac{t^{3}}{3} \biggr) +\frac{t^{3}}{3}(b_{1}+b_{2})b_{2}- \frac{t^{2}}{2}b_{1}b_{2} \biggr\} \biggr]_{0}^{\frac{1}{1+q}}\\ &\qquad{}+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \end{aligned}$$
$$\begin{aligned} &\qquad{}\times \biggl[\frac{b_{2}-b_{1}}{1+q} \biggl(t-\frac{t^{2}}{2} \biggr) - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{ (\frac{t^{3}}{3}-\frac{t^{4}}{4} )(b_{2}-b_{1})^{2}}{q+1}+ \frac{(1-t)^{4}}{4}b_{1}^{2} - \biggl(\frac{t^{3}}{3}-\frac{t^{4}}{4} \biggr)b_{2}^{2} \\ &\qquad{}-2 \biggl(\frac{t^{2}}{2}+\frac{t^{4}}{4}-\frac{2t^{3}}{3} \biggr)b_{1}b_{2}-(b_{1}+b_{2})b_{1} \frac{(1-t)^{3}}{3}+ \biggl(\frac{t^{2}}{2} -\frac{t^{3}}{3} \biggr) (b_{1}+b_{2})b_{2} \\ &\qquad{}-b_{1}b_{2} \biggl(t-\frac{t^{2}}{2} \biggr) \biggr\} \biggr]_{\frac{1}{1+q}}^{1}+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{b_{2}-b_{1}}{2(1+q)}t^{2} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{t^{4}(b_{2}-b_{1})^{2}}{4(q+1)} \\ &\qquad{}- \biggl(\frac{t^{2}}{2}+\frac{t^{4}}{4}-\frac{2t^{3}}{3} \biggr)b_{1}^{2}- \frac{t^{4}}{4}b_{2}^{2}-2 \biggl(\frac{t^{3}}{3} -\frac{t^{4}}{4} \biggr)b_{1}b_{2}+(b_{1}+b_{2})b_{1} \biggl(\frac{t^{2}}{2}-\frac{t^{3}}{3} \biggr) \\ &\qquad{}+\frac{t^{3}}{3}(b_{1}+b_{2})b_{2}- \frac{t^{2}}{2}b_{1}b_{2} \biggr\} \biggr]_{\frac{1}{1+q}}^{1} \\ &\quad =(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \biggl( \frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{2}-b_{1}) - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{q^{4}b_{1}^{2}}{4(1+q)^{4}} \\ &\qquad{}-\frac{b_{1}^{2}}{4}+ \frac{ (\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} )(b_{2}-b_{1})^{2}}{q+1} - \biggl(\frac{1}{3(1+q)^{3}}- \frac{1}{4(1+q)^{4}} \biggr)b_{2}^{2} \\ &\qquad{}-2 \biggl(\frac{1}{2(1+q)^{2}}+\frac{1}{4(1+q)^{4}}- \frac{2}{3(1+q)^{3}} \biggr)b_{1}b_{2}- \frac{q^{3}(b_{1}+b_{2})b_{1}}{3(1+q)^{3}} + \frac{(b_{1}+b_{2})b_{1}}{3(1+q)^{3}} \\ &\qquad{}+ \biggl(\frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{1}+b_{2})b_{2}- \frac{(1+2q)b_{1}b_{2}}{2(1+q)^{2}} \biggr\} \biggr] \\ &\qquad{}+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{b_{2}-b_{1}}{3(1+q)^{3}}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{2}}{4(q+1)^{5}}- \biggl(\frac{1}{2(1+q)^{2}} \\ &\qquad{}+\frac{1}{4(1+q)^{4}}-\frac{2}{3(1+q)^{3}} \biggr)b_{1}^{2}- \frac{1}{4(1+q)^{4}}b_{2}^{2} -2 \biggl(\frac{1}{3(1+q)^{3}}- \frac{1}{4(1+q)^{4}} \biggr)b_{1}b_{2} \\ &\qquad{}+(b_{1}+b_{2})b_{1} \biggl( \frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr)+\frac{(b_{1}+b_{2})b_{2}}{3(1+q)^{3}} - \frac{b_{1}b_{2}}{2(1+q)^{2}} \biggr\} \biggr] \\ &\qquad{}+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{b_{2}-b_{1}}{2(1+q)}-\frac{(b_{2}-b_{1})(1+2q)}{2(1+q)^{3}} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{2}}{12(q+1)} \\ &\qquad{}- \frac{ (\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} )(b_{2}-b_{1})^{2}}{q+1}- \frac{q^{4}b_{1}^{2}}{4(1+q)^{4}} - \biggl(\frac{1}{3}- \frac{1}{4} \biggr)b_{2}^{2} \\ &\qquad{}+ \biggl(\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} \biggr)b_{2}^{2} -2 \biggl(\frac{1}{2}+\frac{1}{4}-\frac{2}{3} \biggr)b_{1}b_{2}+2 \biggl( \frac{1}{2(1+q)^{2}} \\ &\qquad{}+\frac{1}{4(1+q)^{4}}-\frac{2}{3(1+q)^{3}} \biggr)b_{1}b_{2}+ \frac{q^{3}(b_{1}+b_{2})b_{1}}{3(1+q)^{3}}+ \biggl(\frac{1}{2} - \frac{1}{3} \biggr) (b_{1}+b_{2})b_{2} \\ &\qquad{}- \biggl(\frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{1}+b_{2})b_{2}- \frac{b_{1}b_{2}}{2}+\frac{b_{1}b_{2}(1+2q)}{2(1+q)^{2}} \biggr\} \biggr] \\ &\qquad{}+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{b_{2}-b_{1}}{2(1+q)}-\frac{b_{2}-b_{1}}{2(1+q)^{3}} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{2}}{4(q+1)}- \frac{(b_{2}-b_{1})^{2}}{4(q+1)^{5}} \\ &\qquad{}- \biggl(\frac{1}{2}+\frac{1}{4}-\frac{2}{3} \biggr)b_{1}^{2}+ \biggl( \frac{1}{2(1+q)^{2}}+ \frac{1}{4(1+q)^{4}} -\frac{2}{3(1+q)^{3}} \biggr)b_{1}^{2}- \frac{b_{2}^{2}}{4}+\frac{b_{2}^{2}}{4(1+q)^{4}} \\ &\qquad{}-2 \biggl(\frac{1}{3}-\frac{1}{4} \biggr)b_{1}b_{2}+2 \biggl( \frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} \biggr)b_{1}b_{2} +(b_{1}+b_{2})b_{1} \biggl(\frac{1}{2}- \frac{1}{3} \biggr) \end{aligned}$$
$$\begin{aligned} &\qquad-(b_{1}+b_{2})b_{1} \biggl( \frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) +\frac{(b_{1}+b_{2})b_{2}}{3}- \frac{(b_{1}+b_{2})b_{2}}{3(1+q)^{3}}-\frac{b_{1}b_{2}}{2} \\ &\qquad{}+\frac{b_{1}b_{2}}{2(1+q)^{2}} \biggr\} \biggr] \\ &\quad =(b_{2}-b_{1}) \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \biggl( \frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{2}-b_{1}) - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{q^{4}b_{1}^{2}}{4(1+q)^{4}} \\ &\qquad{}-\frac{b_{1}^{2}}{4}+ \frac{ (\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} )(b_{2}-b_{1})^{2}}{q+1} - \biggl(\frac{1}{3(1+q)^{3}}- \frac{1}{4(1+q)^{4}} \biggr)b_{2}^{2} \\ &\qquad{}-2 \biggl(\frac{1}{2(1+q)^{2}}+\frac{1}{4(1+q)^{4}}- \frac{2}{3(1+q)^{3}} \biggr)b_{1}b_{2}- \frac{q^{3}(b_{1}+b_{2})b_{1}}{3(1+q)^{3}} + \frac{(b_{1}+b_{2})b_{1}}{3(1+q)^{3}} \\ &\qquad{}+ \biggl(\frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{1}+b_{2})b_{2}- \frac{(1+2q)b_{1}b_{2}}{2(1+q)^{2}} \biggr\} + \frac{b_{2}-b_{1}}{2(1+q)} \\ &\qquad{}-\frac{(b_{2}-b_{1})(1+2q)}{2(1+q)^{3}}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{2}}{12(q+1)} - \frac{ (\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} )(b_{2}-b_{1})^{2}}{q+1} \\ &\qquad{}-\frac{q^{4}b_{1}^{2}}{4(1+q)^{4}}- \biggl(\frac{1}{3}-\frac{1}{4} \biggr)b_{2}^{2}+ \biggl(\frac{1}{3(1+q)^{3}} - \frac{1}{4(1+q)^{4}} \biggr)b_{2}^{2}-2 \biggl( \frac{1}{2}+\frac{1}{4}-\frac{2}{3} \biggr)b_{1}b_{2} \\ &\qquad{}+2 \biggl(\frac{1}{2(1+q)^{2}}+\frac{1}{4(1+q)^{4}}- \frac{2}{3(1+q)^{3}} \biggr)b_{1}b_{2} + \frac{q^{3}(b_{1}+b_{2})b_{1}}{3(1+q)^{3}}\\ &\qquad{}+ \biggl( \frac{1}{2}- \frac{1}{3} \biggr) (b_{1}+b_{2})b_{2} \\ &\qquad{}- \biggl(\frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{1}+b_{2})b_{2}- \frac{b_{1}b_{2}}{2}+\frac{b_{1}b_{2}(1+2q)}{2(1+q)^{2}}\biggr\} \biggr] \\ &\qquad{}+(b_{2}-b_{1}) \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{b_{2}-b_{1}}{3(1+q)^{3}}-\frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{2}}{4(q+1)^{5}}- \biggl(\frac{1}{2(1+q)^{2}} \\ &\qquad{}+\frac{1}{4(1+q)^{4}}-\frac{2}{3(1+q)^{3}} \biggr)b_{1}^{2}- \frac{1}{4(1+q)^{4}}b_{2}^{2} -2 \biggl(\frac{1}{3(1+q)^{3}}- \frac{1}{4(1+q)^{4}} \biggr)b_{1}b_{2} \\ &\qquad{}+(b_{1}+b_{2})b_{1} \biggl( \frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr)+\frac{(b_{1}+b_{2})b_{2}}{3(1+q)^{3}} - \frac{b_{1}b_{2}}{2(1+q)^{2}} \biggr\} \\ &\qquad{}+\frac{b_{2}-b_{1}}{2(1+q)}-\frac{b_{2}-b_{1}}{2(1+q)^{3}} - \frac{1}{b_{2}-b_{1}} \biggl\{ \frac{(b_{2}-b_{1})^{2}}{4(q+1)}- \frac{(b_{2}-b_{1})^{2}}{4(q+1)^{5}} - \biggl(\frac{1}{2}+ \frac{1}{4}- \frac{2}{3} \biggr)b_{1}^{2} \\ &\qquad{}+ \biggl(\frac{1}{2(1+q)^{2}}+\frac{1}{4(1+q)^{4}} - \frac{2}{3(1+q)^{3}} \biggr)b_{1}^{2}-\frac{b_{2}^{2}}{4}+ \frac{b_{2}^{2}}{4(1+q)^{4}}-2 \biggl(\frac{1}{3}-\frac{1}{4} \biggr)b_{1}b_{2} \\ &\qquad{}+2 \biggl(\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} \biggr)b_{1}b_{2} +(b_{1}+b_{2})b_{1} \biggl(\frac{1}{2}- \frac{1}{3} \biggr)-(b_{1}+b_{2})b_{1} \biggl( \frac{1}{2(1+q)^{2}} \\ &\qquad{}-\frac{1}{3(1+q)^{3}} \biggr) +\frac{(b_{1}+b_{2})b_{2}}{3}- \frac{(b_{1}+b_{2})b_{2}}{3(1+q)^{3}}- \frac{b_{1}b_{2}}{2}+ \frac{b_{1}b_{2}}{2(1+q)^{2}} \biggr\} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \biggl(\frac{1}{2(1+q)^{2}}- \frac{1}{3(1+q)^{3}} \biggr) (b_{2}-b_{1})^{2} - \frac{q^{4}b_{1}^{2}}{4(1+q)^{4}} \\ &\qquad{}+\frac{b_{1}^{2}}{4}- \frac{ (\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} )(b_{2}-b_{1})^{2}}{q+1} + \biggl(\frac{1}{3(1+q)^{3}}+ \frac{1}{4(1+q)^{4}} \biggr)b_{2}^{2} \\ &\qquad{}+2 \biggl(\frac{1}{2(1+q)^{2}}+\frac{1}{4(1+q)^{4}}- \frac{2}{3(1+q)^{3}} \biggr)b_{1}b_{2}+ \frac{q^{3}(b_{1}+b_{2})b_{1}}{3(1+q)^{3}} - \frac{(b_{1}+b_{2})b_{1}}{3(1+q)^{3}} \end{aligned}$$
$$\begin{aligned} &\qquad{}- \biggl(\frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{1}+b_{2})b_{2}+ \frac{(1+2q)b_{1}b_{2}}{2(1+q)^{2}} +\frac{(b_{2}-b_{1})^{2}}{2(1+q)} \\ &\qquad{}-\frac{(b_{2}-b_{1})^{2}(1+2q)}{2(1+q)^{3}}- \frac{(b_{2}-b_{1})^{2}}{12(q+1)} + \frac{ (\frac{1}{3(1+q)^{3}}-\frac{1}{4(1+q)^{4}} )(b_{2}-b_{1})^{2}}{q+1} \\ &\qquad{}+\frac{q^{4}b_{1}^{2}}{4(1+q)^{4}}+ \biggl(\frac{1}{3}-\frac{1}{4} \biggr)b_{2}^{2}- \biggl(\frac{1}{3(1+q)^{3}} - \frac{1}{4(1+q)^{4}} \biggr)b_{2}^{2}+2 \biggl( \frac{1}{2}+\frac{1}{4}-\frac{2}{3} \biggr)b_{1}b_{2} \\ &\qquad{}-2 \biggl(\frac{1}{2(1+q)^{2}}+\frac{1}{4(1+q)^{4}}- \frac{2}{3(1+q)^{3}} \biggr)b_{1}b_{2} - \frac{q^{3}(b_{1}+b_{2})b_{1}}{3(1+q)^{3}}\\ &\qquad{}- \biggl( \frac{1}{2}- \frac{1}{3} \biggr) (b_{1}+b_{2})b_{2} \\ &\qquad{}+ \biggl(\frac{1}{2(1+q)^{2}}-\frac{1}{3(1+q)^{3}} \biggr) (b_{1}+b_{2})b_{2}+ \frac{b_{1}b_{2}}{2}-\frac{b_{1}b_{2}(1+2q)}{2(1+q)^{2}} \biggr] \\ &\qquad{}+ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{(b_{2}-b_{1})^{2}}{3(1+q)^{3}}- \frac{(b_{2}-b_{1})^{2}}{4(q+1)^{5}}+ \biggl(\frac{1}{2(1+q)^{2}}+ \frac{1}{4(1+q)^{4}}- \frac{2}{3(1+q)^{3}} \biggr)b_{1}^{2} \\ &\qquad{}+\frac{1}{4(1+q)^{4}}b_{2}^{2}+2 \biggl( \frac{1}{3(1+q)^{3}}- \frac{1}{4(1+q)^{4}} \biggr)b_{1}b_{2} -(b_{1}+b_{2})b_{1} \biggl( \frac{1}{2(1+q)^{2}} \\ &\qquad{}-\frac{1}{3(1+q)^{3}} \biggr)-\frac{(b_{1}+b_{2})b_{2}}{3(1+q)^{3}} + \frac{b_{1}b_{2}}{2(1+q)^{2}}+ \frac{(b_{2}-b_{1})^{2}}{2(1+q)}+ \frac{(b_{2}-b_{1})^{2}}{2(1+q)^{3}} \\ &\qquad{}-\frac{(b_{2}-b_{1})^{2}}{4(q+1)}+ \frac{(b_{2}-b_{1})^{2}}{4(q+1)^{5}} + \biggl(\frac{1}{2}+ \frac{1}{4}- \frac{2}{3} \biggr)b_{1}^{2}- ( \frac{1}{2(1+q)^{2}}+ \frac{1}{4(1+q)^{4}} \\ &\qquad{}-\frac{2}{3(1+q)^{3}} )b_{1}^{2}+\frac{b_{2}^{2}}{4}- \frac{b_{2}^{2}}{4(1+q)^{4}} +2 \biggl(\frac{1}{3}-\frac{1}{4} \biggr)b_{1}b_{2}-2 \biggl(\frac{1}{3(1+q)^{3}} \\ &\qquad{}-\frac{1}{4(1+q)^{4}} \biggr)b_{1}b_{2}-(b_{1}+b_{2})b_{1} \biggl( \frac{1}{2}-\frac{1}{3} \biggr)+(b_{1}+b_{2})b_{1} \biggl( \frac{1}{2(1+q)^{2}} \\ &\qquad{}-\frac{1}{3(1+q)^{3}} \biggr)-\frac{(b_{1}+b_{2})b_{2}}{3}+ \frac{(b_{1}+b_{2})b_{2}}{3(1+q)^{3}} + \frac{b_{1}b_{2}}{2}- \frac{b_{1}b_{2}}{2(1+q)^{2}} \biggr] \\ &\quad= \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \biggl(\frac{1}{2(1+q)^{2}}- \frac{1}{3(1+q)^{3}} \biggr) (b_{2}-b_{1})^{2} +\frac{b_{1}^{2}}{4}- \frac{(b_{1}+b_{2})b_{1}}{3(1+q)^{3}}+ \frac{(b_{2}-b_{1})^{2}}{2(1+q)} \\ &\qquad{}-\frac{(b_{2}-b_{1})^{2}(1+2q)}{2(1+q)^{3}}- \frac{(b_{2}-b_{1})^{2}}{12(q+1)} + \biggl(\frac{1}{3}- \frac{1}{4} \biggr)b_{2}^{2}+2 \biggl( \frac{1}{2}+\frac{1}{4}-\frac{2}{3} \biggr)b_{1}b_{2} \\ &\qquad{}- \biggl(\frac{1}{2}-\frac{1}{3} \biggr) (b_{1}+b_{2})b_{2} + \frac{b_{1}b_{2}}{2} \biggr]+ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(b_{2}-b_{1})^{2}}{3(1+q)^{3}} +\frac{(b_{2}-b_{1})^{2}}{2(1+q)} \\ &\qquad{}+\frac{(b_{2}-b_{1})^{2}}{2(1+q)^{3}}- \frac{(b_{2}-b_{1})^{2}}{4(q+1)} + \biggl(\frac{1}{2}+ \frac{1}{4}- \frac{2}{3} \biggr)b_{1}^{2}+ \frac{b_{2}^{2}}{4}+2 \biggl(\frac{1}{3}- \frac{1}{4} \biggr)b_{1}b_{2} \\ &\qquad{}-(b_{1}+b_{2})b_{1} \biggl(\frac{1}{2}- \frac{1}{3} \biggr) - \frac{(b_{1}+b_{2})b_{2}}{3}+\frac{b_{1}b_{2}}{2} \biggr] \\ &\quad= \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{(1+3q)(b_{2}-b_{1})^{2}}{6(1+q)^{3}} +\frac{b_{1}^{2}}{4}- \frac{(b_{1}+b_{2})b_{1}}{3(1+q)^{3}}+ \frac{(b_{2}-b_{1})^{2}}{2(1+q)} \\ &\qquad{}-\frac{(b_{2}-b_{1})^{2}(1+2q)}{2(1+q)^{3}}- \frac{(b_{2}-b_{1})^{2}}{12(q+1)} +\frac{b_{2}^{2}}{12}+ \frac{b_{1}b_{2}}{6}-\frac{(b_{1}+b_{2})b_{2}}{6} + \frac{b_{1}b_{2}}{2} \biggr] \\ &\qquad{}+ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{(b_{2}-b_{1})^{2}}{3(1+q)^{3}}+ \frac{(b_{2}-b_{1})^{2}}{2(1+q)} + \frac{(b_{2}-b_{1})^{2}}{2(1+q)^{3}}- \frac{(b_{2}-b_{1})^{2}}{4(q+1)} + \frac{b_{1}^{2}}{12}+\frac{b_{2}^{2}}{4} \end{aligned}$$
$$\begin{aligned} &\qquad{}+\frac{b_{1}b_{2}}{6}-\frac{(b_{1}+b_{2})b_{1}}{6} - \frac{(b_{1}+b_{2})b_{2}}{3}+ \frac{b_{1}b_{2}}{2} \biggr] \\ &\quad = \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{(1+3q)(b_{2}-b_{1})^{2}}{6(1+q)^{3}} - \frac{(b_{1}+b_{2})b_{1}}{3(1+q)^{3}}+ \frac{(b_{2}-b_{1})^{2}}{2(1+q)}- \frac{(b_{2}-b_{1})^{2}(1+2q)}{2(1+q)^{3}} \\ &\qquad{}-\frac{(b_{2}-b_{1})^{2}}{12(q+1)} +\frac{b_{1}^{2}}{4}+ \frac{b_{2}^{2}}{12}+ \frac{b_{1}b_{2}}{6}- \frac{(b_{1}+b_{2})b_{2}}{6} +\frac{b_{1}b_{2}}{2} \biggr] \\ &\qquad{}+ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[\frac{(b_{2}-b_{1})^{2}}{3(1+q)^{3}}+ \frac{(b_{2}-b_{1})^{2}}{2(1+q)} + \frac{(b_{2}-b_{1})^{2}}{2(1+q)^{3}}- \frac{(b_{2}-b_{1})^{2}}{4(q+1)} + \frac{b_{1}^{2}}{12}+\frac{b_{2}^{2}}{4} \\ &\qquad{}+\frac{b_{1}b_{2}}{6}-\frac{(b_{1}+b_{2})b_{1}}{6} - \frac{(b_{1}+b_{2})b_{2}}{3}+ \frac{b_{1}b_{2}}{2} \biggr] \\ &\quad= \bigl\vert \psi ''(b_{1}) \bigr\vert \biggl[ \frac{(b_{2}-b_{1})^{2} (5q^{2}+4q+1 )-4(b_{1}+b_{2})b_{1}}{12(1+q)^{3}} +\frac{3b_{1}^{2}-b_{2}^{2}+6b_{1}b_{2}}{12} \biggr] \\ &\qquad{}+ \bigl\vert \psi ''(b_{2}) \bigr\vert \biggl[ \frac{(b_{2}-b_{1})^{2} (3(1+q)^{2}+10 )}{12(1+q)^{3}}-\frac{(b_{2}-b_{1})^{2}}{12} \biggr]. \end{aligned}$$

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4 Conclusion

We revisited the Hermite–Hadamard inequality in quantum calculus. We deduced some new identities in the way. Using these identities, we obtained new estimates in this regard. Employing the method outlined in this paper, we anticipate that some other inequalities may be reestablished. More results on the Hermite–Hadamard inequality in quantum calculus can be found in [22, 26, 31, 34].