1 Introduction

The brilliant results of Charles Hermite and Jacques Hadamard’s studies, which ended in Hermite–Hadamard inequality, commonly known as Hadamard’s inequality, indicate the fact that if \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) is convex, we have the following double inequality:

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{\omega -\nu} \int _{\nu}^{\omega}\hbar ( x )\,dx \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(1)

When ħ is a concave mapping, the inequality holds in the opposite direction. There has been much research done in the Hermite–Hadamard direction for different kinds of convexities. For example, in [1, 2] the authors established some inequalities linked with midpoint and trapezoidal formulas of numerical integration for convex functions. For more results related to the above inequality and convex functions, the reader can consult [3–7]. There are many generalizations of convex functions, like h-convex functions, preinvex functions, m-convex functions, harmonically convex functions, \(( \alpha ,m ) \)-convex functions, convexity with respect to a pair of functions, etc. These kinds of convexities have a very large role in functional analysis, optimization theory, approximation theory, and fractional mathematical modeling [8–24].

Due to its immediate applications in numerical integration, probability theory, information theory, and integral operator theory, the Hermite–Hadamard inequality is of utmost significance. The field of inequalities has seen a tremendous flow of this remarkable inequality and related outstanding (Hadamard-type) inequalities over the past millennium. These inequalities are partially inspired by the results mentioned above but perhaps even more, so by the difficulty of conducting research in a variety of mathematical subdisciplines like mathematical programming, control theory, variational methods, operation research, probability, and statistics.

Besides this, quantum and postquantum calculus are very important branches of calculus having a vast range of applications in the fields of mathematics and physics. Because of numerous applications of quantum calculus (shortly, q-calculus) and postquantum calculus (shortly, \(( p,q ) \)-calculus) without limit calculus, many researchers began working on them and applying their concepts in differential equations, integral equalities, mathematical modeling, and integral inequalities [25–32].

Alp et al. [33] and Bermudo et al. [34] used q-integrals to prove two different versions of q-Hermite–Hadamard inequalities along with some relevant estimates. The q-Hermite–Hadamard inequalities are described as follows.

Theorem 1.1

([33, 34])

For a convex map \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \), we have the inequalities

$$\begin{aligned}& \hbar \biggl( \frac{q\nu +\omega}{ [ 2 ] _{q}} \biggr) \leq \frac{1}{\omega -\nu}\int _{\nu}^{\omega}\hbar ( x ) \,{}_{\nu}d_{q}x \leq \frac{q\hbar ( \nu ) +\hbar ( \omega ) }{ [ 2 ] _{q}}, \end{aligned}$$
(2)
$$\begin{aligned}& \hbar \biggl( \frac{\nu +q\omega}{ [ 2 ] _{q}} \biggr) \leq \frac{1}{\omega -\nu}\int _{\nu}^{\omega}\hbar ( x ) \,{}^{\omega}d_{q}x \leq \frac{\hbar ( \nu ) +q\hbar ( \omega ) }{ [ 2 ] _{q}}. \end{aligned}$$
(3)

Remark 1.2

It is very easy to observe that by adding (3) and (4) we derive the q-Hermite–Hadamard inequality (see [34])

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2 ( \omega -\nu ) } \biggl[ \int _{\nu}^{\omega} \hbar ( x ) \,{}_{\nu}d_{q}x+ \int _{\nu}^{\omega}\hbar ( x ) \,{}^{\omega}d_{q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(4)

Recently, Ali et al. [35] and Sitthiwirattham et al. [36] used new techniques to prove the following two different and new versions of Hermite–Hadamard-type inequalities in the context of q-operators.

Theorem 1.3

([35, 36])

For a convex map \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \), we have the inequalities

$$\begin{aligned} &\hbar \biggl( \frac{\nu +\omega}{2} \biggr)\leq \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) {}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{ \frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) {}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}, \end{aligned}$$
(5)
$$\begin{aligned} &\hbar \biggl( \frac{\nu +\omega}{2} \biggr)\leq \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\nu}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{ \omega}\hbar ( x ) \,{}^{\omega}d_{q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. \end{aligned}$$
(6)

Remark 1.4

When \(q\rightarrow 1^{-}\) in (3)–(7), we recapture the traditional Hermite–Hadamard inequality (1).

Kunt et al. [37] and Vivas-Cortez et al. [38] extended the previous studies and derived several Hermite–Hadamard-type inequalities with new structures for convex functions using the \(( p,q ) \)-integrals.

Theorem 1.5

([37, 38])

For a convex mapping \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \), we have the inequalities

$$ \hbar \biggl( \frac{q\nu +p\omega}{ [ 2 ] _{p,q}} \biggr) \leq \frac{1}{p ( \omega -\nu ) } \int _{\nu}^{\omega}\hbar ( x ) \,{}_{ \nu}d_{p,q}x \leq \frac{q\hbar ( \nu ) +p\hbar ( \omega ) }{ [ 2 ] _{p,q}} $$
(7)

and

$$ \hbar \biggl( \frac{p\nu +q\omega}{ [ 2 ] _{p,q}} \biggr) \leq \frac{1}{p ( \omega -\nu ) } \int _{\nu}^{\omega}\hbar ( x ) \,{}^{ \omega}d_{p,q}x \leq \frac{p\hbar ( \nu ) +q\hbar ( \omega ) }{ [ 2 ] _{p,q}}. $$
(8)

Remark 1.6

It is also very easy to observe that by adding (7) and (8) we obtain the \(( p,q ) \)-Hermite–Hadamard inequality (see [38])

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2p ( \omega -\nu ) } \biggl[ \int _{\nu}^{\omega} \hbar ( x ) \,{}_{\nu}d_{p,q}x+ \int _{\nu}^{\omega}\hbar ( x ) \,{}^{\omega}d_{p,q}x \biggr] \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(9)

Remark 1.7

It is worth mentioning that inequalities (7)–(9) are generalizations of inequalities (2)–(4), respectively, and for \(p=1\), we can obtain the q-Hermite–Hadamard inequalities.

There has been much research done in the direction of q and \(( p,q ) \)-integral inequalities for different kinds of convexity. For instance, in [37–40], some new midpoint and trapezoidal inequalities via q and \(( p,q ) \)-integrals were established. The authors of [41–48] used q and \(( p,q ) \)-integrals and established Simpson-type inequalities for functions with different forms of convexity. For more recent inequalities in q-calculus, see [49–54].

By considering such advanced level studies we consider the convexity of functions and derive a new variant of Hermite–Hadamard inequality in the setting of \(( p,q ) \)-calculus. Furthermore, we derive some new midpoint and trapezoidal type inequalities for the special class of functions called \(( p,q ) \)-differentiable convex functions in the framework of \(( p,q ) \)-calculus. We also show that our newly established results are an extension of [36], which states the novelty of our research. The results presented here can be helpful in finding the error bounds of numerical integration formulas and variety of mathematical subdisciplines like mathematical programming, control theory, variational methods, operation research, probability, and statistics.

The structure of the paper is as follows. In Sect. 2, we recall some basics of q- and \((p,q)\)-calculus. In Sect. 3, we establish a new variant of q-Hermite–Hadamard-type inequality for some special convex \((p,q)\)-functions. In Sects. 4 and 5, we derive some new midpoint and trapezoidal inequalities for q-differentiable convexity, respectively. Section 6 briefly concludes our work.

2 q- and \(( p,q ) \)-calculus

We recall some basics of quantum calculus in this section. For \(0< q< p\leq 1\), we denote [38, 55]

$$ [ n ] _{q}=\frac{1-q^{n}}{1-q} $$

and

$$ [ n ] _{p,q}=\frac{p^{n}-q^{n}}{p-q}. $$

Definition 2.1

([56])

The left or \(q_{\nu}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}_{\nu}\mathfrak{D}_{q}\hbar ( x )= \frac{\hbar ( x ) -\hbar ( qx+ ( 1-q ) \nu ) }{ ( 1-q ) ( x-\nu ) },\quad x\neq \nu . $$
(10)

Definition 2.2

([34])

The right or \(q^{\omega}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}^{\omega}\mathfrak{D}_{q}\hbar ( x )= \frac{\hbar ( qx+ ( 1-q ) \omega ) -\hbar ( x ) }{ ( 1-q ) ( \omega -x ) },\quad x\neq \omega . $$

Definition 2.3

([57])

The left or \(( p,q ) _{\nu}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}_{\nu}\mathfrak{D}_{p,q}\hbar ( x )= \frac{\hbar ( px+ ( 1-p ) \nu ) -\hbar ( qx+ ( 1-q ) \nu ) }{ ( p-q ) ( x-\nu ) },\quad x \neq \nu . $$

Definition 2.4

([38])

The right or \(( p,q ) ^{\omega}\)-derivative of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ {}^{\omega}\mathfrak{D}_{p,q}\hbar ( x )= \frac{\hbar ( qx+ ( 1-q ) \omega ) -\hbar ( px+ ( 1-p ) \omega ) }{ ( p-q ) ( \omega -x ) },\quad x\neq \omega . $$

Definition 2.5

([56])

The left or \(q_{\nu}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [ \nu ,\omega ] \) is defined as

$$ \int _{\nu}^{x}\hbar ( t ) \,{}_{\nu}d_{q}t= ( 1-q ) ( x-\nu ) \sum_{n=0}^{ \infty }q^{n} \hbar \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) \nu \bigr) . $$

Definition 2.6

([34])

The right or \(q^{\omega}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) at \(x\in [\nu ,\omega ] \) is defined as

$$ \int _{x}^{\omega}\hbar ( t ) \,{}^{\omega}d_{q}t= ( 1-q ) ( \omega -x ) \sum_{n=0}^{ \infty }q^{n} \hbar \bigl( q^{n}x+ \bigl( 1-q^{n} \bigr) \omega \bigr) . $$

Definition 2.7

([57])

The left or \(( p,q ) _{\nu}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) is defined as

$$ \int _{\nu}^{x}\hbar ( t ) \,{}_{\nu}d_{p,q}t= ( p-q ) ( x-\nu ) \sum_{n=0}^{ \infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n+1}}x+ \biggl( 1- \frac{q^{n}}{q^{n+1}} \biggr) \nu \biggr) $$

for \(x\in [ \nu ,p\omega + ( 1-p ) \nu ] \)

Definition 2.8

([38])

The right or \(q^{\omega}\)-integral of \(\hbar : [ \nu ,\omega ] \rightarrow \mathbb{R} \) is defined as

$$ \int _{x}^{\omega}\hbar ( t ) \,{}^{\omega}d_{p,q}t= ( p-q ) ( \omega -x ) \sum_{n=0}^{ \infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n+1}}x+ \biggl( 1- \frac{q^{n}}{p^{n+1}} \biggr) \omega \biggr) $$

for \(x\in [ p\nu + ( 1-p ) \omega ,\omega ] \)

For more properties and details about q- and \(( p,q ) \)-calculus, the reader can consult [34, 38, 56–58].

3 \((p,q)\)-Hermite–Hadamard inequality

In this section, we establish a new version of Hermite–Hadamard inequality for convex functions and the special \((p,q)\)-operators defined in \(( p,q ) \)-calculus.

Theorem 3.1

Let \(\hbar : [ \nu ,\omega ] \subset \mathbb{R} \rightarrow \mathbb{R} \) be convex. Then

$$\begin{aligned} \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq & \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p \omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] \\ \leq & \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. \end{aligned}$$
(11)

Proof

The convexity of ħ implies that

$$ \hbar \biggl( \frac{x+y}{2} \biggr) \leq \frac{1}{2} \bigl[ \hbar ( x ) +\hbar ( y ) \bigr] . $$

Setting \(x=\frac{1-t}{2}\nu +\frac{1+t}{2}\omega \) and \(y=\frac{1+t}{2}\nu +\frac{1-t}{2}\omega \), we get

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2} \biggl[ \hbar \biggl( \frac{1-t}{2}\nu +\frac{1+t}{2}\omega \biggr) +\hbar \biggl( \frac{1+t}{2}\nu + \frac{1-t}{2}\omega \biggr) \biggr] . $$
(12)

By \(( p,q ) \)-integrating (12) with respect to t on \([ 0,p ] \) we get

$$ p\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{2} \biggl[ \int _{0}^{p}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) \,d_{p,q}t+ \int _{0}^{p}\hbar \biggl( \frac{1+t}{2}\nu + \frac{1-t}{2}\omega \biggr) \,d_{p,q}t \biggr] . $$

From Definitions 2.7 and 2.8 we have

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) \leq \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p \omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] . $$

Thus the first inequality in (11) is proved. We again use the convexity to prove the second inequality in (11):

$$ \hbar \biggl( \frac{1-t}{2}\nu +\frac{1+t}{2}\omega \biggr) +\hbar \biggl( \frac{1+t}{2}\nu +\frac{1-t}{2}\omega \biggr) \leq \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$
(13)

By \(( p,q ) \)-integrating (13) with respect to t on \([ 0,p ] \) we get

$$ \int _{0}^{p}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) \,d_{p,q}t+ \int _{0}^{p}\hbar \biggl( \frac{1+t}{2}\nu + \frac{1-t}{2}\omega \biggr) \,d_{p,q}t\leq p \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}. $$

By applying Definitions 2.7 and 2.8 we obtain the required inequality. □

Remark 3.2

By assuming \(p=1\) in Theorem 3.1, we regain inequality (5).

Remark 3.3

By setting \(p=1\) and taking the limit \(q\rightarrow 1^{-}\) we regain the traditional Hermite–Hadamard inequality (1) for the classical convex functions.

Example 3.4

Consider the convex function \(\hbar : [ 0,1 ] \rightarrow \mathbb{R} \) defined as \(\hbar (x)=x^{2}\) with \(q=\frac{1}{3}\) and \(p=\frac{2}{3}\). Then

$$ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) =\frac{1}{4} $$

and

$$ \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}= \frac{1}{2}. $$

On the other hand, by Definitions 2.7 and 2.8 we have

$$\begin{aligned} \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{6}}^{\frac{1}{2}}x^{2}\,{}^{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}.\frac{1}{3}\sum_{n=0}^{\infty } \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \frac{1}{6}+ \biggl( 1-\frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{2} \\ =&\frac{1}{24}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1- \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{2} \\ =&\frac{1}{24}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1-2 \biggl( \frac{1}{2} \biggr) ^{n}+ \biggl( \frac{1}{2} \biggr) ^{2n} \biggr) \\ =&\frac{1}{24} \biggl[ 2-\frac{8}{3}+\frac{8}{7} \biggr] \\ =&\frac{5}{252} \end{aligned}$$

and

$$\begin{aligned} \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{2}}^{\frac{5}{6}}x^{2}\,{}_{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}.\frac{1}{3}\sum _{n=0}^{\infty }\frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n}\frac{5}{6}+ \biggl( 1- \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{2} \\ =&\frac{1}{6}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{1}{2}+ \frac{1}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{2} \\ =&\frac{1}{24} \biggl[ 2+\frac{8}{3}+\frac{8}{7} \biggr] \\ =&\frac{61}{252}. \end{aligned}$$

Thus

$$\begin{aligned}& \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega }{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] \\& \quad = \frac{3}{2} \biggl[ \frac{5}{252}+\frac{61}{252} \biggr] \\& \quad = \frac{99}{252}. \end{aligned}$$

It is clear that

$$ \frac{1}{4}< \frac{99}{252}< \frac{1}{2}. $$

4 \((p,q)\)-Midpoint inequalities

In this section, we establish some new inequalities of midpoint type for \(( p,q ) \)-differentiable functions in the setting of \(( p,q ) \)-calculus. We begin with a lemma, which has a great role in establishing the inequalities of this section.

Lemma 4.1

For \(\hbar : [ \nu ,\omega ] \subset \mathbb{R} \rightarrow \mathbb{R} \), if \({}_{\nu}\mathfrak{D}_{p,q}\hbar \) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar \) are continuous and integrable mappings over \([ \nu ,\omega ] \), then we have the following identity:

$$\begin{aligned}& \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega }{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \\& \quad = \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \\& \qquad {}-{}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr) \,d_{p,q}t \biggr] . \end{aligned}$$
(14)

Proof

Definitions 2.3 and 2.4 give

$$ {}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) =2p \biggl[ \frac{\hbar ( \frac{q}{p}t\nu + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) -\hbar ( t\nu + ( 1-t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \omega -\nu ) t} \biggr] $$
(15)

and

$$ {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) =2p \biggl[ \frac{\hbar ( t\omega + ( 1-t ) \frac{\nu +\omega}{2} ) -\hbar ( \frac{q}{p}t\omega + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \omega -\nu ) t} \biggr] . $$
(16)

By Definition 2.8 from (15) we have

$$\begin{aligned}& \int _{0}^{p} ( 1-qt ) {}^{\omega} \mathfrak{D}_{p,q} \hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \int _{0}^{p} ( 1-qt )p \frac{\hbar ( \frac{q}{p}t\nu + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) -\hbar ( t\nu + ( 1-t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \frac{\omega -\nu}{2} ) t}\,d_{p,q}t \\& \quad = \frac{2p^{2}}{\omega -\nu} \Biggl[ \sum_{n=0}^{\infty } \hbar \biggl( \frac{q^{n+1}}{p^{n+1}}\nu + \biggl( 1-\frac{q^{n+1}}{p^{n+1}} \biggr) \frac{\nu +\omega}{2} \biggr) \\& \qquad {} -\sum_{n=0}^{ \infty } \hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1-\frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \qquad {} -\frac{2p^{2}q}{\omega -\nu} \Biggl[ \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n+1}}{p^{n+1}}\nu + \biggl( 1- \frac{q^{n+1}}{p^{n+1}} \biggr) \frac{\nu +\omega}{2} \biggr) \\& \qquad {} -\sum _{n=0}^{\infty }\frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}}{\omega -\nu} \biggl[ \hbar \biggl( \frac{\nu +\omega}{2} \biggr) -\hbar ( \nu ) \biggr] \\& \qquad {}-\frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{1}{q}\sum _{n=0}^{ \infty }\frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) -\frac{1}{q} \hbar ( \nu ) \\& \qquad {}-\frac{1}{p}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& = \frac{2p^{2}}{\omega -\nu}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) -\frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{p-q}{pq}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}}{\omega -\nu}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) -\frac{4p}{ ( \omega -\nu ) ^{2}} \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x, \end{aligned}$$

which implies that

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( 1-qt ) {}^{ \omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p} \omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{2}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) - \frac{1}{p ( \omega -\nu ) }\int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x. \end{aligned}$$
(17)

Similarly, from Definition 2.7 and relation (16) we have

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( 1-qt ) {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu +\frac{p+t}{2p} \omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{p ( \omega -\nu ) } \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x- \frac{1}{2}\hbar \biggl( \frac{\nu +\omega}{2} \biggr) . \end{aligned}$$
(18)

Thus we derive the required identity (14) by subtracting (17) from (18). □

Remark 4.2

In Lemma 4.1, for \(p=1\), we obtain the following identity:

$$\begin{aligned}& \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}} \hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \\& \quad = \frac{\omega -\nu}{4} \biggl[ \int _{0}^{1} ( 1-qt ) \biggl( {}_{\nu} \mathfrak{D}_{q}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) \\& \qquad {}-{}^{\omega}\mathfrak{D}_{q}\hbar \biggl( \frac{1+t}{2}\nu +\frac{1-t}{2}\omega \biggr) \biggr) \,d_{q}t \biggr] , \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

Theorem 4.3

If Lemma 4.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \) are convex, then

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q\frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr] . \end{aligned}$$
(19)

Proof

Taking the modulus in (14), by the convexity properties of \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \), we estimate

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {}+ \biggl\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p} \nu + \frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \frac{p-t}{2p} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert +\frac{p+t}{2p} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr) \,d_{p,q}t \\& \qquad {} + \int _{0}^{p} ( 1-qt ) \biggl( \frac{p+t}{2p} \big\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert + \frac{p-t}{2p} \big\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert \biggr) \,d_{p,q}t \biggr] \\& \quad = \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \biggl\vert {}_{\nu} \mathfrak{D}_{p,q} \hbar ( \nu ) \biggl\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert { }_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q\frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr] , \end{aligned}$$

and our proof is completed. □

Remark 4.4

If we set \(p=1\) in the previous theorem, then

$$\begin{aligned}& \biggl\vert \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega} \hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{8} \biggl[ \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{ \nu}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert + \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{q} \hbar ( \omega ) \bigl\vert \\& \qquad {}+ \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert + \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \big\vert {} ^{\omega} \mathfrak{D}_{q}\hbar ( \omega ) \big\vert \biggr] , \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

Example 4.5

Consider \(\hbar : [ 0,1 ] \rightarrow \mathbb{R} \) defined by \(\hbar (x)=x^{3}\). Let also \(q=\frac{1}{3}\) and \(p=\frac{2}{3}\). Then we have the convex functions \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{7x^{2}}{27}\) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{1}{3} ( 7x^{2}+13x+7 ) \), which gives

$$\begin{aligned} \int _{pa+ ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{6}}^{\frac{1}{2}}x^{3}\,{}^{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}\frac{1}{3}\sum_{n=0}^{\infty } \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \frac{1}{6}+ \biggl( 1-\frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{3} \\ =&\frac{1}{48}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1- \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{3} \\ =&\frac{1}{48} \biggl[ 2-4+\frac{24}{7}-\frac{16}{15} \biggr] \\ =&\frac{19}{2520} \end{aligned}$$

and

$$\begin{aligned} \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x =& \int _{\frac{1}{2}}^{\frac{5}{6}}x^{3}\,{}_{ \frac{1}{2}}d_{\frac{2}{3},\frac{1}{3}}x \\ =&\frac{1}{3}\frac{1}{3}\sum_{n=0}^{\infty } \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \biggl( \frac{3}{2} \biggl( \frac{1}{2} \biggr) ^{n} \frac{5}{6}+ \biggl( 1- \biggl( \frac{1}{2} \biggr) ^{n} \biggr) \frac{1}{2} \biggr) ^{3} \\ =&\frac{1}{48}\sum_{n=0}^{\infty } \biggl( \frac{1}{2} \biggr) ^{n} \biggl( 1+ \biggl( \frac{1}{2} \biggr) ^{n} \biggr) ^{3} \\ =&\frac{1}{48} \biggl[ 2+4+\frac{24}{7}+\frac{16}{15} \biggr] \\ =&\frac{551}{2520}. \end{aligned}$$

Thus the left-hand side of inequality (19) reduces to

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad = \biggl\vert \frac{3}{2} \biggl[ \frac{18}{2520}+ \frac{551}{2520} \biggr] -\frac{1}{8} \biggr\vert \\& \quad = \frac{9}{42}. \end{aligned}$$

On the other hand, since \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( a ) =0\), \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =\frac{7}{27}\), \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \vert =\frac{7}{3}\), and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =9\), The righ-hand side of inequality (19) becomes

$$\begin{aligned}& \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {} + \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigl\vert \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q\frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr] \\& \quad = \frac{27}{64} \biggl[ \biggl( \frac{8}{9}-\frac{128}{567} \biggr) \frac{7}{27}+ \biggl( \frac{8}{9}- \frac{128}{567} \biggr) \frac{7}{3}+ \biggl( 0+ \frac{16}{567} \biggr) 9 \biggr] \\& \quad = \frac{472}{567}. \end{aligned}$$

It is clear that

$$ \frac{9}{42}< \frac{472}{567}. $$

Theorem 4.6

If Lemma 4.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s\geq 1\), then

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{p^{2}+pq-qp^{2}}{ [ 2 ] _{p,q}} \biggr) ^{1-\frac{1}{s}} \\& \qquad {} \times \biggl[ \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {} + \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \nu ) \big\vert ^{s} \\& \qquad {} +\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

Proof

Taking the modulus in (14), by the power mean inequality we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {} + \biggl\vert {} ^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p} \nu + \frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( 1-qt ) \,d_{p,q}t \biggr) ^{1-\frac{1}{s}} \biggl[ \biggl( \int _{0}^{p} ( 1-qt ) \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} ( 1-qt ) \bigg\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \bigg\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \biggr] . \end{aligned}$$

By the convexity of \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{p^{2}+pq-qp^{2}}{ [ 2 ] _{p,q}} \biggr) ^{1-\frac{1}{s}} \\& \qquad {} \times \biggl[ \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {} _{\nu} \mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {} + \biggl( \frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q+p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \nu ) \big\vert ^{s} \\& \qquad {} +\frac{1}{2p} \biggl( \frac{p^{3}+p^{2}q-p^{2}}{ [ 2 ] _{p,q}}-p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

The proof is completed. □

Remark 4.7

In Theorem 4.6, for \(p=1\), we have the inequality

$$\begin{aligned}& \biggl\vert \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega} \hbar ( x ) \,{}_{\frac{\nu +\omega }{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4} \biggl( \frac{1}{ [ 2 ] _{q}} \biggr) ^{1-\frac{1}{s}} \biggl[ \biggl( \frac{1}{2} \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+ \frac{1}{2} \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \bigl\vert {}_{ \nu}\mathfrak{D}_{q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{1}{2} \biggl( \frac{2+q}{ [ 2 ] _{q}}-q \frac{ [ 3 ] _{q}+ [ 2 ] _{q}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \big\vert {} ^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \big\vert ^{s}+\frac{1}{2} \biggl( \frac{q}{ [ 2 ] _{q}}- \frac{q^{3}}{ [ 2 ] _{q} [ 3 ] _{q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr], \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

Theorem 4.8

If Lemma 4.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s>1\), then

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{1- ( 1-pq ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr], \end{aligned}$$

where \(s^{-1}+r^{-1}=1\).

Proof

Taking the modulus in (14), by the Hölder inequality we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( 1-qt ) \biggl( \biggl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {}+ \biggl\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p} \nu + \frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( 1-qt )^{r}\,d_{p,q}t \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \int _{0}^{p} \biggl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} \biggl\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \biggr] . \end{aligned}$$

By the convexity of \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert ^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) we have

$$\begin{aligned}& \biggl\vert \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p \nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{1- ( 1-pq ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {} _{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \big\vert {} ^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

 □

Remark 4.9

In Theorem 4.8, for \(p=1\), we have the inequality

$$\begin{aligned}& \biggl\vert \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega} \hbar ( x )\, {}_{\frac{\nu +\omega }{2}}d_{q}x \biggr] -\hbar \biggl( \frac{\nu +\omega}{2} \biggr) \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4} \biggl( \frac{1- ( 1-q ) ^{r+1}}{q [ r+1 ] _{q}} \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \frac{q}{2 ( 1+q ) } \bigl\vert {}_{\nu} \mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{2+q}{2 ( 1+q ) } \bigl\vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{2+q}{2 ( 1+q ) } \big\vert {}^{ \omega} \mathfrak{D}_{q}\hbar ( \nu ) \big\vert ^{s}+ \frac{q}{2 ( 1+q ) } \big\vert {}^{\omega}\mathfrak{D}_{q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] , \end{aligned}$$

which was obtained by Sitthiwirattham et al. [36].

5 \((p,q)\)-Trapezoid inequalities

Now we obtain some \((p,q)\)-trapezoidal inequalities. Let us begin by the following important equality.

Lemma 5.1

For \(\hbar : [ \nu ,\omega ] \subset \mathbb{R} \rightarrow \mathbb{R} \), if \({}_{\nu}\mathfrak{D}_{p,q}\hbar \) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar \) are continuous and integrable mappings over \([ \nu ,\omega ] \), then we have the identity

$$\begin{aligned}& \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{p,q}x+ \int _{\frac{\nu +\omega}{2}}^{p \omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{\frac{\nu +\omega}{2}}d_{p,q}x \biggr] \\& \quad = \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( qt ) \biggl( {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \\& \qquad {} -{}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr) \,d_{p,q}t \biggr]. \end{aligned}$$
(20)

Proof

By using (15) and Definition 2.8 we get

$$\begin{aligned}& \int _{0}^{p} ( qt ) {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \int _{0}^{p} ( qt ) {p} \frac{\hbar ( \frac{q}{p}t\nu + ( 1-\frac{q}{p}t ) \frac{\nu +\omega}{2} ) -\hbar ( t\nu + ( 1-t ) \frac{\nu +\omega}{2} ) }{ ( p-q ) ( \frac{\omega -\nu}{2} ) t}\,d_{p,q}t \\& \quad = \frac{2p^{2}}{\omega -\nu} \Biggl[ \sum_{n=0}^{\infty } \frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n+1}}{p^{n+1}}\nu + \biggl( 1- \frac{q^{n+1}}{p^{n+1}} \biggr) \frac{\nu +\omega}{2} \biggr) \\& \qquad {}-\sum _{n=0}^{\infty }\frac{q^{n}}{p^{n+1}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1-\frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{1}{q}\sum _{n=0}^{ \infty }\frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) -\frac{1}{q}\hbar ( \nu ) \\& \qquad {} -\frac{1}{p}\sum_{n=0}^{\infty }\frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1-\frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = \frac{2p^{2}}{\omega -\nu}\hbar ( \nu ) + \frac{2p^{2}q}{\omega -\nu} \Biggl[ \frac{p-q}{pq}\sum_{n=0}^{\infty } \frac{q^{n}}{p^{n}}\hbar \biggl( \frac{q^{n}}{p^{n}}\nu + \biggl( 1- \frac{q^{n}}{p^{n}} \biggr) \frac{\nu +\omega}{2} \biggr) \Biggr] \\& \quad = -\frac{2p^{2}}{\omega -\nu}\hbar ( \nu ) + \frac{4p}{ ( \omega -\nu ) ^{2}}\int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{ \frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x. \end{aligned}$$

This gives

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( qt ) {}^{\omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{p ( \omega -\nu ) } \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{p,q}x- \frac{1}{2}\hbar ( \nu ) . \end{aligned}$$
(21)

Similarly, by (16) and Definition 2.7 it becomes

$$\begin{aligned}& \frac{\omega -\nu}{4p^{2}} \int _{0}^{p} ( qt ) {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \,d_{p,q}t \\& \quad = \frac{1}{2}\hbar ( \omega ) - \frac{1}{p ( \omega -\nu ) } \int _{ \frac{\nu +\omega }{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{\frac{\nu +\omega}{2}}d_{p,q}x. \end{aligned}$$
(22)

Therefore we establish the required identity (20) by equalities (21) and (22). □

Corollary 5.2

In Lemma 5.1, for \(p=1\), we obtain the new identity

$$\begin{aligned}& \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}} \hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{q}x+ \int _{ \frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \\& \quad = \frac{\omega -\nu}{4} \biggl[ \int _{0}^{1} ( qt ) \biggl( {}_{\nu} \mathfrak{D}_{q}\hbar \biggl( \frac{1-t}{2}\nu + \frac{1+t}{2}\omega \biggr) -{}^{\omega}\mathfrak{D}_{q}\hbar \biggl( \frac{1+t}{2}\nu +\frac{1-t}{2}\omega \biggr) \biggr)\, d_{q}t \biggr] . \end{aligned}$$

This identity helps us to find some estimates of q-trapezoidal inequalities.

Theorem 5.3

If Lemma 5.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \) are convex, then

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{8 [ 2 ] _{p,q} [ 3 ] _{p,q}} \bigl[ \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigl\vert \\& \qquad {}+ \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \omega ) \big\vert \bigr] . \end{aligned}$$
(23)

Proof

By taking the modulus in equality (20) we may write

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( qt ) \biggl( \biggl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert \\& \qquad {} + \biggl\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr\vert \biggr) \,d_{p,q}t \biggr] . \end{aligned}$$
(24)

Since the functions \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert \) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert \) are convex, we have

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl[ \int _{0}^{p} ( qt ) \biggl( \frac{p-t}{2p} \bigl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert +\frac{p+t}{2p} \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigr\vert \biggr) \,d_{p,q}t \\& \qquad {} + \int _{0}^{p} ( qt ) \biggl( \frac{p+t}{2p} \big\vert {} ^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert + \frac{p-t}{2p} \big\vert {} ^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert \biggr) \,d_{p,q}t \biggr] \\& \quad = \frac{\omega -\nu}{8p^{3}} \biggl[ \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {} _{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigl\vert + \biggl( p^{3}q\frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \omega ) \bigl\vert \\& \qquad {}+ \biggl( p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert + \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert \biggr] . \end{aligned}$$

Thus the proof is completed. □

Corollary 5.4

In Theorem 5.3, for \(p=1\), we derive the new q-trapezoidal inequality

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{8 [ 2 ] _{q} [ 3 ] _{q}} \bigl[ q^{2} \bigl\vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{q}+ [ 2 ] _{q} \bigr) \bigl\vert {}_{\nu} \mathfrak{D}_{q}\hbar ( \omega ) \bigl\vert \\& \qquad {}+ \bigl( [ 3 ] _{q}+ [ 2 ] _{q} \bigr) \bigr\vert {}^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert +q^{2} \big\vert {}^{\omega}\mathfrak{D}_{q} \hbar ( \omega ) \big\vert \bigr]. \end{aligned}$$

Example 5.5

Consider \(\hbar : [ 0,1 ] \rightarrow \mathbb{R} \) defined by \(\hbar (x)=x^{3}\). Let also \(q=\frac{1}{3}\) and \(p=\frac{2}{3}\). Then we have the convex functions \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{7x^{2}}{27}\) and \({}^{\omega}\mathfrak{D}_{p,q}\hbar ( x ) =\frac{1}{3} ( 7x^{2}+13x+7 ) \). So the left-hand side of (23) can be written as

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad = \biggl\vert \frac{1}{2}-\frac{3}{2} \biggl[ \frac{18}{2520}+ \frac{551}{2520} \biggr] \biggr\vert \\& \quad = \frac{271}{1680}. \end{aligned}$$

On the other hand, since \({}_{\nu}\mathfrak{D}_{p,q}\hbar ( \nu ) =0\), \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =\frac{7}{27}\), \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \vert =\frac{7}{3}\), and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \omega ) \vert =9\), the right-hand side of inequality (23) becomes

$$\begin{aligned}& \frac{q ( \omega -\nu ) }{8 [ 2 ] _{p,q} [ 3 ] _{p,q}} \bigl[ \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigl\vert \\& \qquad {}+ \bigl( [ 3 ] _{p,q}+ [ 2 ] _{p,q} \bigr) \bigr\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert + \bigl( [ 3 ] _{p,q}- [ 2 ] _{p,q} \bigr) \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \omega ) \big\vert \bigr] \\& \quad = \frac{3}{56} \biggl[ \biggl( \frac{7}{9}+1 \biggr) \frac{7}{27}+ \biggl( \frac{7}{9}+1 \biggr) \frac{7}{3}+ \biggl( \frac{7}{9}-1 \biggr) 9 \biggr] \\& \quad = \frac{3}{56} \biggl[ \frac{16}{9}\frac{7}{27}+ \frac{16}{9} \frac{7}{3}-2 \biggr] \\& \quad = \frac{3}{56} \biggl[ \frac{16}{9}\frac{7}{27}+ \frac{16}{9} \frac{7}{3}-2 \biggr] \\& \quad = \frac{877}{4536}. \end{aligned}$$

It is clear that

$$ \frac{271}{1680}< \frac{877}{4536}. $$

Theorem 5.6

If Lemma 5.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s\geq 1\), then

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{4p^{2} [ 2 ] _{p,q}} \\& \qquad {} \times \biggl[ \biggl( \biggl( \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \biggl( \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \biggl( \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \nu ) \big\vert ^{s}+ \biggl( \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{2 [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega )\big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

Proof

In (24), by the power mean inequality we get

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( qt ) \,d_{p,q}t \biggr) ^{1-\frac{1}{s}} \biggl[ \biggl( \int _{0}^{p} ( qt ) \biggl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu +\frac{p+t}{2p} \omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} ( qt ) \bigg\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p+t}{2p}\nu + \frac{p-t}{2p}\omega \biggr) \bigg\vert ^{s}\,d_{p,q}t \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

By the convexity of the functions \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) we obtain

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{qp^{2}}{ [ 2 ] _{p,q}} \biggr) ^{1-\frac{1}{s}} \\& \qquad {}\times \biggl[ \biggl( \frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu}\mathfrak{D}_{p,q} \hbar ( \nu ) \bigr\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q} + [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {} {}+ \biggl( \frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q}+ [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar ( \nu )\big\vert ^{s} \\& \qquad {}+\frac{1}{2p} \biggl( p^{3}q \frac{ [ 3 ] _{p,q}- [ 2 ] _{p,q}}{ [ 2 ] _{p,q} [ 3 ] _{p,q}} \biggr) \big\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] , \end{aligned}$$

which completes the proof. □

Corollary 5.7

For \(p=1\) in Theorem 5.6, we derive the new q-trapezoidal inequality

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x )\, {}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \biggr\vert \\& \quad \leq \frac{q ( \omega -\nu ) }{4 [ 2 ] _{q}} \biggl[ \biggl( \frac{q^{2} \vert {}_{\nu}\mathfrak{D}_{q}\hbar ( \nu ) \vert ^{s}+ ( [ 3 ] _{q}+ [ 2 ] _{q} ) \vert {} _{\nu}\mathfrak{D}_{q}\hbar ( \omega ) \vert ^{s}}{2 [ 3 ] _{q}} \biggr) ^{ \frac{1}{s}} \\& \qquad {}+ \biggl( \frac{ ( [ 3 ] _{q}+ [ 2 ] _{q} ) \vert {}^{\omega}\mathfrak{D}_{q}\hbar ( \nu ) \vert ^{s}+q^{2} \vert {} ^{\omega}\mathfrak{D}_{q}\hbar ( \omega ) \vert ^{s}}{2 [ 3 ] _{q}} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

Theorem 5.8

If Lemma 5.1holds and \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex for \(s>1\), then

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{ ( qp ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {} _{ \nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{ \omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}}\big\vert {} ^{\omega}\mathfrak{D}_{p,q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr], \end{aligned}$$

where \(s^{-1}+r^{-1}=1\).

Proof

Applying the Hölder inequality to (24), we establish

$$\begin{aligned}& \biggl\vert \frac{\hbar (\nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \int _{0}^{p} ( qt ) ^{r}\,d_{p,q}t \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \int _{0}^{p} \biggl\vert {}_{\nu} \mathfrak{D}_{p,q}\hbar \biggl( \frac{p-t}{2p}\nu + \frac{p+t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \\& \qquad {}+ \biggl( \int _{0}^{p} \biggl\vert {}^{\omega} \mathfrak{D}_{p,q} \hbar \biggl( \frac{p+t}{2p}\nu +\frac{p-t}{2p}\omega \biggr) \biggr\vert ^{s}\,d_{p,q}t \biggr) ^{ \frac{1}{s}} \biggr] . \end{aligned}$$

Since the functions \(\vert {}_{\nu}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) and \(\vert {}^{\omega}\mathfrak{D}_{p,q}\hbar \vert ^{s}\) are convex, we have

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{p ( \omega -\nu ) } \biggl[ \int _{p\nu + ( 1-p ) \frac{\nu +\omega}{2}}^{\frac{\nu +\omega}{2}}\hbar ( x ) \,{}^{\frac{\nu +\omega}{2}}d_{p,q}x+ \int _{ \frac{\nu +\omega}{2}}^{p\omega + ( 1-p ) \frac{\nu +\omega}{2}}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{p,q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4p^{2}} \biggl( \frac{ ( qp ) ^{r+1}}{q [ r+1 ] _{p,q}} \biggr) ^{\frac{1}{r}} \\& \qquad {}\times \biggl[ \biggl( \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {} _{\nu} \mathfrak{D}_{p,q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \bigl\vert {}_{ \nu}\mathfrak{D}_{p,q}\hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{p^{3}+p^{2}q+p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \nu ) \big\vert ^{s}+ \frac{p^{3}+p^{2}q-p^{2}}{2p [ 2 ] _{p,q}} \big\vert {}^{\omega} \mathfrak{D}_{p,q}\hbar ( \omega )\big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr] . \end{aligned}$$

The proof is ended. □

Corollary 5.9

For \(p=1\) in Theorem 5.8, we derive the new q-trapezoidal inequality

$$\begin{aligned}& \biggl\vert \frac{\hbar ( \nu ) +\hbar ( \omega ) }{2}- \frac{1}{\omega -\nu} \biggl[ \int _{\nu}^{\frac{\nu +\omega}{2}}\hbar ( x )\, {}^{ \frac{\nu +\omega}{2}}d_{q}x+ \int _{\frac{\nu +\omega}{2}}^{\omega}\hbar ( x ) \,{}_{ \frac{\nu +\omega}{2}}d_{q}x \biggr] \biggr\vert \\& \quad \leq \frac{\omega -\nu}{4} \biggl( \frac{q^{r}}{ [ r+1 ] _{q}} \biggr) ^{\frac{1}{r}} \biggl[ \biggl( \frac{q}{2 ( 1+q ) } \bigl\vert {}_{\nu} \mathfrak{D}_{q}\hbar ( \nu ) \bigr\vert ^{s}+ \frac{2+q}{2 ( 1+q ) } \bigl\vert {} _{\nu}\mathfrak{D}_{q} \hbar ( \omega ) \bigr\vert ^{s} \biggr) ^{\frac{1}{s}} \\& \qquad {}+ \biggl( \frac{2+q}{2 ( 1+q ) } \big\vert {} ^{ \omega} \mathfrak{D}_{q}\hbar ( \nu ) \big\vert ^{s}+ \frac{q}{2 ( 1+q ) }\big\vert {}^{\omega}\mathfrak{D}_{q} \hbar ( \omega ) \big\vert ^{s} \biggr) ^{\frac{1}{s}} \biggr]. \end{aligned}$$

6 Conclusions

In the present research work, we analyzed a new variant of Hermite–Hadamard inequality in relation to convex functions in the framework of \(( p,q ) \)-calculus. Moreover, we derived some new estimates for \(( p,q ) \)-midpoint and \(( p,q ) \)-trapezoidal inequalities for \(( p,q ) \)-differentiable convex functions using the left and right \(( p,q ) \)-integrals. The upcoming researchers can obtain similar inequalities for different kinds of convexity and coordinated convexity in the context of \(( p,q ) \)-calculus theory in their future research works.