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New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions

Abstract

In this research, we derive two generalized integral identities involving the \(q^{\varkappa _{2}}\)-quantum integrals and quantum numbers, the results are then used to establish some new quantum boundaries for quantum Simpson’s and quantum Newton’s inequalities for q-differentiable preinvex functions. Moreover, we obtain some new and known Simpson’s and Newton’s type inequalities by considering the limit \(q\rightarrow 1^{-}\) in the key results of this paper.

Introduction

In the domain of q-analysis, much work is being carried out initiating from Euler in order to attain adeptness in mathematics that constructs quantum computing q-calculus considered as a relationship between physics and mathematics. In different areas of mathematics, it has numerous applications such as combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, and other sciences, mechanics, the theory of relativity, and quantum theory [15]. Apparently, Euler invented this important mathematics branch. He used the q parameter in Newton’s work with infinite series. Later, in a methodical manner, the q-calculus that knew without limit calculus was firstly given by Jackson [2]. In 1908–1909, the general form of the q-integral and q-difference operator is defined by Jackson [4]. In 1969, for the first time Agarwal [6] defined the q-fractional derivative. In 1966–1967, Al-Salam [7] introduced a q-analog of the q-fractional integral and q-Riemann–Liouville fractional. In 2004, Rajkovic [8] gave a definition of the Riemann-type q-integral which was generalized to Jackson q-integral. In 2013, Tariboon introduced \(_{\varkappa _{1}}D_{q}\)-difference operator [9].

Many integral inequalities are well known in classical analysis, such as the Hölder inequality, Simpson’s inequality, Newton’s inequality, the Hermite–Hadamard inequality and the Ostrowski inequality, Cauchy–Bunyakovsky–Schwarz, Gruss, Gruss–Cebysev, and other integral inequalities have been proved and applied in the setup of q-calculus using classical convexity. Many mathematicians have done studies in q-calculus analysis; the interested reader can check [1024].

Thomas Simpson has developed crucial methods for the numerical integration and estimation of definite integrals considered as Simpson’s rule during (1710–1761). Nevertheless, a similar approximation was used by Kepler almost 10 decades earlier, so it is also known as Kepler’s rule. Simpson’s rule includes the three-point Newton–Cotes quadrature rule, so an estimation based on a three steps quadratic kernel is sometimes called a Newton-type result.

  1. 1.

    Simpson’s \(1/3\) formula is given as

    $$ \frac{1}{\varkappa _{2}-\varkappa _{1}} \int _{\varkappa _{1}}^{ \varkappa _{2}}\varphi ( \varkappa ) \,d\varkappa \approx \frac{1}{6} \biggl[ \varphi ( \varkappa _{1} ) +4\varphi \biggl( \frac{\varkappa _{1}+\varkappa _{2}}{2} \biggr) +\varphi ( \varkappa _{2} ) \biggr]. $$
  2. 2.

    Simpson’s \(3/8\) formula is given as follows

    $$ \frac{1}{\varkappa _{2}-\varkappa _{1}} \int _{\varkappa _{1}}^{ \varkappa _{2}}\varphi ( \varkappa ) \,d\varkappa \approx \frac{1}{8} \biggl[ \varphi ( \varkappa _{1} ) +3\varphi \biggl( \frac{2\varkappa _{1}+\varkappa _{2}}{3} \biggr) +3\varphi \biggl( \frac{\varkappa _{1}+2\varkappa _{2}}{3} \biggr) + \varphi ( \varkappa _{2} ) \biggr]. $$

There are a large number of estimations related to these quadrature rules in the literature, one of them is the following estimation known as Simpson’s inequality:

Theorem 1

Let \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) be a four times continuously differentiable function on \(( \varkappa _{1},\varkappa _{2} ) \), and

$$ \bigl\Vert \varphi ^{ ( 4 ) } \bigr\Vert _{\infty }= \sup _{\varkappa \in ( \varkappa _{1},\varkappa _{2} ) } \bigl\vert \varphi ^{ ( 4 ) } ( \varkappa ) \bigr\vert < \infty. $$

Then we have the following inequality:

$$\begin{aligned} &\biggl\vert \frac{1}{3} \biggl[ \frac{\varphi ( \varkappa _{1} ) +\varphi ( \varkappa _{2} ) }{2}+2\varphi \biggl( \frac{\varkappa _{1}+\varkappa _{2}}{2} \biggr) \biggr] - \frac{1}{\varkappa _{2}-\varkappa _{1}} \int _{\varkappa _{1}}^{ \varkappa _{2}}\varphi ( \varkappa ) \,d\varkappa \biggr\vert \\ &\quad \leq \frac{1}{2880} \bigl\Vert \varphi ^{ ( 4 ) } \bigr\Vert _{\infty } ( \varkappa _{2}-\varkappa _{1} ) ^{4}. \end{aligned}$$

In the recent era, Simpson’s type of inequalities has been emphasized by many authors for numerous types of functions. Convexity is useful and potent for solving different problems that appear within various branches of applied and pure mathematics. For an instance, Dragomir demonstrated novel Simpson’s type consequences and their applications to quadratic formulas in numerical integration in [25]. Furthermore, Alomari [26] has presented Simpson’s type of inequalities for s-convex functions. The refinements of Simpson’s type of inequalities depending on convexity have been visualized by Sarikaya et al. in [27]. For the further studies of this area, one can consult [2830].

The main objective of this paper is to study Newton’s and Simpson’s type inequalities for preinvex functions by using the notions of quantum calculus.

Preliminaries of q-calculus and some inequalities

The basic notions and findings which are needed in the sequel to prove our crucial results are reviewed in this section. Throughout this paper, we assume that \(\varkappa _{1}<\varkappa _{2}\) and \(0< q<1\). Let ω be a nonempty closed set in \(\mathbb{R} ^{n}\), \(\varphi:\omega \rightarrow \mathbb{R} \) be a continuous function, and \(\eta (\cdot,\cdot):\omega \times \omega \rightarrow \mathbb{R} ^{n}\) be a continuous bifunction.

Definition 1

([13])

A set ω is called an invex set respecting a bifunction \(\eta (\cdot,\cdot) \) if

$$ \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) \in \omega , \quad\forall \varkappa _{1}, \varkappa _{2}\in \omega, t\in [ 0,1 ]. $$

The invex set ω is more commonly referred to as η-connected set.

Definition 2

([13])

A mapping φ is called a preinvex respecting an arbitrary bifunction \(\eta (\cdot,\cdot) \) if the following inequality holds:

$$ \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \leq t\varphi ( \varkappa _{1} ) + ( 1-t ) \varphi ( \varkappa _{2} ),\quad\forall \varkappa _{1},\varkappa _{2}\in \omega ,t\in [ 0,1 ]. $$

The function φ is called preconcave if −φ is preinvex.

Remark 1

If we set \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\), then the definition of preinvex function reduces to the definition of a convex function given below;

$$ \varphi \bigl( \varkappa _{2}+t ( \varkappa _{1}-\varkappa _{2} ) \bigr) \leq t\varphi ( \varkappa _{1} ) + ( 1-t ) \varphi ( \varkappa _{2} ) ,\quad \forall \varkappa _{1},\varkappa _{2}\in \omega ,t\in [ 0,1 ]. $$

Now we present some well-known concepts and theorems for q-derivatives and q-integrals of a function φ on \([ \varkappa _{1},\varkappa _{2} ] \).

Definition 3

([5, 9])

We assume that a function \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) is continuous. Then the \(q_{\varkappa _{1}}\)-derivative of φ at \(\varkappa \in [ \varkappa _{1},\varkappa _{2} ] \) is defined by

$$ {} _{\varkappa _{1}}D_{q}\varphi ( \varkappa ) = \frac{\varphi ( \varkappa ) -\varphi ( q\varkappa + ( 1-q ) \varkappa _{1} ) }{ ( 1-q ) ( \varkappa -\varkappa _{1} ) }, \quad\varkappa \neq \varkappa _{1}. $$
(2.1)

Since \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) is a continuous function, we can define

$$ {} _{\varkappa _{1}}D_{q}\varphi ( \varkappa _{1} ) =\underset{\varkappa \rightarrow \varkappa _{1}}{\lim }{} _{\varkappa _{1}}D_{q} \varphi ( \varkappa ) . $$

The function φ is called \(q_{\varkappa _{1}}\)-differentiable on \([ \varkappa _{1},\varkappa _{2} ] \) if \({} _{\varkappa _{1}}D_{q}\varphi ( \varkappa ) \) exists for all \(\varkappa \in [ \varkappa _{1},\varkappa _{2} ] \). If we assume \(\varkappa _{1}=0\) in (2.1), then \({} _{0}D_{q}\varphi ( \varkappa ) =D_{q}\varphi ( \varkappa ) \), where \({} D_{q}\varphi ( \varkappa ) \) is the familiar q-derivative of φ at \(\varkappa \in [ 0,\varkappa _{2} ] \) defined as follows (see [5]):

$$ {} D_{q}\varphi ( \varkappa ) = \frac{\varphi ( \varkappa ) -\varphi ( q\varkappa ) }{ ( 1-q ) \varkappa },\quad \varkappa \neq 0. $$
(2.2)

Definition 4

([31])

We assume that a function \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) is continuous. Then the \(q_{\varkappa _{1}}\)-definite integral on \([ \varkappa _{1},\varkappa _{2} ] \) is defined by

$$ \int _{\varkappa _{1}}^{\varkappa }\varphi ( t ) {} _{\varkappa _{1}}\,d_{q}t= ( 1-q ) ( \varkappa -\varkappa _{1} ) \sum_{n=0}^{\infty }q^{n} \varphi \bigl( q^{n}\varkappa + \bigl( 1-q^{n} \bigr) \varkappa _{1} \bigr) ,\quad\varkappa \in [ \varkappa _{1},\varkappa _{2} ]. $$
(2.3)

Remark 2

If \(\varkappa _{1}=0\) in (2.3), then \(\int _{0}^{\varkappa }\varphi ( t ) {} _{0}\,d_{q}t=\int _{0}^{\varkappa }\varphi ( t ) {} \,d_{q}t\), where \(\int _{0}^{\varkappa }\varphi ( t ) {} \,d_{q}t\) is the familiar q-definite integral (see, [5]) on \([ 0,\varkappa ] \) defined by

$$ \int _{0}^{\varkappa }\varphi ( t ) {} _{0}\,d_{q}t= \int _{0}^{\varkappa }\varphi ( t ) {} \,d_{q}t= ( 1-q ) \varkappa \sum _{n=0}^{\infty }q^{n} \varphi \bigl( q^{n}\varkappa \bigr). $$
(2.4)

Definition 5

If \(c\in ( \varkappa _{1},\varkappa ) \), then the q-definite integral on \([ c,\varkappa ] \) is expressed as

$$ \int _{c}^{\varkappa }\varphi ( t ) {} _{\varkappa _{1}}\,d_{q}t= \int _{\varkappa _{1}}^{\varkappa }\varphi ( t ) {} _{\varkappa _{1}}\,d_{q}t- \int _{\varkappa _{1}}^{c}\varphi ( t ) {} _{\varkappa _{1}}\,d_{q}t. $$
(2.5)

Alp et al. [9] proved the following q-Hermite–Hadamard inequality:

Theorem 2

(\(q_{a}\)-Hermite–Hadamard inequality)

We assume that a function \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) is convex differentiable on \([ \varkappa _{1},\varkappa _{2} ] \) and \(0< q<1\). Then we have the following inequality:

$$ \varphi \biggl( \frac{q\varkappa _{1}+\varkappa _{2}}{ [ 2 ] _{q}} \biggr) \leq \frac{1}{\varkappa _{2}-\varkappa _{1}} \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} _{\varkappa _{1}}\,d_{q}\varkappa \leq \frac{q\varphi ( \varkappa _{1} ) +\varphi ( \varkappa _{2} ) }{ [ 2 ] _{q}}. $$

On the other hand, Bermudo et al. [32] gave the following new definitions of a quantum integral and derivative. In the same paper, the authors proved a new variant of the quantum Hermite–Hadamard type inequality linked to their newly defined quantum integral.

Definition 6

([32])

Let \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) be a continuous function. Then the \(q^{\varkappa _{2}}\)-definite integral on \([ \varkappa _{1},\varkappa _{2} ] \) is given by

$$\begin{aligned} \int _{\varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa &= ( 1-q ) ( \varkappa _{2}-\varkappa _{1} ) \sum _{n=0}^{\infty }q^{n}\varphi \bigl( q^{n}\varkappa _{1}+ \bigl( 1-q^{n} \bigr) \varkappa _{2} \bigr) \\ &= ( \varkappa _{2}- \varkappa _{1} ) \int _{0}^{1}\varphi \bigl( t \varkappa _{1}+ ( 1-t ) \varkappa _{2} \bigr) {} \,d_{q}t. \end{aligned}$$

Definition 7

([32])

Let \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) be a continuous function. Then the \(q^{\varkappa _{2}}\)-derivative of φ at \(\varkappa \in [ \varkappa _{1},\varkappa _{2} ] \) is given by

$$ {} ^{\varkappa _{2}}D_{q}\varphi ( \varkappa )= \frac{\varphi ( q\varkappa + ( 1-q ) \varkappa _{2} ) -\varphi ( \varkappa ) }{ ( 1-q ) ( \varkappa _{2}-\varkappa ) }, \quad \varkappa \neq \varkappa _{2}. $$

Theorem 3

(\(q^{b}\)-Hermite–Hadamard inequality, [32])

We consider that a function \(\varphi: [ \varkappa _{1},\varkappa _{2} ] \rightarrow \mathbb{R} \) is convex differentiable on \([ \varkappa _{1},\varkappa _{2} ] \) and \(0< q<1\). Then we have the following inequality:

$$ \varphi \biggl( \frac{\varkappa _{1}+q\varkappa _{2}}{ [ 2 ] _{q}} \biggr) \leq \frac{1}{\varkappa _{2}-\varkappa _{1}} \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa \leq \frac{\varphi ( \varkappa _{1} ) +q\varphi ( \varkappa _{2} ) }{ [ 2 ] _{q}}. $$
(2.6)

Let us set the following notations:

$$ [ n ] _{q}= \textstyle\begin{cases} \frac{q^{n}-1}{q-1}=\sum_{i=0}^{n-1}q^{i}, & n\in N, \\ \frac{q^{n}-1}{q-1}, & n\in C,\end{cases} $$

and

$$ ( 1-t ) _{q}^{n}= ( t,q ) _{n}=\prod _{i=0}^{n-1} \bigl( 1-q^{i}t \bigr). $$
(2.7)

Lemma 1

([9])

For \(a\in \mathbb{R} \backslash \{ -1 \} \), the following formula holds:

$$ \int _{\varkappa _{1}}^{\varkappa } ( t-\varkappa _{1} ) ^{a}{} _{\varkappa _{1}}\,d_{q}t= \frac{ ( \varkappa -\varkappa _{1} ) ^{a+1}}{ [ a+1 ] _{q}}. $$
(2.8)

Quantum integral identities

In this section, we will prove two equalities which will help us to obtain our main results.

Lemma 2

Let \(\varphi:I= [ \varkappa _{2}+\eta ( \varkappa _{1}, \varkappa _{2} ),\varkappa _{2} ] \rightarrow \mathbb{R} \) be a \(q^{\varkappa _{2}}\)-differentiable function on I with \(-\eta ( \varkappa _{1},\varkappa _{2} ) =\eta ( \varkappa _{2},\varkappa _{1} ) >0\) and \(0< q<1\). If \({}^{\varkappa _{2}}D_{q} \varphi \) is continuous and integrable on I, then the following identity holds for \(q^{\varkappa _{2}}\)-integrals:

$$\begin{aligned} &\frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) } \int _{ \varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{ \varkappa _{2}}\varphi ( \varkappa ) {}^{\varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{}-\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + [ 4 ] _{q} q^{2} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \\ &\quad=q \eta ( \varkappa _{2},\varkappa _{1} ) \int _{0}^{1} \Psi _{q} ( t ) {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) \bigr) \,d_{q} t, \end{aligned}$$
(3.1)

where

$$ \Psi _{q} ( t ) = \textstyle\begin{cases} t-\frac{1}{ [ 6 ] _{q}}&\textit{if }t \in [ 0,\frac{1}{ [ 2 ] _{q}} ), \\ t-\frac{ [ 5 ] _{q}}{ [ 6 ] _{q}}& \textit{if }t\in [ \frac{1}{ [ 2 ] _{q}},1 ].\end{cases} $$

Proof

Using the basic properties of q-integral and the definition of \(\Psi _{q} ( t ) \), we have

$$\begin{aligned} & \int _{0}^{1}\Psi _{q} ( t ) {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q} t \\ &\quad=\frac{ [ 5 ] _{q}-1}{ [ 6 ] _{q}} \int _{0}^{ \frac{1}{ [ 2 ] _{q}}}{}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t+ \int _{0}^{1}t{}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t \\ &\qquad{}-\frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \int _{0}^{1} {}^{ \varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t. \end{aligned}$$
(3.2)

From Definition 7, we have

$$ ^{\varkappa _{2}}D_{q}\varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) \bigr) = \frac{\varphi ( \varkappa _{2}+tq\eta ( \varkappa _{1},\varkappa _{2} ) ) -\varphi ( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) ) }{ ( 1-q ) t\eta ( \varkappa _{2},\varkappa _{1} ) }. $$

Now, we compute the integrals on the right side of (3.2). Using Definition 6, we obtain

$$\begin{aligned} & \int _{0}^{\frac{1}{ [ 2 ] _{q}}}{}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t \end{aligned}$$
(3.3)
$$\begin{aligned} &\quad= \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \frac{\varphi ( \varkappa _{2}+tq\eta ( \varkappa _{1},\varkappa _{2} ) ) -\varphi ( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) ) }{ ( 1-q ) t\eta ( \varkappa _{2},\varkappa _{1} ) }\,d_{q}t \\ &\quad=\frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) } \Biggl[ \sum_{n=0}^{\infty } \varphi \biggl( \varkappa _{2}+ \frac{q^{n+1}}{ [ 2 ] _{q}}\eta ( \varkappa _{1},\varkappa _{2} ) \biggr) -\sum _{n=0}^{\infty }\varphi \biggl( \varkappa _{2}+ \frac{q^{n}}{ [ 2 ] _{q}}\eta ( \varkappa _{1},\varkappa _{2} ) \biggr) \Biggr] \\ &\quad=\frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) } \biggl[ \varphi ( \varkappa _{2} ) -\varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) \biggr], \\ &\int _{0}^{1}{}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t \eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t= \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) } \bigl[ \varphi ( \varkappa _{2} ) -\varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) \bigr], \end{aligned}$$
(3.4)

and

$$\begin{aligned} & \int _{0}^{1}t{}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t \eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t \\ &\quad= \int _{0}^{1} \frac{\varphi ( \varkappa _{2}+tq\eta ( \varkappa _{1},\varkappa _{2} ) ) -\varphi ( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) ) }{ ( 1-q ) \eta ( \varkappa _{2},\varkappa _{1} ) }\,d_{q}t \\ &\quad=\frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) } \Biggl[ \frac{1-q}{q}\sum _{n=0}^{\infty }q^{n}\varphi \bigl( \varkappa _{2}+q^{n} \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) - \frac{\varphi ( \varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) ) }{q} \Biggr] \\ &\quad=\frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) } \biggl[ \frac{1}{q\eta ( \varkappa _{2},\varkappa _{1} ) } \int _{ \varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{ \varkappa _{2}}\varphi ( \varkappa ) {}^{\varkappa _{2}} \,d_{q} \varkappa -\frac{\varphi ( \varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) ) }{q} \biggr]. \end{aligned}$$
(3.5)

Finally, by substituting (3.3)–(3.5) in (3.2) and multiplying the resultant equality by \(\eta ( \varkappa _{2},\varkappa _{1} ) \), we obtain the required identity, which completes the proof. □

Remark 3

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Lemma 2, then we have the following identity:

$$\begin{aligned} & \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{\varkappa _{1}}^{\varkappa _{2}}\varphi ( t ) {} ^{\varkappa _{2}}\,d_{q}t-\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +q^{2} [ 4 ] _{q}\varphi \biggl( \frac{\varkappa _{1}+q\varkappa _{2}}{ [ 2 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \\ &\quad =q ( \varkappa _{2}-\varkappa _{1} ) \int _{0}^{1} \Psi _{q} ( t ) {} ^{\varkappa _{2}}D_{q}\varphi \bigl( t\varkappa _{1}+ ( 1-t ) \varkappa _{2} \bigr){} \,d_{q}t, \end{aligned}$$
(3.6)

where

$$ \Psi _{q} ( t ) = \textstyle\begin{cases} t-\frac{1}{ [ 6 ] _{q}}&\text{if }t \in [ 0,\frac{1}{ [ 2 ] _{q}} ), \\ t-\frac{ [ 5 ] _{q}}{ [ 6 ] _{q}}& \text{if }t\in [ \frac{1}{ [ 2 ] _{q}},1 ],\end{cases} $$

which is given by Ali et al. in [33].

Lemma 3

Let \(\varphi:I= [ \varkappa _{2}+\eta ( \varkappa _{1}, \varkappa _{2} ),\varkappa _{2} ] \rightarrow \mathbb{R} \) be a \(q^{\varkappa _{2}}\)-differentiable function on I with \(-\eta ( \varkappa _{1},\varkappa _{2} ) =\eta ( \varkappa _{2},\varkappa _{1} ) >0\) and \(0< q<1\). If \({}^{\varkappa _{2}}D_{q} \varphi \) is continuous and integrable on I, then one has the identity

$$\begin{aligned} &\frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) } \int _{ \varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{ \varkappa _{2}}\varphi ( \varkappa ) {}^{\varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{}-\frac{1}{ [ 8 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + \frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) \\ &\qquad{} + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}} \varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+ [ 2 ] _{q}\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \\ &\quad= q\eta ( \varkappa _{2},\varkappa _{1} ) \int _{0}^{1} \Delta _{q} ( t ) {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) \bigr) \,d_{q} t, \end{aligned}$$

where

$$ \Delta _{q} ( t ) = \textstyle\begin{cases} t-\frac{1}{ [ 8 ] _{q}}&\textit{if }t \in [ 0,\frac{1}{ [ 3 ] _{q}} ), \\ t-\frac{1}{ [ 2 ] _{q}}&\textit{if }t \in [ \frac{1}{ [ 3 ] _{q}}, \frac{ [ 2 ] _{q}}{ [ 3 ] _{q}} ), \\ t-\frac{ [ 7 ] _{q}}{ [ 8 ] _{q}}& \textit{if }t\in [ \frac{ [ 2 ] _{q}}{ [ 3 ] _{q}},1 ].\end{cases} $$

Proof

By the fundamental properties of q-integrals and the definition of \(\Delta _{q} ( t ) \), we obtain

$$\begin{aligned} & \int _{0}^{1}\Delta _{q} ( t ) {}^{\varkappa _{2}}D_{q} \bigl( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) \bigr) \,d_{q}t\\ &\quad= \frac{ [ 8 ] _{q}- [ 2 ] _{q}}{ [ 8 ] _{q} [ 2 ] _{q}} \int _{0}^{\frac{1}{ [ 3 ] _{q}}} {}^{ \varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t \\ &\qquad{}+ \frac{ [ 7 ] _{q} [ 2 ] _{q}- [ 8 ] _{q}}{ [ 8 ] _{q} [ 2 ] _{q}} \int _{0}^{ \frac{ [ 2 ] _{q}}{ [ 3 ] _{q}}}{}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \,d_{q}t \\ &\qquad{}+ \int _{0}^{1} \biggl( t- \frac{ [ 7 ] _{q}}{ [ 8 ] _{q}} \biggr) {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) \,d_{q}t. \end{aligned}$$

Following arguments similar to those in the proof of Lemma 2, the required identity can be proved. □

Remark 4

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Lemma 3, then we have the following identity:

$$\begin{aligned} & \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{\varkappa _{1}}^{\varkappa _{2}}\varphi ( t ) {} ^{\varkappa _{2}}\,d_{q}t{} \\ &\qquad{} -\frac{1}{ [ 8 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) + \frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}} \varphi \biggl( \frac{\varkappa _{1}+q [ 2 ] _{q}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{1}+q^{2}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \\ &\quad =q ( \varkappa _{2}-\varkappa _{1} ) \int _{0}^{1} \Delta _{q} ( t ) {} ^{\varkappa _{2}}D_{q}\varphi \bigl( t\varkappa _{1}+ ( 1-t ) \varkappa _{2} \bigr){} \,d_{q}t, \end{aligned}$$

where

$$ \Delta _{q} ( t ) = \textstyle\begin{cases} t-\frac{1}{ [ 8 ] _{q}}&\text{if }t \in [ 0,\frac{1}{ [ 3 ] _{q}} ), \\ t-\frac{1}{ [ 2 ] _{q}}&\text{if }t \in [ \frac{1}{ [ 3 ] _{q}}, \frac{ [ 2 ] _{q}}{ [ 3 ] _{q}} ), \\ t-\frac{ [ 7 ] _{q}}{ [ 8 ] _{q}}& \text{if }t\in [ \frac{ [ 2 ] _{q}}{ [ 3 ] _{q}},1 ], \end{cases} $$

which is proved by Ali et al. in [33].

Main results

In this section, we present some new Simpson’s and Newton’s type inequalities for preinvex functions by using Lemma 2 and Lemma 3, respectively. For brevity, we start this section with some notations which will be used in our new results:

$$\begin{aligned} A_{1} ( q ) ={}&\frac{2q^{2} [ 2 ] _{q}^{2}+ [ 6 ] _{q}^{2} ( [ 6 ] _{q}- [ 3 ] _{q} ) }{ [ 2 ] _{q}^{3} [ 3 ] _{q} [ 6 ] _{q}^{3}}, \end{aligned}$$
(4.1)
$$\begin{aligned} B_{1} ( q ) ={}&2 \frac{q [ 3 ] _{q} [ 6 ] _{q}-q^{2}}{ [ 2 ] _{q} [ 3 ] _{q} [ 6 ] _{q}^{3}}+\frac{1}{ [ 2 ] _{q}^{3}} \biggl( \frac{q+q^{2}}{ [ 3 ] _{q}}-\frac{q^{2}+2q}{ [ 6 ] _{q}} \biggr), \end{aligned}$$
(4.2)
$$\begin{aligned} A_{2} ( q ) ={}&\frac{2q^{2} [ 5 ] _{q}^{3}}{ [ 2 ] _{q} [ 3 ] _{q} [ 6 ] _{q}^{3}}+ \frac{ [ 6 ] _{q} ( 1+ [ 2 ] _{q}^{3} ) - [ 3 ] _{q} [ 5 ] _{q} ( 1+ [ 2 ] _{q}^{2} ) }{ [ 2 ] _{q}^{3} [ 3 ] _{q} [ 6 ] _{q}}, \end{aligned}$$
(4.3)
$$\begin{aligned} B_{2} ( q ) ={}&2 \frac{q [ 5 ] _{q}^{2} [ 6 ] _{q} [ 3 ] _{q}-q^{2} [ 5 ] _{q}^{3}}{ [ 2 ] _{q} [ 3 ] _{q} [ 6 ] _{q}^{3}}+ \frac{q^{2}}{ [ 2 ] _{q} [ 3 ] _{q}}- \frac{q [ 5 ] _{q}}{ [ 2 ] _{q} [ 6 ] _{q}} \end{aligned}$$
(4.4)
$$\begin{aligned} &{}-\frac{1}{ [ 2 ] _{q}^{3}} \biggl[ \frac{ [ 5 ] _{q} ( 2q+q^{2} ) }{ [ 6 ] _{q}}- \frac{q+q^{2}}{ [ 3 ] _{q}} \biggr], \\ A_{3} ( q ) ={}&\frac{2q^{2} [ 3 ] _{q}^{3}+ [ 8 ] _{q}^{2} ( [ 8 ] _{q} [ 2 ] _{q}- [ 3 ] _{q}^{2} ) }{ [ 8 ] _{q}^{3} [ 3 ] _{q}^{4} [ 2 ] _{q}}, \end{aligned}$$
(4.5)
$$\begin{aligned} B_{3} ( q ) ={}&2 \frac{q [ 8 ] _{q} [ 3 ] _{q}-q^{2}}{ [ 8 ] _{q}^{3} [ 2 ] _{q} [ 3 ] _{q}}+ \frac{ [ 3 ] _{q}^{2}- [ 2 ] _{q}}{ [ 3 ] _{q}^{4} [ 2 ] _{q}} \end{aligned}$$
(4.6)
$$\begin{aligned} &{}+ \frac{1- [ 3 ] _{q} [ 2 ] _{q}}{ [ 8 ] _{q} [ 3 ] _{q}^{2} [ 2 ] _{q}}, \\ A_{4} ( q ) ={}&\frac{2q^{2}}{ [ 2 ] _{q}^{4} [ 3 ] _{q}}+ \frac{ [ 2 ] _{q}^{2} ( 1+ [ 2 ] _{q}^{3} ) - [ 3 ] _{q}^{2} ( 1+ [ 2 ] _{q}^{2} ) }{ [ 3 ] _{q}^{4} [ 2 ] _{q}^{2}}, \end{aligned}$$
(4.7)
$$\begin{aligned} B_{4} ( q ) ={}&\frac{2q}{ [ 2 ] _{q}^{3}}- \frac{q}{ [ 3 ] _{q}^{2}}- \frac{q^{2}}{ [ 3 ] _{q}^{2}}-A_{4} ( q ), \end{aligned}$$
(4.8)
$$\begin{aligned} A_{5} ( q ) ={}&\frac{2q^{2} [ 7 ] _{q}^{3}}{ [ 8 ] _{q}^{3} [ 2 ] _{q} [ 3 ] _{q}}+ \frac{ [ 2 ] _{q} [ 8 ] _{q} ( [ 2 ] _{q}^{3}+ [ 3 ] _{q}^{3} ) - [ 7 ] _{q} [ 3 ] _{q}^{2} ( [ 2 ] _{q}^{2}+ [ 3 ] _{q}^{2} ) }{ [ 3 ] _{q}^{4} [ 8 ] _{q} [ 2 ] _{q}}, \end{aligned}$$
(4.9)

and

$$\begin{aligned} B_{5} ( q ) ={}&2 \frac{q [ 7 ] _{q}^{2} [ 8 ] _{q} [ 3 ] _{q}-q^{2} [ 7 ] _{q}^{3}}{ [ 8 ] _{q}^{3} [ 2 ] _{q} [ 3 ] _{q}}+ \frac{q^{2}}{ [ 2 ] _{q} [ 3 ] _{q}}- \frac{q [ 7 ] _{q}}{ [ 2 ] _{q} [ 8 ] _{q}} \\ &{}+ \frac{ [ 2 ] _{q} ( [ 3 ] _{q}^{2}- [ 2 ] _{q}^{2} ) }{ [ 3 ] _{q}^{4}}- \frac{ ( q+q^{2} ) [ 7 ] _{q} [ 2 ] _{q}}{ [ 3 ] _{q}^{2} [ 8 ] _{q}}. \end{aligned}$$
(4.10)

Simpson’s type inequalities

In this subsection, we will prove some quantum Simpson’s type inequalities for preinvex functions.

Theorem 4

We assume that the conditions of Lemma 2hold. If \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert \) is preinvex and integrable on I, then the following inequality holds for \(q^{\varkappa _{2}}\)-integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{} -\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + [ 4 ] _{q} q^{2} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \bigl\{ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert \bigl[ A_{1} ( q ) +A_{2} ( q ) \bigr] + \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert \bigl[ B_{1} ( q ) +B_{2} ( q ) \bigr] \bigr\} , \end{aligned}$$
(4.11)

where \(0< q<1\) and \(A_{1} ( q ) \), \(A_{2} ( q ) \), \(B_{1} ( q ) \), \(B_{2} ( q ) \) are given as in (4.1)(4.4), respectively.

Proof

By taking the modulus in Lemma 2 and using the properties of the modulus, we obtain

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{} -\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + [ 4 ] _{q} q^{2} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \int _{0}^{ \frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert \bigl\vert {} ^{ \varkappa _{2}}D_{q}\varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) \bigr) \bigr\vert \,d_{q}t \\ &\qquad{}+q\eta ( \varkappa _{2},\varkappa _{1} ) \int _{ \frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1},\varkappa _{2} ) \bigr) \bigr\vert \,d_{q}t. \end{aligned}$$
(4.12)

Since \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert \) is preinvex, by Lemma 1, we get

$$\begin{aligned} & \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \bigr\vert \,d_{q}t \\ &\quad\leq \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert \biggl( \frac{2q^{2} [ 2 ] _{q}^{2}+ [ 6 ] _{q}^{2} ( [ 6 ] _{q}- [ 3 ] _{q} ) }{ [ 2 ] _{q}^{3} [ 3 ] _{q} [ 6 ] _{q}^{3}} \biggr) \\ &\qquad{}+ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert \biggl( 2 \frac{q [ 3 ] _{q} [ 6 ] _{q}-q^{2}}{ [ 2 ] _{q} [ 3 ] _{q} [ 6 ] _{q}^{3}}+ \frac{1}{ [ 2 ] _{q}^{3}} \biggl( \frac{q+q^{2}}{ [ 3 ] _{q}}- \frac{q^{2}+2q}{ [ 6 ] _{q}} \biggr) \biggr). \end{aligned}$$
(4.13)

Similarly, using the preinvexity of \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert \) and Lemma 1, we have

$$\begin{aligned} & \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+t\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \bigr\vert \,d_{q}t \\ &\quad\leq \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert \biggl( \frac{2q^{2} [ 5 ] _{q}^{3}}{ [ 2 ] _{q} [ 3 ] _{q} [ 6 ] _{q}^{3}}+ \frac{ [ 6 ] _{q} ( 1+ [ 2 ] _{q}^{3} ) - [ 3 ] _{q} [ 5 ] _{q} ( 1+ [ 2 ] _{q}^{2} ) }{ [ 2 ] _{q}^{3} [ 3 ] _{q} [ 6 ] _{q}} \biggr) \\ &\qquad{}+ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert \biggl( 2 \frac{q [ 5 ] _{q}^{2} [ 6 ] _{q} [ 3 ] _{q}-q^{2} [ 5 ] _{q}^{3}}{ [ 2 ] _{q} [ 3 ] _{q} [ 6 ] _{q}^{3}}+ \frac{q^{2}}{ [ 2 ] _{q} [ 3 ] _{q}}- \frac{q [ 5 ] _{q}}{ [ 2 ] _{q} [ 6 ] _{q}} \\ &\qquad{} -\frac{1}{ [ 2 ] _{q}^{3}} \biggl[ \frac{ [ 5 ] _{q} ( 2q+q^{2} ) }{ [ 6 ] _{q}}- \frac{q+q^{2}}{ [ 3 ] _{q}} \biggr] \biggr). \end{aligned}$$
(4.14)

By putting (4.13) and (4.14) in (4.12), we obtain the inequality (4.11), which finishes the proof. □

Corollary 1

In Theorem 4, if we take the limit \(q\rightarrow 1^{-}\), then we obtain the following Simpson type inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) \,d \varkappa \\ &\qquad{}- \frac{1}{6} \biggl[ \varphi \bigl( \varkappa _{2}+\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) +4\varphi \biggl( \frac{2\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{2} \biggr) +\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq \frac{5\eta ( \varkappa _{2},\varkappa _{1} ) }{72} \bigl[ \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert + \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert \bigr], \end{aligned}$$

which can be viewed as a special case of the inequality derived in [34].

Therefore, we can deduce the following results for convex functions.

Remark 5

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Theorem 4, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa - \frac{1}{ [ 6 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +q^{2} [ 4 ] _{q}\varphi \biggl( \frac{\varkappa _{1}+q\varkappa _{2}}{ [ 2 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q ( \varkappa _{2}-\varkappa _{1} ) \bigl\{ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert \bigl[ A_{1} ( q ) +A_{2} ( q ) \bigr] + \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert \bigl[ B_{1} ( q ) +B_{2} ( q ) \bigr] \bigr\} , \end{aligned}$$

which is given by Ali et al. [33].

Remark 6

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\), \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\), and \(q\rightarrow 1^{-}\) in Theorem 4, then Theorem 4 reduces to [26, Corollary 1].

Remark 7

In Theorem 4, if \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\), \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\), \(\varphi ( \varkappa _{1} ) =\varphi ( \frac{\varkappa _{1}+\varkappa _{2}}{2} ) =\varphi ( \varkappa _{2} ) \), and \(q\rightarrow 1^{-}\), then Theorem 4 reduces to [26, Corollary 3].

The corresponding version of Simpson’s inequality for powers in terms of the first q-derivative is incorporated in the following result.

Theorem 5

We assume that the assumptions of Lemma 2hold. If \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert ^{p_{1}}\) is preinvex and integrable on I, where \(p_{1}>1\) with \(\frac{1}{r_{1}}+\frac{1}{p_{1}}=1\), then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{} -\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + [ 4 ] _{q} q^{2} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \frac{q^{2r_{1}} [ 4 ] _{q}^{r_{1}}}{ [ 2 ] _{q}^{r_{1}+1} [ 6 ] _{q}^{r_{1}}} \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{1}{ [ 2 ] _{q}^{3}} \bigl\vert {} ^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+\frac{q^{2}+2q}{ [ 2 ] _{q}^{3}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( \frac{ [ 2 ] _{q}^{r_{1}+1} [ 5 ] _{q}^{r_{1}}-q^{r_{1}} [ 4 ] _{q}^{r_{1}}}{ [ 2 ] _{q}^{r_{1}+1} [ 6 ] _{q}^{r_{1}}} \biggr) ^{ \frac{1}{r_{1}}} \\ &\qquad{} \times \biggl( \frac{q^{2}+2q}{ [ 2 ] _{q}^{3}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{q^{3}+q^{2}-q}{ [ 2 ] _{q}^{3}} \bigl\vert {}^{ \varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$
(4.15)

where \(0< q<1\).

Proof

From the integrals in the right side of (4.12) and applying the well-known quantum Hölder integral inequality, it is found that

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{} -\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + [ 4 ] _{q} q^{2} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}}\,d_{q}t \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{} \times \biggl( \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \bigr\vert ^{p_{1}}\,d_{q}t \biggr) ^{\frac{1}{p_{1}}} \biggr] \\ &\qquad{}\times \biggl[ +q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl( \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}}\,d_{q}t \biggr) ^{ \frac{1}{r_{1}}} \\ & \qquad{}\times \biggl( \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \bigr\vert ^{p_{1}} \,d_{q}t \biggr) ^{\frac{1}{p_{1}}} \biggr]. \end{aligned}$$

By using the preinvexity of \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert ^{p_{1}}\), we obtain

$$\begin{aligned} & \biggl\vert \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) }\int _{\varkappa _{1}}^{\varkappa _{2}}\varphi ( t ) {} ^{\varkappa _{2}}\,d_{q}t- \frac{1}{ [ 6 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +q^{2} [ 4 ] _{q}\varphi \biggl( \frac{\varkappa _{1}+q\varkappa _{2}}{ [ 2 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}}\,d_{q}t \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}} \int _{0}^{ \frac{1}{ [ 2 ] _{q}}}t\,d_{q}t+ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \int _{0}^{ \frac{1}{ [ 2 ] _{q}}} ( 1-t ) \,d_{q}t \biggr) ^{\frac{1}{p_{1}}} \biggr] \\ &\qquad{}+q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}}\,d_{q}t \biggr) ^{ \frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}} \int _{ \frac{1}{ [ 2 ] _{q}}}^{1}t\,d_{q}t+ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \int _{ \frac{1}{ [ 2 ] _{q}}}^{1} ( 1-t ) \,d_{q}t \biggr) ^{\frac{1}{p_{1}}} \biggr]. \end{aligned}$$
(4.16)

To calculate the integrals in the right side of (4.16), if we first use the definition of the quantum integral, then we obtain

$$\begin{aligned} \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}}\,d_{q}t &=\frac{1-q}{ [ 2 ] _{q}}\sum_{\varkappa _{2}=0}^{\infty }q^{\varkappa _{2}} \biggl\vert \frac{q^{\varkappa _{2}}}{ [ 2 ] _{q}}- \frac{1}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}} \\ &\leq \frac{1-q}{ [ 2 ] _{q}}\sum_{\varkappa _{2}=0}^{ \infty }q^{\varkappa _{2}} \biggl\vert \frac{1}{ [ 2 ] _{q}}-\frac{1}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}} \\ &= \biggl( \frac{1}{ [ 2 ] _{q}}- \frac{1}{ [ 6 ] _{q}} \biggr) ^{r_{1}}\frac{1}{ [ 2 ] _{q}} \\ &=\frac{q^{2r_{1}} [ 4 ] _{q}^{r_{1}}}{ [ 2 ] _{q}^{r_{1}+1} [ 6 ] _{q}^{r_{1}}}. \end{aligned}$$
(4.17)

Similarly, we have

$$ \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert ^{r_{1}}\,d_{q}t\leq \frac{ [ 2 ] _{q}^{r_{1}+1} [ 5 ] _{q}^{r_{1}}-q^{r_{1}} [ 4 ] _{q}^{r_{1}}}{ [ 2 ] _{q}^{r_{1}+1} [ 6 ] _{q}^{r_{1}}}. $$
(4.18)

For the other integrals in the right side of (4.16) we consider the case when \(\varkappa _{1}=0\) of Lemma 1, we find that

$$\begin{aligned} &\int _{0}^{\frac{1}{ [ 2 ] _{q}}}t\,d_{q}t = \frac{1}{ [ 2 ] _{q}^{3}}, \end{aligned}$$
(4.19)
$$\begin{aligned} &\int _{0}^{\frac{1}{ [ 2 ] _{q}}} ( 1-t ) \,d_{q}t = \frac{q^{2}+2q}{ [ 2 ] _{q}^{3}}. \end{aligned}$$
(4.20)

Similarly, we get

$$\begin{aligned} &\int _{\frac{1}{ [ 2 ] _{q}}}^{1}t\,d_{q}t = \frac{q^{2}+2q}{ ( [ 2 ] _{q} ) ^{3}} \end{aligned}$$
(4.21)
$$\begin{aligned} &\int _{\frac{1}{ [ 2 ] _{q}}}^{1} ( 1-t ) \,d_{q}t = \frac{q^{3}+q^{2}-q}{ ( [ 2 ] _{q} ) ^{3}}. \end{aligned}$$
(4.22)

By substituting (4.17)–(4.22) in (4.16), we obtain the desired inequality (4.15), which completes the proof. □

Remark 8

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Theorem 5, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa - \frac{1}{ [ 6 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +q^{2} [ 4 ] _{q}\varphi \biggl( \frac{\varkappa _{1}+q\varkappa _{2}}{ [ 2 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q ( \varkappa _{2}-\varkappa _{1} ) \biggl[ \biggl( \frac{q^{2r_{1}} [ 4 ] _{q}^{r_{1}}}{ [ 2 ] _{q}^{r_{1}+1} [ 6 ] _{q}^{r_{1}}} \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{1}{ [ 2 ] _{q}^{3}} \bigl\vert {}^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} )\bigr\vert ^{p_{1}}+\frac{q^{2}+2q}{ [ 2 ] _{q}^{3}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( \frac{ [ 2 ] _{q}^{r_{1}+1} [ 5 ] _{q}^{r_{1}}-q^{r_{1}} [ 4 ] _{q}^{r_{1}}}{ [ 2 ] _{q}^{r_{1}+1} [ 6 ] _{q}^{r_{1}}} \biggr) ^{ \frac{1}{r_{1}}} \\ &\qquad{} \times \biggl( \frac{q^{2}+2q}{ [ 2 ] _{q}^{3}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{q^{3}+q^{2}-q}{ [ 2 ] _{q}^{3}} \bigl\vert {} ^{ \varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$

which is given by Ali et al. in [33].

Another version of Simpson’s inequality for powers in terms of the first q-derivative is obtained as follows.

Theorem 6

Suppose that the assumptions of Lemma 2hold. If \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert ^{p_{1}}\) is preinvex and integrable on I, where \(p_{1}> 1\), then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{} -\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + [ 4 ] _{q} q^{2} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \frac{2q}{ [ 2 ] _{q} [ 6 ] _{q}^{2}}+ \frac{q^{3} [ 3 ] _{q}-q}{ [ 6 ] _{q} [ 2 ] _{q}^{3}} \biggr) ^{1- \frac{1}{p_{1}}} \\ &\qquad{}\times \bigl( A_{1} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{1} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( 2q \frac{ [ 5 ] _{q}^{2}}{ [ 2 ] _{q} [ 6 ] _{q}^{2}}+\frac{1}{ [ 2 ] _{q}}- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}}- \frac{ [ 5 ] _{q} [ 2 ] _{q}^{2}- [ 6 ] _{q}}{ [ 6 ] _{q} [ 2 ] _{q}^{3}} \biggr) ^{1-\frac{1}{p_{1}}} \\ &\qquad{} \times \bigl( A_{2} ( q )\bigl\vert {}^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{2} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \biggr], \end{aligned}$$
(4.23)

where \(0< q<1\) and \(A_{1} ( q ) \), \(A_{2} ( q ) \), \(B_{1} ( q ) \), \(B_{2} ( q ) \) are given as in (4.1)(4.4), respectively.

Proof

Utilizing the results in the proof of Theorem 4, after applying the well-known power mean inequality to the integrals in the right side of (4.12), owing to the preinvexity of \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert ^{p_{1}}\), we find that

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa \\ &\qquad{} -\frac{1}{ [ 6 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) + [ 4 ] _{q} q^{2} \varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 2 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \biggr) ^{1-\frac{1}{p_{1}}} \\ &\qquad{}\times \biggl( \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}} \int _{0}^{ \frac{1}{ [ 2 ] _{q}}}t \biggl\vert t-\frac{1}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \\ &\qquad{} + \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \int _{0}^{ \frac{1}{ [ 2 ] _{q}}} ( 1-t ) \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \biggr) ^{\frac{1}{p_{1}}} \biggr] \\ &\qquad{}+q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \biggr) ^{1- \frac{1}{p_{1}}} \\ &\qquad{}\times \biggl( \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}} \int _{ \frac{1}{ [ 2 ] _{q}}}^{1}t \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \\ &\qquad{} + \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \int _{ \frac{1}{ [ 2 ] _{q}}}^{1} ( 1-t ) \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \biggr) ^{\frac{1}{p_{1}}} \biggr] \\ &\quad=q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \biggr) ^{1-\frac{1}{p_{1}}} \\ &\qquad{} \times \bigl( A_{1} ( q ) \bigl\vert {} ^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{1} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \biggr] \\ &\qquad{}+ \biggl[ q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl( \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t \biggr) ^{1- \frac{1}{p_{1}}} \\ &\qquad{} \times \bigl( A_{2} ( q ) \bigl\vert {}^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{2} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \biggr]. \end{aligned}$$
(4.24)

We also observe that

$$\begin{aligned} \int _{0}^{\frac{1}{ [ 2 ] _{q}}} \biggl\vert t- \frac{1}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t &=2 \int _{0}^{ \frac{1}{ [ 6 ] _{q}}} \biggl( \frac{1}{ [ 6 ] _{q}}-t \biggr) \,d_{q}t+ \int _{0}^{ \frac{1}{ [ 2 ] _{q}}} \biggl( t-\frac{1}{ [ 6 ] _{q}} \biggr) \,d_{q}t \\ &=\frac{2q}{ [ 2 ] _{q} [ 6 ] _{q}^{2}}+ \frac{q^{3} [ 3 ] _{q}-q}{ [ 6 ] _{q} [ 2 ] _{q}^{3}}, \end{aligned}$$
(4.25)

and by using similar operations, we have

$$ \int _{\frac{1}{ [ 2 ] _{q}}}^{1} \biggl\vert t- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}} \biggr\vert \,d_{q}t=2q \frac{ [ 5 ] _{q}^{2}}{ [ 2 ] _{q} [ 6 ] _{q}^{2}}+ \frac{1}{ [ 2 ] _{q}}- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}}- \frac{ [ 5 ] _{q} [ 2 ] _{q}^{2}- [ 6 ] _{q}}{ [ 6 ] _{q} [ 2 ] _{q}^{3}}. $$
(4.26)

By substituting (4.25) and (4.26) in (4.24), we obtain the required inequality (4.23). Hence, the proof is completed. □

Corollary 2

If we take the limit \(q\rightarrow 1^{-}\) in Theorem 6, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) \,d \varkappa \\ &\qquad{}- \frac{1}{6} \biggl[ \varphi \bigl( \varkappa _{2}+\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) +4\varphi \biggl( \frac{2\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{2} \biggr) +\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq \frac{5^{1-\frac{1}{p_{1}}}}{72}\eta ( \varkappa _{2}, \varkappa _{1} ) \biggl[ \biggl( \frac{29}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{61}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}}\\ &\qquad{}+ \biggl( \frac{61}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+\frac{29}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$

which can be viewed as a special case of the inequality derived in [34].

Remark 9

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Theorem 6, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa - \frac{1}{ [ 6 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +q^{2} [ 4 ] _{q}\varphi \biggl( \frac{\varkappa _{1}+q\varkappa _{2}}{ [ 2 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad \leq q ( \varkappa _{2}-\varkappa _{1} ) \biggl[ \biggl( \frac{2q}{ [ 2 ] _{q} [ 6 ] _{q}^{2}}+ \frac{q^{3} [ 3 ] _{q}-q}{ [ 6 ] _{q} [ 2 ] _{q}^{3}} \biggr) ^{1- \frac{1}{p_{1}}} \\ &\qquad{}\times \bigl( A_{1} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{1} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( 2q \frac{ [ 5 ] _{q}^{2}}{ [ 2 ] _{q} [ 6 ] _{q}^{2}}+\frac{1}{ [ 2 ] _{q}}- \frac{ [ 5 ] _{q}}{ [ 6 ] _{q}}- \frac{ [ 5 ] _{q} [ 2 ] _{q}^{2}- [ 6 ] _{q}}{ [ 6 ] _{q} [ 2 ] _{q}^{3}} \biggr) ^{1-\frac{1}{p_{1}}} \\ &\qquad{} \times \bigl( A_{2} ( q )\bigl\vert {}^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{2} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \biggr], \end{aligned}$$

which is proved by Ali et al. in [33].

Remark 10

In Theorem 6, if we take \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\), \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\), and \(q\rightarrow 1^{-}\), then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{6} \biggl[ \varphi ( \varkappa _{1} ) +4\varphi \biggl( \frac{\varkappa _{1}+\varkappa _{2}}{2} \biggr) +\varphi ( \varkappa _{2} ) \biggr] - \frac{1}{\varkappa _{2}-\varkappa _{1}}\int _{\varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) \,d\varkappa \biggr\vert \\ &\quad\leq \frac{5^{1-\frac{1}{p_{1}}}}{72} ( \varkappa _{2}- \varkappa _{1} ) \biggl[ \biggl( \frac{29}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{61}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}}\\ &\qquad{}+ \biggl( \frac{61}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+\frac{29}{18} \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \biggr], \end{aligned}$$

which can be proved as a special case of inequality derived in [26].

Newton’s type inequalities

In this subsection, we will present some quantum Newton’s type inequalities for preinvex functions.

Theorem 7

We assume that the assumptions of Lemma 3hold. If \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert \) is preinvex and integrable on I, then the following inequality holds for \(q^{\varkappa _{2}}\)-integrals:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa -\frac{1}{ [ 8 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \\ &\qquad{} + \frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}} \varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+ [ 2 ] _{q}\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad \leq q\eta ( \varkappa _{2},\varkappa _{1} ) \bigl\{ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert \bigl[ A_{3} ( q ) +A_{4} ( q ) +A_{5} ( q ) \bigr] \\ &\qquad{} + \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert \bigl[ B_{3} ( q ) +B_{4} ( q ) +B_{5} ( q ) \bigr] \bigr\} , \end{aligned}$$
(4.27)

where \(0< q<1\) and \(A_{3} ( q ) \), \(A_{4} ( q ) \), \(A_{5} ( q ) \), \(B_{3} ( q ) \), \(B_{4} ( q ) \), \(B_{5} ( q ) \) are given as in (4.5)(4.10), respectively.

Proof

Following arguments similar to those in the proof of Theorem 4, by taking into account Lemma 3, the desired inequality (4.27) can be obtained. □

Corollary 3

If we take the limit \(q\rightarrow 1^{-}\) in Theorem 7, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) \,d \varkappa \\ &\qquad{} -\frac{1}{8} \biggl[ \varphi \bigl( \varkappa _{2}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) +3\varphi \biggl( \frac{3\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{3} \biggr) \\ &\qquad{}+3\varphi \biggl( \frac{3\varkappa _{2}+2\eta ( \varkappa _{1},\varkappa _{2} ) }{3} \biggr) +\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq \frac{25\eta ( \varkappa _{2},\varkappa _{1} ) }{576} \bigl[ \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert + \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert \bigr], \end{aligned}$$

which can be viewed as a special case of the inequality given in [34].

Remark 11

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Theorem 7, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa -\frac{1}{ [ 8 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +\frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{\varkappa _{1}+q [ 2 ] _{q}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) \\ & \qquad{} + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{1}+q^{2}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad \leq q ( \varkappa _{2}-\varkappa _{1} ) \bigl\{ \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert \bigl[ A_{3} ( q ) +A_{4} ( q ) +A_{5} ( q ) \bigr] \\ &\qquad{} + \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert \bigl[ B_{3} ( q ) +B_{4} ( q ) +B_{5} ( q ) \bigr] \bigr\} , \end{aligned}$$

which is proved by Ali et al. in [33].

Remark 12

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\), \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\), and \(q\rightarrow 1^{-}\) in Theorem 7, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\varkappa _{2}-\varkappa _{1}} \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) \,d \varkappa \\ &\qquad{} -\frac{1}{8} \biggl[ \varphi ( \varkappa _{1} ) +3 \varphi \biggl( \frac{\varkappa _{1}+2\varkappa _{2}}{3} \biggr) +3 \varphi \biggl( \frac{2\varkappa _{1}+\varkappa _{2}}{3} \biggr) + \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq \frac{25 ( \varkappa _{2}-\varkappa _{1} ) }{576} \bigl[ \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert + \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert \bigr], \end{aligned}$$

which was derived as a special case of an inequality proved in [30].

Theorem 8

We assume that the assumptions of Lemma 3hold. If \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert ^{p_{1}}\) is preinvex and integrable on I, where \(p_{1}>1\) with \(\frac{1}{r_{1}}+\frac{1}{p_{1}}=1\), then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa -\frac{1}{ [ 8 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \\ &\qquad{} + \frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) \\ &\qquad{} + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}} \varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+ [ 2 ] _{q}\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl( \frac{q^{3r_{1}} [ 5 ] _{q}^{r_{1}}}{ [ 3 ] _{q}^{r_{1}+1} [ 8 ] _{q}^{r_{1}}} \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{1}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{ [ 3 ] _{q} [ 2 ] _{q}-1}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad{}+q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl( \frac{q^{r_{1}} [ 2 ] _{q}-q^{2r_{1}}}{ [ 3 ] _{q}^{r_{1}+1} [ 2 ] _{q}^{r_{1}}} \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{q^{2}+2}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{q [ 3 ] _{q} [ 2 ] _{q}- ( q^{2}+2q ) }{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad{}+q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl( \frac{q^{7r_{1}}}{ [ 8 ] _{q}^{r_{1}}}- \frac{ [ 2 ] _{q} ( [ 7 ] _{q} [ 3 ] _{q}- [ 8 ] _{q} [ 2 ] _{q} ) ^{r_{1}}}{ [ 8 ] _{q}^{r_{1}} [ 3 ] _{q}^{r_{1}+1}} \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{ [ 3 ] _{q}^{2}- [ 2 ] _{q}^{2}}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} )\bigr\vert ^{p_{1}}+ \frac{q^{2} [ 3 ] _{q} [ 2 ] _{q}+ [ 2 ] _{q}^{2}- [ 3 ] _{q}^{2}}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}}, \end{aligned}$$
(4.28)

where \(0< q<1\).

Proof

If the techniques used in the proof of Theorem 5 are applied by taking into account the Lemma 3, the desired inequality (4.28) can be obtained. □

Corollary 4

In Theorem 8, if we take the limit \(q\rightarrow 1^{-}\), then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) \,d \varkappa \\ & \qquad{}-\frac{1}{8} \biggl[ \varphi \bigl( \varkappa _{1}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) +3\varphi \biggl( \frac{3\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{3} \biggr) \\ &\qquad{}+3\varphi \biggl( \frac{3\varkappa _{2}+2\eta ( \varkappa _{1},\varkappa _{2} ) }{3} \biggr) +\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq \frac{\eta ( \varkappa _{2},\varkappa _{1} ) }{3} \biggl[ \frac{5}{8} \biggl( \frac{ \vert \varphi ^{\prime } ( \varkappa _{1} ) \vert ^{p_{1}}+5 \vert \varphi ^{\prime } ( \varkappa _{2} ) \vert ^{p_{1}}}{6} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( \frac{2.3^{r_{1}}-1}{6^{r_{1}}} \biggr) ^{\frac{1}{r_{1}}} \biggl( \frac{ \vert \varphi ^{\prime } ( \varkappa _{1} ) \vert ^{p_{1}}+ \vert \varphi ^{\prime } ( \varkappa _{2} ) \vert ^{p_{1}}}{2} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad{} + \biggl( \frac{3.7^{r_{1}}-2}{8^{r_{1}}} \biggr) ^{ \frac{1}{r_{1}}} \biggl( \frac{5 \vert \varphi ^{\prime } ( \varkappa _{1} ) \vert ^{p_{1}}+ \vert \varphi ^{\prime } ( \varkappa _{2} ) \vert ^{p_{1}}}{6} \biggr) ^{\frac{1}{p_{1}}} \biggr]. \end{aligned}$$
(4.29)

Remark 13

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in (4.29), then the inequality (4.29) reduces to the inequality presented in [12, Remark 4].

Remark 14

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Theorem 8, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa - \frac{1}{ [ 8 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +\frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{\varkappa _{1}+q [ 2 ] _{q}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) \\ &\qquad{} + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{1}+q^{2}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q ( b-a ) \biggl( \frac{q^{3r_{1}} [ 5 ] _{q}^{r_{1}}}{ [ 3 ] _{q}^{r_{1}+1} [ 8 ] _{q}^{r_{1}}} \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{1}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{ [ 3 ] _{q} [ 2 ] _{q}-1}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+q ( b-a ) \biggl( \frac{q^{r_{1}} [ 2 ] _{q}-q^{2r_{1}}}{ [ 3 ] _{q}^{r_{1}+1} [ 2 ] _{q}^{r_{1}}} \biggr) ^{\frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{q^{2}+2}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{q [ 3 ] _{q} [ 2 ] _{q}- ( q^{2}+2q ) }{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}} \\ &\qquad{}+q ( b-a ) \biggl( \frac{q^{7r_{1}}}{ [ 8 ] _{q}^{r_{1}}}-\frac{ [ 2 ] _{q} ( [ 7 ] _{q} [ 3 ] _{q}- [ 8 ] _{q} [ 2 ] _{q} ) ^{r_{1}}}{ [ 8 ] _{q}^{r_{1}} [ 3 ] _{q}^{r_{1}+1}} \biggr) ^{ \frac{1}{r_{1}}} \\ &\qquad{}\times \biggl( \frac{ [ 3 ] _{q}^{2}- [ 2 ] _{q}^{2}}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} )\bigr\vert ^{p_{1}}+ \frac{q^{2} [ 3 ] _{q} [ 2 ] _{q}+ [ 2 ] _{q}^{2}- [ 3 ] _{q}^{2}}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{\frac{1}{p_{1}}}, \end{aligned}$$

which is given by Ali et al. in [33].

Theorem 9

Suppose that the assumptions of Lemma 3hold. If \(\vert {}^{\varkappa _{2}}D_{q}\varphi \vert ^{p_{1}}\) is preinvex and integrable on I, where \(p_{1}> 1\), then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) ^{ \varkappa _{2}} \,d_{q} \varkappa -\frac{1}{ [ 8 ] _{q}} \biggl[ \varphi \bigl( \varkappa _{2}+\eta ( \varkappa _{1}, \varkappa _{2} ) \bigr) \\ &\qquad{} + \frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}} \varphi \biggl( \frac{ [ 3 ] _{q}\varkappa _{2}+ [ 2 ] _{q}\eta ( \varkappa _{1},\varkappa _{2} ) }{ [ 3 ] _{q}} \biggr) +q \varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q\eta ( \varkappa _{2},\varkappa _{1} ) \biggl[ \biggl( \frac{2q}{ [ 8 ] _{q}^{2} [ 2 ] _{q}}+ \frac{ [ 8 ] _{q}- [ 3 ] _{q} [ 2 ] _{q}}{ [ 3 ] _{q}^{2} [ 2 ] _{q} [ 8 ] _{q}} \biggr) ^{1-\frac{1}{p_{1}}} \\ &\qquad{}\times \bigl( A_{3} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{3} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( \frac{2q}{ [ 2 ] _{q}^{3}}+ \frac{q}{ [ 3 ] _{q}^{2} [ 2 ] _{q}}+ \frac{1- [ 3 ] _{q} [ 2 ] _{q}}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \biggr) ^{1- \frac{1}{p_{1}}} \\ &\qquad{}\times \bigl( A_{4} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{4} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( 2 \frac{q [ 7 ] _{q}^{2}}{ [ 8 ] _{q}^{2} [ 2 ] _{q}}+ \frac{ [ 3 ] _{q}^{2}+ [ 2 ] _{q}^{2}}{ [ 2 ] _{q} [ 3 ] _{q}^{2}}- \frac{ [ 7 ] _{q} ( [ 3 ] _{q}+ [ 2 ] _{q} ) }{ [ 8 ] _{q} [ 3 ] _{q}} \biggr) ^{1-\frac{1}{p_{1}}} \\ & \qquad{}\times \bigl( A_{5} ( q )\bigl\vert {} ^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{5} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \biggr], \end{aligned}$$
(4.30)

where \(0< q<1\) and \(A_{3} ( q ) \), \(A_{4} ( q ) \), \(A_{5} ( q ) \), \(B_{3} ( q ) \), \(B_{4} ( q ) \), \(B_{5} ( q ) \) are given as in (4.5)(4.10), respectively.

Proof

The proof follows along the same lines used in the proof of Theorem 6 by taking into account Lemma 3. □

Corollary 5

In Theorem 9, if we take the limit \(q\rightarrow 1^{-}\), then we have the following Newton-type inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{\eta ( \varkappa _{2},\varkappa _{1} ) }\int _{\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }^{\varkappa _{2}}\varphi ( \varkappa ) \,d \varkappa \\ & \qquad{}-\frac{1}{8} \biggl[ \varphi \bigl( \varkappa _{1}+ \eta ( \varkappa _{1},\varkappa _{2} ) \bigr) +3\varphi \biggl( \frac{3\varkappa _{2}+\eta ( \varkappa _{1},\varkappa _{2} ) }{3} \biggr) \\ &\qquad{}+3\varphi \biggl( \frac{3\varkappa _{2}+2\eta ( \varkappa _{1},\varkappa _{2} ) }{3} \biggr) +\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq \frac{\eta ( \varkappa _{2},\varkappa _{1} ) }{36} \biggl[ \biggl( \frac{17}{16} \biggr) ^{1-\frac{1}{p_{1}}} \biggl( \frac{251}{1152} \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+ \frac{973}{1152} \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad{} + \biggl( \frac{ \vert \varphi ^{\prime } ( \varkappa _{1} ) \vert ^{p_{1}}+ \vert \varphi ^{\prime } ( \varkappa _{2} ) \vert ^{p_{1}}}{2} \biggr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( \frac{17}{16} \biggr) ^{1-\frac{1}{p_{1}}} \biggl( \frac{973}{1152} \bigl\vert \varphi ^{\prime } ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+\frac{251}{1152} \bigl\vert \varphi ^{\prime } ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \biggr) ^{ \frac{1}{p_{1}}} \biggr]. \end{aligned}$$
(4.31)

Remark 15

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in (4.31), then the inequality (4.31) reduces to the inequality presented in [12, Remark 5].

Remark 16

If we set \(\eta ( \varkappa _{2},\varkappa _{1} ) =\varkappa _{2}- \varkappa _{1}\) and \(\eta ( \varkappa _{1},\varkappa _{2} ) =\varkappa _{1}- \varkappa _{2}\) in Theorem 9, then we have the following inequality:

$$\begin{aligned} & \biggl\vert \frac{1}{ ( \varkappa _{2}-\varkappa _{1} ) } \int _{ \varkappa _{1}}^{\varkappa _{2}}\varphi ( \varkappa ) {} ^{\varkappa _{2}}\,d_{q}\varkappa - \frac{1}{ [ 8 ] _{q}} \biggl[ \varphi ( \varkappa _{1} ) +\frac{q^{3} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{\varkappa _{1}+q [ 2 ] _{q}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) \\ &\qquad{} + \frac{q^{2} [ 6 ] _{q}}{ [ 2 ] _{q}}\varphi \biggl( \frac{ [ 2 ] _{q}\varkappa _{1}+q^{2}\varkappa _{2}}{ [ 3 ] _{q}} \biggr) +q\varphi ( \varkappa _{2} ) \biggr] \biggr\vert \\ &\quad\leq q ( \varkappa _{2}-\varkappa _{1} ) \biggl[ \biggl( \frac{2q}{ [ 8 ] _{q}^{2} [ 2 ] _{q}}+ \frac{ [ 8 ] _{q}- [ 3 ] _{q} [ 2 ] _{q}}{ [ 3 ] _{q}^{2} [ 2 ] _{q} [ 8 ] _{q}} \biggr) ^{1-\frac{1}{p_{1}}} \\ &\qquad{}\times \bigl( A_{3} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{3} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( \frac{2q}{ [ 2 ] _{q}^{3}}+ \frac{q}{ [ 3 ] _{q}^{2} [ 2 ] _{q}}+ \frac{1- [ 3 ] _{q} [ 2 ] _{q}}{ [ 3 ] _{q}^{2} [ 2 ] _{q}} \biggr) ^{1- \frac{1}{p_{1}}} \\ &\qquad{}\times \bigl( A_{4} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{4} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q}\varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \\ &\qquad{}+ \biggl( 2 \frac{q [ 7 ] _{q}^{2}}{ [ 8 ] _{q}^{2} [ 2 ] _{q}}+ \frac{ [ 3 ] _{q}^{2}+ [ 2 ] _{q}^{2}}{ [ 2 ] _{q} [ 3 ] _{q}^{2}}- \frac{ [ 7 ] _{q} ( [ 3 ] _{q}+ [ 2 ] _{q} ) }{ [ 8 ] _{q} [ 3 ] _{q}} \biggr) ^{1-\frac{1}{p_{1}}} \\ &\qquad{} \times \bigl( A_{5} ( q ) \bigl\vert {} ^{ \varkappa _{2}}D_{q} \varphi ( \varkappa _{1} ) \bigr\vert ^{p_{1}}+B_{5} ( q ) \bigl\vert {}^{\varkappa _{2}}D_{q} \varphi ( \varkappa _{2} ) \bigr\vert ^{p_{1}} \bigr) ^{ \frac{1}{p_{1}}} \biggr], \end{aligned}$$

which is derived by Ali et al. in [33].

Conclusion

In this paper, we proved some new inequalities of Simpson’s and Newton’s type for q-differentiable preinvex functions by using the notion of \(q^{\varkappa _{2}}\)-quantum integral. It is also shown that some classical results can be obtained by the results presented in the current research by taking the limit \(q\rightarrow 1^{-}\). It will be an interesting problem to prove similar inequalities for the functions of two variables.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.

Funding

The work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, 11601485, 11971241).

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Ali, M.A., Abbas, M., Budak, H. et al. New quantum boundaries for quantum Simpson’s and quantum Newton’s type inequalities for preinvex functions. Adv Differ Equ 2021, 64 (2021). https://doi.org/10.1186/s13662-021-03226-x

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Keywords

  • Simpson’s \(\frac{1}{3}\) formula
  • Simpson’s \(\frac{3}{8}\) formula
  • Integral inequalities
  • Quantum calculus
  • Preinvex functions
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