Post-quantum Ostrowski type integral inequalities for functions of two variables

1.   Introduction

A. M. Ostowski established the following intriguing integral inequality in 1938, which is known in the literature as the Ostrowski inequality.

Theorem 1.1. ^[47] Let $\digamma :[\pi _{1}, \pi _{2}]\rightarrow \mathbb{R}$ be a differentiable function on $(\pi _{1}, \pi _{2})$ whose derivative isbounded on $(\pi _{1}, \pi _{2}),$ i.e., $\left \Vert \digamma ^{^{\prime}}(\tau)\right \Vert _{\infty }: = \sup \left \vert \digamma ^{^{\prime}}(\tau)\right \vert < \infty,$ for all $\tau \in (\pi _{1}, \pi _{2}).$Then we have the following integral inequality:

for all $x\in \lbrack \pi _{1}, \pi _{2}].$ The $\frac{1}{4}$ is the bestpossible.

Inequality (1.1) can be rewritten in the following way:

Since 1938, numerous mathematicians have worked on and around the Ostrowski inequality, in a variety of ways and with a variety of applications in Numerical Analysis and Probability, etc.

Many authors investigate several versions of the Ostrowski integral inequality for bounded variation mappings, Lipschitzian mappings, monotonic mappings, absolutely continuous mappings, convex mappings, and $n$-times differentiable mappings with error estimates for various particular means and numerical quadrature techniques. For recent results and generalizations concerning Ostrowski's inequality, one can consult ^{[8,9,15,24,26,27,42,48,49,50,52]} and the references therein.

The following is a formal definition of co-ordinated convex (concave) functions:

Definition 1.2. A function $\digamma :\Delta \rightarrow \mathbb{R}$ is called co-ordinated convex on $\Delta,$ for all $(x, u), (y, v)\in \Delta$ and $\tau, s\in \lbrack 0, 1]$, if it satisfies the following inequality:

The mapping $\digamma$ is a co-ordinated concave on $\Delta$ if the inequality (1.3) holds in reversed direction for all $\tau, s\in \lbrack 0, 1]$ and $(x, u), (y, v)\in \Delta$.

For co-ordinated convex functions, M. A. Latif et al. established the following Ostrowski type inequalities in ^[41]:

Theorem 1.3. ^[41] Let $\digamma :\Delta : = [\pi _{1}, \pi _{2}]\times\lbrack \pi _{3}, \pi _{4}]\rightarrow \mathbb{R}$ be a twice partial differentiable mapping on $\Delta ^{\circ }$ with $\pi_{1} < \pi _{2}, \; \pi _{3} < \pi _{4}, \; \pi _{1}, \pi _{3}\geq 0$ such that $\frac{\partial ^{2}\digamma }{\partial s\partial \tau }\in L(\Delta).$ If $\left\vert \frac{\partial ^{2}\digamma }{\partial s\partial \tau }\right\vert$ is co-ordinated convex on $\Delta$ and $\left\vert \frac{\partial ^{2}\digamma }{\partial s\partial \tau }\right\vert \leq M, \; (x, y)\in \Delta$, then the following inequality holds:

where

Theorem 1.4. ^[41] Let $\digamma :\Delta : = [\pi _{1}, \pi _{2}]\times\lbrack \pi _{3}, \pi _{4}]\rightarrow \mathbb{R}$ be a twice partial differentiable mapping on $\Delta ^{\circ }$ with $\pi_{1} < \pi _{2}, \; \pi _{3} < \pi _{4}, \; \pi _{1}, \pi _{3}\geq 0$ such that $\frac{\partial ^{2}\digamma }{\partial s\partial \tau }\in L(\Delta).$ If $\left\vert \frac{\partial ^{2}\digamma }{\partial s\partial \tau }\right\vert ^{s}$ is co-ordinated convex on $\Delta,$ $s > 1, \; \frac{1}{s}+\frac{1}{r} = 1$ and $\left\vert \frac{\partial ^{2}\digamma }{\partial s\partial \tau }(x, y)\right\vert \leq M, \; (x, y)\in \Delta$, then thefollowing inequality holds:

where $\pi _{11}$ is defined in Theorem 1.3.

Theorem 1.5. ^[41] Let $\digamma :\Delta : = [\pi _{1}, \pi _{2}]\times\lbrack \pi _{3}, \pi _{4}]\rightarrow \mathbb{R}$ be a twice partial differentiable mapping on $\Delta ^{\circ }$ with $\pi_{1} < \pi _{2}, \; \pi _{3} < \pi _{4}, \; \pi _{1}, \pi _{3}\geq 0$ such that $\frac{\partial ^{2}\digamma }{\partial s\partial \tau }\in L(\Delta).$ If $\left\vert \frac{\partial ^{2}\digamma }{\partial s\partial \tau }\right\vert ^{p}$ is co-ordinated convex on $\Delta,$ $p\geq 1$ and $\left \vert\frac{\partial ^{2}\digamma }{\partial s\partial \tau }(x, y)\right \vert\leq M, \; (x, y)\in \Delta$, then the following inequality holds:

where $\pi _{11}$ is defined in Theorem 1.3.

On the other side, in the domain of $q$-analysis, many works are 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 ^[31,35]. Quantum calculus also has many applications in quantum information theory which is an interdisciplinary area that encompasses computer science, information theory, philosophy, and cryptography, among other areas ^[16,17]. Apparently, Euler invented this important mathematics branch. He used the $q$ parameter in Newton's work on infinite series. Later, in a methodical manner, the $q$-calculus that knew without limits calculus was firstly given by F. H. Jackson ^[30,33]. In 1966, W. Al-Salam ^[12] introduced a $q$-analogue of the $q$ -fractional integral and $q$-Riemann-Liouville fractional. Since then, the related research has gradually increased. In particular, in 2013, J. Tariboon and S. K. Ntouyas introduced $_{\pi _{1}}D_{q}$-difference operator and $q_{\pi _{1}}$-integral in ^[54]. In 2020, S. Bermudo et al. introduced the notion of $^{\pi _{2}}D_{q}$ derivative and $q^{\pi _{2}}$ -integral in ^[14]. T. Acar et al. generalized to quantum calculus and introduced the notions of post-quantum calculus or shortly $\left(p, q\right)$-calculus in ^[1]. In ^[53], M. Tunç and E. Gö v gave the post-quantum variant of $_{\pi _{1}}D_{q}$-difference operator and $q_{\pi _{1}}$-integral. Recently, in 2021, Y. M. Chu et al. introduced the notions of $^{\pi _{2}}D_{p, q}$ derivative and $\left(p, q\right) ^{\pi _{2}}$-integral in ^[25].

Many integral inequalities have been studied using quantum and post-quantum integrals for various types of functions. For example, in ^{[3,6,10,11,14,18,19,34,43,44]}, the authors used $_{\pi _{1}}D_{q}, ^{\pi _{2}}D_{q}$-derivatives and $q_{\pi _{1}}, q^{\pi _{2}}$-integrals to prove Hermite-Hadamard integral inequalities and their left-right estimates for convex and coordinated convex functions. In ^[45], M. A. Noor et al. presented a generalized version of quantum integral inequalities. For generalized quasi-convex functions, E. R. Nwaeze et al. proved certain parameterized quantum integral inequalities in ^[46]. M. A. Khan et al. proved quantum Hermite-Hadamard inequality using the green function in ^[37]. H. Budak et al. ^[20], M. A. Ali et al. ^[2,4] and M. Vivas-Cortez et al. ^[55] developed new quantum Simpson's and quantum Newton's type inequalities for convex and coordinated convex functions. For quantum Ostrowski's inequalities for convex and co-ordinated convex functions on can consult ^[5,7,23]. M. Kunt et al. ^[38] generalized the results of ^[10] and proved Hermite-Hadamard type inequalities and their left estimates using $_{\pi _{1}}D_{p, q}$-difference operator and $\left(p, q\right) _{\pi _{1}}$ -integral. Recently, M. A. Latif et al. ^[39] found the right estimates of Hermite-Hadamard type inequalities proved by M. Kunt et al. ^[38]. To prove Ostrowski's inequalities, Y.-M. Chu et al. ^[25] used the concepts of $^{\pi _{2}}D_{p, q}$-difference operator and $\left(p, q\right) ^{\pi _{2}}$-integral.

The following quantum variants of inequalities (1.4)–(1.6) proved my H. Budak et al. in ^[21].

Theorem 1.6. ^[21] Let $\digamma :\Delta \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a twice partially $q_{1}q_{2}$-differentiable function on $\Delta^{\circ }$ and partial $q_{1}q_{2}$-derivatives $\frac{^{\pi _{2}, \; \pi_{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{^{\pi_{2}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s},$ $\frac{_{\pi_{1}}^{\pi _{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi _{1}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s},$ $\frac{_{\pi _{3}}^{\pi _{2}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau _{\; \pi _{3}}\partial _{q_{2}}s}$, $\frac{_{\pi _{1}, \; \pi _{3}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi _{1}}\partial _{q_{1}}\tau \; _{\pi _{3}}\partial_{q_{2}}s}$ be continuous and integrable on $\left[\pi _{1}, \pi _{2}\right]\times \left[\pi _{3}, \pi _{4}\right] \subseteq \Delta ^{\circ }.$ If $\left \vert \frac{^{\pi _{2}, \; \pi _{4}}\partial _{q_{1}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau \; ^{\pi_{4}}\partial _{q_{2}}s}\right \vert,$ $\left \vert \frac{_{\pi _{1}}^{\pi_{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s}\right \vert,$ $\left \vert \frac{_{\pi _{3}}^{\pi _{2}}\partial _{q_{1}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau _{\; \pi_{3}}\partial _{q_{2}}s}\right \vert$, $\left \vert \frac{_{\pi _{1}, \; \pi_{3}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{q_{1}}\tau \; _{\pi _{3}}\partial _{q_{2}}s}\right \vert \leq M$ for all $\left(\tau, s\right) \in \left[\pi _{1}, \pi _{2}\right]\times \left[\pi _{3}, \pi _{4}\right]$, then we have the following quantumOstrowski's type inequality:

for all $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$ where $q_{1}, q_{2}\in \left(0, 1\right).$

Theorem 1.7. ^[21] Let $\digamma :\Delta \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a twice partially $q_{1}q_{2}$-differentiable function on $\Delta^{\circ }$ and partial $q_{1}q_{2}$-derivatives $\frac{^{\pi _{2}, \; \pi_{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{^{\pi_{2}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s},$ $\frac{_{\pi_{1}}^{\pi _{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi _{1}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s},$ $\frac{_{\pi _{3}}^{\pi _{2}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau _{\; \pi _{3}}\partial _{q_{2}}s}$, $\frac{_{\pi _{1}, \; \pi _{3}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi _{1}}\partial _{q_{1}}\tau \; _{\pi _{3}}\partial_{q_{2}}s}$ be continuous and integrable on $\left[\pi _{1}, \pi _{2}\right]\times \left[\pi _{3}, \pi _{4}\right] \subseteq \Delta ^{\circ }.$ If $\left \vert \frac{^{\pi _{2}, \; \pi _{4}}\partial _{q_{1}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau \; ^{\pi_{4}}\partial _{q_{2}}s}\right \vert,$ $\left \vert \frac{_{\pi _{1}}^{\pi_{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s}\right \vert,$ $\left \vert \frac{_{\pi _{3}}^{\pi _{2}}\partial _{q_{1}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau _{\; \pi_{3}}\partial _{q_{2}}s}\right \vert$, $\left \vert \frac{_{\pi _{1}, \; \pi_{3}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{q_{1}}\tau \; _{\pi _{3}}\partial _{q_{2}}s}\right \vert \leq M$ for all $\left(\tau, s\right) \in \left[\pi _{1}, \pi _{2}\right]\times \left[\pi _{3}, \pi _{4}\right]$, then we have the following quantumOstrowski's type inequality:

for all $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$ where $q_{1}, q_{2}\in \left(0, 1\right)$ and $\frac{1}{r}+\frac{1}{s} = 1,$ $s > 1.$

Theorem 1.8. ^[21] Let $\digamma :\Delta \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a twice partially $q_{1}q_{2}$-differentiable function on $\Delta^{\circ }$ and partial $q_{1}q_{2}$-derivatives $\frac{^{\pi _{2}, \; \pi_{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{^{\pi_{2}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s},$ $\frac{_{\pi_{1}}^{\pi _{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi _{1}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s},$ $\frac{_{\pi _{3}}^{\pi _{2}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau _{\; \pi _{3}}\partial _{q_{2}}s}$, $\frac{_{\pi _{1}, \; \pi _{3}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi _{1}}\partial _{q_{1}}\tau \; _{\pi _{3}}\partial_{q_{2}}s}$ be continuous and integrable on $\left[\pi _{1}, \pi _{2}\right]\times \left[\pi _{3}, \pi _{4}\right] \subseteq \Delta ^{\circ }.$ If $\left \vert \frac{^{\pi _{2}, \; \pi _{4}}\partial _{q_{1}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau \; ^{\pi_{4}}\partial _{q_{2}}s}\right \vert,$ $\left \vert \frac{_{\pi _{1}}^{\pi_{4}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{q_{1}}\tau \; ^{\pi _{4}}\partial _{q_{2}}s}\right \vert,$ $\left \vert \frac{_{\pi _{3}}^{\pi _{2}}\partial _{q_{1}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{q_{1}}\tau _{\; \pi_{3}}\partial _{q_{2}}s}\right \vert$, $\left \vert \frac{_{\pi _{1}, \; \pi_{3}}\partial _{q_{1}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{q_{1}}\tau \; _{\pi _{3}}\partial _{q_{2}}s}\right \vert \leq M$ for all $\left(\tau, s\right) \in \left[\pi _{1}, \pi _{2}\right]\times \left[\pi _{3}, \pi _{4}\right]$, then we have the following quantumOstrowski's type inequality:

for all $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$ where $q_{1}, q_{2}\in \left(0, 1\right)$ and $s\geq 1.$

Inspired by this ongoing studies, we introduce some new notions of post-quantum partial derivatives and prove some new ostrowski type inequalities for the functions of two variables by using the post-quantum double integrals and newly introduced post-quantum partial derivatives. Moreover, we show that the results presented in this paper are the extensions of results proved in ^[21,40].

The following is the structure of this paper: A brief overview of the concepts of $q$-calculus, as well as some related works, is given in Section 2. In Section 3, we recall the notions of $\left(p, q\right)$-calculus and give some realted works. In Section 4, we show the relationship between the results presented here and comparable results in the literature by proving some new post-quantum Ostrowski type inequalities for the functions of two variables. Section 5 concludes with some recommendations for future studies.

2.   Quantum calculus and some inequalities

In this section, we present some required definitions and inequalities.

In ^[33], F. H. Jackson gave the $q$-Jackson integral from $0$ to $\pi _{2}$ for $0 < q < 1$ as follows:

provided the sum converge absolutely. Moreover, he gave the $q$-Jackson integral in an arbitrary interval $[\pi _{1}, \pi _{2}]$ as:

Definition 2.1. ^[54] For a continuous function $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R},$ then $q_{\pi _{1}}$-derivative of $\digamma$ at $x\in \left[\pi _{1}, \pi _{2}\right]$ is characterized by the expression:

For $x = \pi _{1}$, we state $_{\pi _{1}}D_{q}\digamma \left(\pi _{1}\right) = \lim_{x\rightarrow \pi _{1}}\; _{\pi _{1}}D_{q}\digamma \left(x\right)$ if it exists and it is finite.

Definition 2.2. ^[14] For a continuous function $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R},$ then $q^{\pi _{2}}$-derivative of $\digamma$ at $x\in \left[\pi _{1}, \pi _{2}\right]$ is characterized by the expression:

For $x = \pi _{2}$, we state $^{\pi _{2}}D_{q}\digamma \left(\pi _{2}\right) = \lim_{x\rightarrow \pi _{2}}\; ^{\pi _{2}}D_{q}\digamma \left(x\right)$ if it exists and it is finite.

Definition 2.3. ^[54] Let $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R}$ be a continuous function. Then, the $q_{\pi _{1}}$-definite integral on $\left[\pi _{1}, \pi _{2}\right]$ is defined as:

On the other hand, S. Bermudo et al. gave the following new definition:

Definition 2.4. ^[14] Let $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R}$ be a continuous function. Then, the $q^{\pi _{2}}$-definite integral on $\left[\pi _{1}, \pi _{2}\right]$ is defined as:

For more details about $q^{\pi _{2}}$-integrals and corresponding inequalities one can see ^[14].

Now, let's give the following notation which will be used many times in the next sections (see, ^[35]):

Moreover, we give the following Lemma for our main results:

Lemma 2.5. ^[54] We have the equality

for $\alpha \in \mathbb{R} \backslash \left \{ -1\right \}.$

In ^[23], H. Budak et al. proved the following variant of quantum Ostrowski inequality using the $q_{\pi _{1}}$ and $q^{\pi _{2}}$-integrals:

Theorem 2.6. ^[23] Let $\digamma :\left[\pi _{1}, \pi _{2}\right] \subset \mathbb{R} \rightarrow \mathbb{R}$ be a function and $^{\pi _{2}}D_{q}\digamma$, $_{\pi _{1}}D_{q}\digamma$be two continuous and integrable functions on $\left[\pi _{1}, \pi _{2}\right].$ If $\left \vert\begin{array}{c}^{\pi _{2}}D_{q}\end{array}\digamma \left(\tau \right) \right \vert, \left \vert\begin{array}{c}_{\pi _{1}}D_{q}\end{array}\digamma \left(\tau \right) \right \vert \leq M$ for all $\tau \in \left[\pi _{1}, \pi _{2}\right],$ then we have the following quantum Ostrowskitype inequality:

for all $x\in \left[\pi _{1}, \pi _{2}\right]$ where $0 < q < 1.$

On the other hand, the authors gave the following definitions of $q_{\pi _{1}\pi _{3}}$, $q_{\pi _{1}}^{\pi _{4}}$, $q_{\pi _{2}}^{\pi _{3}}$ and $q^{\pi _{2}\pi _{4}{\rm{ }}}$integrals and related inequalities of Hermite-Hadamard type:

Definition 2.7. ^[19,40] Suppose that $\digamma :\left[\pi _{1}, \pi _{2} \right] \times \left[\pi _{3}, \pi _{4}\right] \subset \mathbb{R} ^{2}\rightarrow \mathbb{R}$ is a continuous function. Then, the following $q_{\pi _{1}\pi _{3}}$, $q_{\pi _{1}}^{\pi _{4}},$ $q_{\pi _{3}}^{\pi _{2}}$ and $q^{\pi _{2}\pi _{4}}$ integrals on $\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$ are defined by

and

respectively, for $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right].$

^[40,57] Let $\digamma :\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right] \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a continuous function of two variables. Then, the partial $q_{1}$ -derivatives, $q_{2}$-derivatives and $q_{1}q_{2}$-derivatives at $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$ can be given as follows:

3.   Post-quantum calculus and some inequalities

In this section, we review some fundamental notions and notations of $\left(p, q\right)$-calculus.

The $\left[n\right] _{p, q}$ is said to be ($p, q$)-integers and expressed as ^[13,28,29]:

with $0 < q < p\leq 1.$ The $\left[n\right] _{p, q}!$ and $\left[\begin{array}{c} n \\ k \end{array} \right]!$ are called ($p, q$)-factorial and ($p, q$)-binomial, respectively, and expressed as:

Definition 3.1. ^[1] The $\left(p, q\right)$-derivative of mapping $\digamma : \left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R}$ is given as:

with $0 < q < p\leq 1.$

Definition 3.2. ^[53] The $\left(p, q\right) _{\pi _{1}}$-derivative of mapping $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R}$ is given as:

with $0 < q < p\leq 1.$ For $x = \pi _{1}$, we state $_{\pi _{1}}D_{p, q}\digamma \left(\pi _{1}\right) = \lim_{x\rightarrow \pi _{1}}\; _{\pi _{1}}D_{p, q}\digamma \left(x\right)$ if it exists and it is finite.

Definition 3.3. ^[25] The $\left(p, q\right) ^{\pi _{2}}$-derivative of mapping $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R}$ is given as:

with $0 < q < p\leq 1.$ For $x = \pi _{2}$, we state $^{\pi _{2}}D_{p, q}\digamma \left(\pi _{2}\right) = \lim_{x\rightarrow \pi _{2}}\; ^{\pi _{2}}D_{p, q}\digamma \left(x\right)$ if it exists and it is finite.

Remark 3.4. It is clear that if we use $p = 1$ in (3.1) and (3.2), then the equalities (3.1) and (3.2) reduce to (2.2) and (2.3), respectively.

Definition 3.5. ^[53] The definite ($p, q$)$_{\pi _{1}}$-integral of mapping $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R}$ on $\left[\pi _{1}, \pi _{2}\right]$ is stated as:

with $0 < q < p\leq 1.$

Definition 3.6. ^[25] The definite ($p, q$)$^{\pi _{2}}$-integral of mapping $\digamma :\left[\pi _{1}, \pi _{2}\right] \rightarrow \mathbb{R}$ on $\left[\pi _{1}, \pi _{2}\right]$ is stated as:

with $0 < q < p\leq 1.$

Remark 3.7. It is evident that if we pick $p = 1$ in (3.3) and (3.4), then the equalities (3.3) and (3.4) change into (2.4) and (2.5), respectively.

Remark 3.8. If we take $\pi _{1} = 0$ and $x = \pi _{2} = 1$ in (3.3), then we have

Similarly, by taking $x = \pi _{1} = 0$ and $\pi _{2} = 1$ in (3.4), then we obtain that

Lemma 3.9. ^[56] We have the following equalities

where $\alpha \in \mathbb{R} \backslash \left\{ -1\right\}.$

For more details in $\left(p, q\right)$-calculus, one can consult ^[22,32,51].

In ^[38], M. Kunt et al. proved the following HH type inequalities for convex functions via ($p, q$)$_{\pi _{1}}$-integral:

Theorem 3.10. ^[38] For a convex mapping $\digamma :\left[\pi _{1}, \pi _{2}\right]\rightarrow \mathbb{R}$ which is differentiable on $\left[\pi _{1}, \pi _{2}\right],$ thefollowing inequalities hold for $\left(p, q\right) _{\pi _{1}}$-integral:

where $0 < q < p\leq 1.$

Recently, M. Vivas-Cortez et al. ^[56] proved the following HH type inequalities for convex functions using the $\left(p, q\right) ^{\pi _{2}}$ -integral:

Theorem 3.11. ^[56] For a convex mapping $\digamma :\left[\pi _{1}, \pi _{2}\right]\rightarrow \mathbb{R}$ which is differentiable on $\left[\pi _{1}, \pi _{2}\right],$ thefollowing inequalities hold for $\left(p, q\right) ^{\pi _{2}}$-integral:

where $0 < q < p\leq 1.$

In ^[36] and ^[58], the authors gave the following notions of post-quantum integrals for the functions of two variables.

Definition 3.12. ^[36,58] For a function $\digamma :\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right] \rightarrow \mathbb{R}$:

(1) The $\left(p, q\right) _{\pi _{1}}^{\pi _{4}}$ integral of $\digamma$ is given as:

where $x, y\in \left[\pi _{1}, p_{1}\pi _{2}+\left(1-p_{1}\right) \pi _{1} \right] \times \left[p_{2}\pi _{3}+\left(1-p_{2}\right) \pi _{4}, \pi _{4} \right].$

(2) The $\left(p, q\right) _{\pi _{3}}^{\pi _{2}}$ integral of $\digamma$ is given as:

where $x, y\in \left[p_{1}\pi _{1}+\left(1-p_{1}\right) \pi _{1}, \pi _{2} \right] \times \left[\pi _{3}, p_{2}\pi _{4}+\left(1-p_{2}\right) \pi _{3} \right].$

(3) The $\left(p, q\right) ^{\pi _{2}\pi _{4}}$ integral of $\digamma$ is given as:

where $x, y\in \left[p_{1}\pi _{1}+\left(1-p_{1}\right) \pi _{2}, \pi _{2} \right] \times \left[p_{2}\pi _{3}+\left(1-p_{2}\right) \pi _{4}, \pi _{4} \right].$

(4) The $\left(p, q\right) _{\pi _{1}\pi _{3}}$ integral of $\digamma$ is given as:

where $x, y\in \left[\pi _{1}, p_{1}\pi _{2}+\left(1-p_{1}\right) \pi _{3} \right] \times \left[\pi _{3}, p_{2}\pi _{4}+\left(1-p_{2}\right) \pi _{3} \right].$

Remark 3.13. It is obvious that if we use $p_{1} = p_{2} = 1$, then Definition 3.12 transforms into Definition 2.7.

In ^[36], H. Kalsoom et al. introduced the following notions of post-quantum partial derivatives.

Definition 3.14. ^[36] Let $\digamma :\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right] \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a continuous function of two variables. Then the partial $p_{1}q_{1}$ -derivatives, $p_{2}q_{2}$-derivatives and $p_{1}q_{1}p_{2}q_{2}$ -derivatives at $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$ can be given as follows:

Now, from the above given concepts, we give the following new post-quantum partial derivatives.

Definition 3.15. Let $\digamma :\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right] \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a continuous function of two variables. Then the partial $p_{1}q_{1}$ -derivatives, $p_{2}q_{2}$-derivatives and $p_{1}q_{1}p_{2}q_{2}$ -derivatives at $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$ can be given as follows:

Remark 3.16. It is obvious that if we set $p_{1} = p_{2} = 1$ in Definitions 3.14 and 3.15, then we obtain the Definition 2.8.

4.   Quantum Ostrowski type inequalities for function of two variables

In this section, we prove some new post-quantum Ostrowski type inequalities for the functions of two variables.

Lemma 4.1. Let $\digamma :\Delta \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R}$ be a twice partially $p_{1}q_{1}p_{2}q_{2}$-differentiable function on $\Delta ^{\circ }$. If partial $p_{1}q_{1}p_{2}q_{2}$-derivatives $\frac{^{\pi _{2}, \; \pi _{4}}\partial _{p\, _{1}, q_{1}, p_{2}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{p_{1}, q_{1}}\tau \; ^{\pi_{4}}\partial _{p_{2}, q_{2}}s},$ $\frac{_{\pi _{1}}^{\pi _{4}}\partial_{p_{1}, q_{1}, p_{2}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{p_{1}, q_{1}}\tau \; ^{\pi _{4}}\partial _{p_{2}, q_{2}}s},$ $\frac{_{\pi _{3}}^{\pi _{2}}\partial _{p_{1}, q_{1}, p_{2}, q_{2}}^{2}\digamma\left(\tau, s\right) }{^{\pi _{2}}\partial _{p_{1}, q_{1}}\tau _{\; \pi_{3}}\partial _{p_{2}, q_{2}}s}$ and $\frac{_{\pi _{1}, \; \pi _{3}}\partial_{p_{1}, q_{1}, p_{2}, q_{2}}^{2}\digamma \left(\tau, s\right) }{_{\pi_{1}}\partial _{p_{1}, q_{1}}\tau \; _{\pi _{3}}\partial _{p_{2}, q_{2}}s}$ arecontinuous and integrable on $\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right] \subseteq \Delta ^{\circ }$, then followingidentity holds for $p_{1}q_{1}p_{2}q_{2}$-integrals:

where

for all $\left(x, y\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, d\right]$ and $0 < q_{i} < p_{i}\leq 1.$

Proof. From Definitions 3.14 and 3.15, we have

By the equality (4.2) and Definition 3.12, we have

Similarly, by the equalities (4.3)–(4.5) we obtain the identities

and

Thus, we have

which completes the proof.

Remark 4.2. In Lemma 4.1, if we set $p_{1} = p_{2} = 1$, then the Lemma 4.1 reduces to [21,Lemma 2].

Remark 4.3. In Lemma 4.1, if we set $p_{1} = p_{2} = 1$ and $q_{1}, q_{2}\rightarrow 1^{-},$ then Lemma 4.1 reduces to [41,Lemma 1].

In terms of brevity, we will use the following notations

Theorem 4.4. Suppose that the assumptions of Lemma 4.1 hold. If $\left\vert \Phi (\tau, s)\right \vert,$ $\left \vert \Theta (\tau, s)\right\vert,$ $\left \vert \Psi (\tau, s)\right \vert$ and $\left \vert \Omega(\tau, s)\right \vert$ are co-ordinated convex on $\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$, then we have the inequality

Proof. Taking modulus in (4.1), we have

Since $\left \vert \Phi (\tau, s)\right \vert$ is co-ordinated convex, we obtain

By the similar way, as $\left \vert \Theta (\tau, s)\right \vert,$ $\left \vert \Psi (\tau, s)\right \vert$ and $\left \vert \Omega (\tau, s)\right \vert$ are co-ordinated convex, we establish

and

If we substitute the inequalities (4.9)–(4.12) in (4.8), then we obtain the desired result.

Remark 4.5. In Theorem 4.4, if we set $p_{1} = p_{2} = 1$, then Theorem 4.4 reduces to [21,Theorem 5].

Corollary 4.6. In Theorem 4.4, if we choose $\left \vert \Phi (\tau, s)\right \vert,$ $\left \vert \Theta (\tau, s)\right \vert,$ $\left \vert\Psi (\tau, s)\right \vert$, $\left \vert \Omega (\tau, s)\right \vert \leq M$ for all $\left(\tau, s\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right],$ then we obtain the following post-quantumOstrowski type inequality

Remark 4.7. In Corollary 4.6, if we put $p_{1} = p_{2} = 1$, then we recapture the inequality (1.7).

Remark 4.8. In Corollary 4.6, if we set $p_{1} = p_{2} = 1$ and $q_{1}, q_{2}\rightarrow 1^{-},$ then Corollary 4.6 reduces to Theorem 1.3.

Theorem 4.9. Suppose that the assumptions of Lemma 4.1 are hold. If $\left \vert \Phi (\tau, s)\right \vert ^{s},$ $\left \vert \Theta (\tau, s)\right \vert ^{s},$ $\left \vert \Psi (\tau, s)\right \vert ^{s}$ and $\left \vert \Omega (\tau, s)\right \vert ^{s}$ are co-ordinated convex on $\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$, then we have the inequality

where $\frac{1}{r}+\frac{1}{s} = 1,$ $s > 1.$

Proof. From the Lemma 4.1, we have

By using the well-known Hölder inequality and the co-ordinated convexity of $\left \vert \Phi (\tau, s)\right \vert ^{s}$, we obtain

Similarly, we have

and

By substituting the inequalities (4.14)–(4.17) in (4.13), then we obtain the required result.

Remark 4.10. In Theorem 4.9, if we use $p_{1} = p_{2} = 1$, then Theorem 4.9 reduces to [21,Theorem 6].

Corollary 4.11. In Theorem 4.9, if we choose $\left \vert \Phi (\tau, s)\right \vert,$ $\left \vert \Theta (\tau, s)\right \vert,$ $\left \vert\Psi (\tau, s)\right \vert$, $\left \vert \Omega (\tau, s)\right \vert \leq M$ for all $\left(\tau, s\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right],$ then we obtain the following post-quantumOstrowski type inequality

Remark 4.12. In Corollary 4.11, if we set $p_{1} = p_{2} = 1$, then we recpature the inequality (1.8).

Remark 4.13. In Corollary 4.11, if we set $p_{1} = p_{2} = 1$ and $q_{1}, q_{2}\rightarrow 1^{-},$ then Corollary 4.11 reduces to Theorem 1.4.

Theorem 4.14. Suppose that the assumptions of Lemma 4.1 hold. If $\left\vert \Phi (\tau, s)\right \vert ^{s},$ $\left \vert \Theta (\tau, s)\right\vert ^{s},$ $\left \vert \Psi (\tau, s)\right \vert ^{s}$ and $\left \vert\Omega (\tau, s)\right \vert ^{s}$, $s\geq 1$ are co-ordinated convex on $\left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right]$, then we have the inequality

Proof. By using the power mean inequality and the co-ordinated convexity of $\left \vert \Phi (\tau, s)\right \vert ^{s}$, we obtain

Similarly, since $\left \vert \Theta (\tau, s)\right \vert ^{s},$ $\left \vert \Psi (\tau, s)\right \vert ^{s}$ and $\left \vert \Omega (\tau, s)\right \vert ^{s}$ are co-ordinated convex, we establish

and

If we substitute the inequalities (4.18)–(4.21) in (4.13), then we obtain the desired result.

Remark 4.15. In Theorem 4.14, if we assume $p_{1} = p_{2} = 1$, then Theorem 4.14 becomes [21,Theorem 7].

Corollary 4.16. In Theorem 4.14, if we choose $\left \vert \Phi (\tau, s)\right \vert,$ $\left \vert \Theta (\tau, s)\right \vert,$ $\left \vert\Psi (\tau, s)\right \vert$, $\left \vert \Omega (\tau, s)\right \vert \leq M$ for all $\left(\tau, s\right) \in \left[\pi _{1}, \pi _{2}\right] \times \left[\pi _{3}, \pi _{4}\right],$ then we obtain the following post-quantumOstrowski type inequality

Remark 4.17. In Corollary 4.16, if we consider $p_{1} = p_{2} = 1$, then we recapture the inequality (1.9).

Remark 4.18. In Corollary 4.16, if we consider $p_{1} = p_{2} = 1$ and $q_{1}, q_{2}\rightarrow 1^{-},$ then Corollary 4.16 reduces to Theorem 1.5.

5.   Conclusions

In this study, we proved some new post-quantum variants of Ostrowski type inequalities for the differentiable functions of two variables. We also proved that the results proved in this study are the refinements of some existing results in the field of integral inequalities. It is an interesting and new problem that the upcoming researchers can obtain the similar inequalities for the other kind of convexity in their future work.

Acknowledgments

We want to give thanks to the Dirección de investigación from Pontificia Universidad Católica del Ecuador for technical support to our research project entitled: "Algunas desigualdades integrales para funciones convexas generalizadas y aplicaciones".

Conflict of interest

The authors declare no conflicts of interest.