Abstract
In this investigation, we demonstrate the quantum version of Montgomery identity for the functions of two variables. Then we use the result to derive some new Ostrowski-type inequalities for the functions of two variables via quantum integrals. We also consider the particular cases of the key results and offer some new integral inequalities.
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1 Introduction
In the field of q-analysis, many studies have recently been carried out, starting with Euler owing to a vast requirement for mathematics that models quantum computing q-calculus occurred for the relationship between physics and mathematics. In different areas of mathematics, it has numerous applications such as combinatorics, number theory, basic hypergeometric functions, orthogonal polynomials, mechanics, the theory of relativity, and quantum theory [1, 2]. 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 without limit calculus was firstly given by Jackson [3]. In 1908–1909 the general form of the q-integral and q-difference operator was defined by Jackson [4]. In 1969, for the first time, Agarwal [5] defined the q-fractional derivative. In 1966–1967, Al-Salam [6] introduced a q-analog of the q-fractional integral and q-Riemann–Liouville fractional. In 2004, Rajkovic gave a definition of the Riemann-type q-integral, which was generalized to Jackson q-integral. In 2013, Tariboon [7] introduced the \({}_{a}D_{q}\)-difference operator. Recently, in 2020, Bermudo et al. [8] introduced the notions of the \({}^{b}D_{q}\)-derivative and integral.
Many well-known integral inequalities, such as the Hölder, Hermite–Hadamard, Simpson, Newton, Ostrowski, Cauchy–Bunyakovsky–Schwarz, Gruss, Gruss–Chebyshev, and other integral inequalities, have been studied in the setup of q-calculus using the concept of classical convexity. For more results in this direction, we refer to [9–20].
In 1938, Ostrowski [21] established the following interesting integral inequality.
Theorem 1
Let \(F:[a,b]\rightarrow \mathbb{R} \) be a differentiable function on \((a,b)\) with bounded derivative, that is, \(\Vert F' \Vert _{\infty }:=\sup_{x \in (a,b)} \vert F'(x ) \vert <\infty \). Then we have the following integral inequality:
for all \(\tau \in {}[ a,b]\). The constant \(\frac{1}{4}\) is the best possible.
Inequality (1.1) can be rewritten in the equivalent form
Since 1938, when Ostrowski proved his famous inequality (see [21]), this inequality has been studied by many mathematicians in various fields, such as numerical analysis and probability.
Various generalizations and extensions of the Ostrowski integral inequality for bounded-variation, monotonic, Lipschitzian, convex, absolutely continuous, and n times differentiable mappings with error estimates for some special means and some numerical quadrature rules were considered by many scientists. For more recent results, we refer to [22–32] and the references therein.
A formal definition of coordinated convex (concave) functions may be expressed as follows.
Definition 1
A function \(F:\Delta \rightarrow \mathbb{R} \) is said to be coordinated convex on Δ if it satisfies the following inequality for all \((x ,y),(z,w)\in \Delta \) and \(\lambda ,\mu \in {}[ 0,1]\):
The mapping F is coordinated concave on Δ if inequality (1.3) holds in the reversed direction for all \((x ,y),(z,w)\in \Delta \) and \(\lambda ,\mu \in {}[ 0,1]\).
Latif et al. [33] established the following Ostrowski-type inequalities for coordinated convex functions.
Theorem 2
Let \(F:\Delta :=[a,b]\times {}[ c,d]\rightarrow \mathbb{R} \) be a twice differentiable mapping on \(\Delta ^{\circ }\) with \(a< b\), \(c< d\), \(a,c\geq 0\) such that \(\frac{\partial ^{2}F}{\partial s\,\partial t}\in L(\Delta )\). If \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert \) is coordinated convex on Δ and \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert \leq M\), \((x,y)\in \Delta \), then we have the following inequality:
where
Theorem 3
Let \(F:\Delta :=[a,b]\times {}[ c,d]\rightarrow \mathbb{R} \) be a twice differentiable mapping on \(\Delta ^{\circ }\) with \(a< b\), \(c< d\), \(a,c\geq 0\) such that \(\frac{\partial ^{2}F}{\partial s\,\partial t}\in L(\Delta )\). If \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert ^{p}\) is coordinated convex on Δ, \(p>1\), \(\frac{1}{p}+\frac{1}{r}=1\), and \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t}(x,y) \vert \leq M\), \((x,y)\in \Delta \), then we have the following inequality:
where \(A_{1}\) is defined in Theorem 2.
Theorem 4
Let \(F:\Delta :=[a,b]\times {}[ c,d]\rightarrow \mathbb{R} \) be a twice differentiable mapping on \(\Delta ^{\circ }\) with \(a< b\), \(c< d\), \(a,c\geq 0\) such that \(\frac{\partial ^{2}F}{\partial s\,\partial t}\in L(\Delta )\). If \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t} \vert ^{p}\) is coordinated convex on Δ, \(p> 1\), and \(\vert \frac{\partial ^{2}F}{\partial s\,\partial t}(x,y) \vert \leq M\), \((x,y)\in \Delta \), then we have the following inequality:
where \(A_{1}\) is defined in Theorem 2.
Inspired by this ongoing study, we establish some new quantum Ostrowski inequalities for \(q_{1}q_{2}\)-differentiable coordinated convex functions. This is the primary motivation of this paper. The ideas and strategies of the paper may open new venues for further research in this field.
2 Preliminaries of q-calculus and some inequalities
In this section, we review the basic notions and findings needed to prove our crucial results. Moreover, we use the following notation (see [34]):
Jackson [4] has defined the q-Jackson integral from 0 to b for \(0< q<1\) as follows:
provided that the series converges absolutely.
Moreover, he defined the q-Jackson integral in a general interval \([a,b]\) as
Definition 2
([35])
For a continuous function \(F: [ a,b ] \rightarrow \mathbb{R} \), the \(q_{a}\)-derivative of F at \(x\in [ a,b ] \) is defined by the expression
The function F is said to be \(q_{a}\)-differentiable on \([ a,b ] \) if \({} _{a}D_{q}F ( x ) \) exists for all \(x\in [ a,b ] \). If \(a=0\) in (2.2), then \({} _{0}D_{q}F ( x ) =D_{q}F ( x ) \), where \(D_{q}F ( x ) \) is the familiar q-derivative of F at \(x\in [ a,b ] \) defined by the expression (see [34])
Definition 3
([8])
For a continuous function \(F: [ a,b ] \rightarrow \mathbb{R} \), the \(q^{b}\)-derivative of F at \(x\in [ a,b ] \) is characterized by the expression
The function F is said to be \(q^{b}\)-differentiable on \([ a,b ] \) if \({} ^{b}D_{q}F ( x ) \) exists for all \(x\in [ a,b ] \). If \(b=0\) in (2.2), then \({} ^{0}D_{q}F ( x ) =D_{q}F ( x )\), where \(D_{q}F ( x ) \) is the familiar q-derivative of F at \(x\in [ a,b ] \) defined by the expression (see [34])
Definition 4
([35])
Let \(F: [ a,b ] \rightarrow \mathbb{R} \) be a continuous function. Then the \(q_{a}\)-definite integral on \([ a,b ] \) is defined as
On the other hand, Bermudo et al. [8] gave the following new definition of the quantum integral.
Definition 5
Let \(F: [ a,b ] \rightarrow \mathbb{R} \) be a continuous function. Then the \(q^{b}\)-definite integral on \([ a,b ] \) is defined as
For more detail about \(q^{b}\)-integrals and corresponding inequalities, we refer to [8].
We have to give the following notation, which will be used many times in the next sections (see [34]):
Lemma 1
([36])
We have the equality
for \(\alpha \in \mathbb{R} \backslash \{ -1 \} \).
Latif et al. [37] defined the \(q_{ac}\)-integral and partial q-derivatives for two-variable functions as follows.
Definition 6
Suppose that \(F: [ a,b ] \times [ c,d ] \subset \mathbb{R} ^{2}\rightarrow \mathbb{R} \) is a continuous function. Then the definite \(q_{ac}\)-integral on \([ a,b ] \times [ c,d ] \) is defined as
for \(( x ,y ) \in [ a,b ] \times [ c,d ] \).
Lemma 2
If the assumptions of Definition 6hold, then
Definition 7
([37])
Let \(F: [ a,b ] \times [ c,d ] \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R} \) be a continuous function of two variables. Then the partial \(q_{1}\)-derivatives, \(q_{2}\)-derivatives, and \(q_{1}q_{2}\)-derivatives at \(( x,y ) \in [ a,b ] \times [ c,d ] \) can be given as follows:
For more detail on the related to q-integrals and derivatives for the functions of two variables, we refer to [37].
On the other hand, Budak et al. [38] gave the following definitions of \(q_{a}^{d}\)-, \(q_{b}^{c}\)-, and \(q^{bd}\)-integrals.
Definition 8
Suppose that \(F: [ a,b ] \times [ c,d ] \subset \mathbb{R} ^{2}\rightarrow \mathbb{R} \) is continuous function. Then the following \(q_{a}^{d}\)-, \(q_{c}^{b}\)-, and \(q^{bd}\)-integrals on \([ a,b ] \times [ c,d ] \) are defined by
and
respectively, for \(( x ,y ) \in [ a,b ] \times [ c,d ] \).
Definition 9
([39])
Let \(F: [ a,b ] \times [ c,d ] \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R} \) be a continuous function of two variables. Then the partial \(q_{1}\)-derivatives, \(q_{2}\)-derivatives, and \(q_{1}q_{2}\)-derivatives at \(( x,y ) \in [ a,b ] \times [ c,d ] \) can be given as follows:
3 Quantum Montgomery identity for the functions of two variables
In this section, we prove a quantum Montgomery identity via newly defined quantum integrals for functions of two variables.
Lemma 3
Let \(F:\Delta \subseteq \mathbb{R} ^{2}\rightarrow \mathbb{R} \) be a twice \(q_{1}q_{2}\)-differentiable function on \(\Delta ^{\circ }\). If the partial \(q_{1}q_{2}\)-derivatives \(\frac{{}^{b, d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s}\) are continuous and integrable on \([ a,b ] \times [ c,d ] \subseteq \Delta ^{\circ }\), then we have the following identity for \(q_{1}q_{2}\)-integrals:
where
and
for \(q_{1},q_{2}\in ( 0,1 ) \).
Proof
By Lemma 2 and the definitions of \(\Psi _{q_{1}} ( t ) \) and \(\Psi _{q_{2}} ( s ) \) we obtain
From Definition 9 we have
To conclude the proof, we need to calculate the integrals in the right side of (3.2). By the definition of \(q_{1}q_{2}\)-integrals we obtain that
Similarly, we have
Additionally, we have
By similar operations we have
and
Now from (3.4)–(3.12) we obtain the following relations:
which finishes the proof. □
4 Some new quantum Ostrowski-type integral inequalities
In this section, we prove some new quantum Ostrowski-type inequalities for \(q_{1}q_{2}\)-differentiable coordinated convex functions using the lemma proved in the last section.
Theorem 5
Suppose that the assumptions of Lemma 3hold. If \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\), \(p_{1} > 1\), is coordinated convex on \([ a,b ] \times [ c,d ] \), then we have the inequality
where
and \(q_{1},q_{2}\in ( 0,1 ) \).
Proof
By taking the modulus in Lemma 3 we have
Applying the power mean inequality for quantum integrals, we obtain
Now using the convexity of \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\), we obtain
By using similar operations we find
and
We also observe that
By (4.3)–(4.10) we obtain the desired inequality, which finishes the proof. □
Theorem 6
Suppose that the assumptions of Lemma 3hold. If \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\) is coordinated convex on \([ a,b ] \times [ c,d ] \), then we have the inequality
where \(q_{1},q_{2}\in ( 0,1 ) \) and \(\frac{1}{r_{1}}+\frac{1}{p_{1}}=1\), \(p_{1}>1\).
Proof
Applying the well-known Hölder inequality for \(q_{1}q_{2}\)-integrals to the integrals in the right side of (4.2), we find
Now applying the convexity of \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert ^{p_{1}}\), we obtain that
From (4.13)–(4.17) we get the desired inequality, and the proof is accomplished. □
5 Some particular cases
In this section, we present some particular cases of the results given in Sect. 4.
Remark 1
In Theorem 5,
(i) By taking \(p_{1}=1\) and \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert \leq M\), we have the following new Ostrowski-type inequality:
Particularly, taking the limit as \(q_{1},q_{2}\rightarrow 1^{-}\) in (5.1), we reduce inequality (5.1) to (1.4).
(ii) By taking \(x=\frac{a+q_{1}b}{ [ 2 ] _{q_{1}}}\) and \(y=\frac{c+q_{2}d}{ [ 2 ] _{q_{2}}}\) we obtain the following new midpoint inequality:
where
and \(q_{1},q_{2}\in ( 0,1 ) \). Particularly, taking the limit as \(q_{1},q_{2}\rightarrow 1^{-}\) in (5.2), we reduce inequality (5.2) to [37, Theorem 2, inequality (2.4)]
(iii) By taking \(p_{1}=1\) we have the following inequality:
Particularly, taking the limit as \(q_{1},q_{2}\rightarrow 1^{-}\) in (5.4), we reduce inequality (5.4) reduces to [40, Theorem 2].
Remark 2
Consider Theorem 6.
(i) If \(\vert \frac{{}^{b,d}\partial _{q_{1},q_{2}}^{2}F ( t,s ) }{{}^{b}\partial _{q_{1}}t\,{}^{d}\partial _{q_{2}}s} \vert \leq M\), then as \(q_{1},q_{2}\rightarrow 1^{-}\), we obtain inequality (1.5).
(ii) Taking \(x =\frac{a+q_{1}b}{ [ 2 ] _{q_{1}}}\) and \(y=\frac{c+q_{2}d}{ [ 2 ] _{q_{2}}}\), we obtain the following new midpoint inequality:
6 Conclusion
In this research, we proved some new quantum Ostrowski-type inequalities for \(q_{1}q_{2}\)-differentiable coordinated convex functions using the \(q_{1}q_{2}\)-integrals. We also showed that the results proved in this research transformed into some new and known inequalities by considering the limits as \(q_{1},q_{2}\rightarrow 1^{-}\) in the main results. It is interesting that the forthcoming researchers can offer similar inequalities for different kinds of convexities in their future work.
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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., Chu, YM., Budak, H. et al. Quantum variant of Montgomery identity and Ostrowski-type inequalities for the mappings of two variables. Adv Differ Equ 2021, 25 (2021). https://doi.org/10.1186/s13662-020-03195-7
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DOI: https://doi.org/10.1186/s13662-020-03195-7