1. Introduction
In mathematics,
q-calculus, also known as quantum calculus, is the study of calculus with no limits. In quantum calculus, we obtain
q-analogues of mathematical objects that can be recaptured as
. The concept was presented by renowned mathematician Euler in the Eighteenth Century, introducing the number in
q-infinite series defined by Newton. Later on, Jackson [
1] introduced the concept of
q-definite integrals extending the concept of
q-calculus. Quantum calculus has numerous applications in various fields of mathematics and physics, especially in orthogonal polynomials, number theory, hypergeometric functions, mechanics, and relativity theory. For the background study of quantum calculus theory and the theory of inequalities in quantum calculus, interested, readers are referred to [
2,
3,
4].
The following double inequality is well known in the literature as Hermit-Hadamard’s inequality. Let
be convex on
, where
with
:
Due to its role in various fields of modern mathematics, such as numerical analysis and functional and mathematical analyses, the validity of Hermit-Hadamard’s inequality is important; see [
5,
6,
7].
The concept of
q-derivatives over the definite interval
was attended by Tariboon et al. [
8,
9] and addressed numerous problems on quantum analogues such as the
q-Hölder inequality, the
q-Ostrowski inequality, the
q-Cauchy-Schwarz inequality, the
q-Grüss-Čebyšev integral inequality, the
q-Grüss inequality, and other integral inequalities by classical convexity. Most recently, Alp et al. [
10] proved the
q-Hermite-Hadamard inequality and then acquired the generalized
q-Hermite-Hadamard inequality; also, they studied some integral inequalities, which provide quantum estimates for the left part of the quantum analogue of the
q-Hermite-Hadamard inequality through
q-differentiable convex and quasi-convex functions, for more details and interesting applications see References [
11,
12,
13,
14,
15,
16]. A recent development in the context of the above concept was presented by Tunç and Göv [
17,
18,
19], named as
-derivatives and
-integrals over
. Some well-known results that depend on
-calculus are the
-Minkowski inequality, the
-Hölder inequality, the
-Ostrowski inequality, the
-Cauchy-Schwarz inequality, the
-Grüss-Čebyšev integral inequality, the
-Grüss inequality, and other integral inequalities by classical convexity. Dragomir et al. [
20] studied some integral inequalities that provide
estimates for the right part of the quantum analogue of the Hermit-Hadamard inequality through
- differentiable convex and quasi-convex functions.
Dragomir [
20] proved the following inequalities of the Hermite-Hadamard type for coordinated convex functions on a rectangle from the plane
.
Theorem 1. Let be convex on the coordinates on with and . Then, one has the inequalities: Kunt et al. [
21] proved the following
-Hermite-Hadamard inequalities for convex functions by the quantum integral approach:
Theorem 2. Let and h: be a convex differentiable function on . Then, the following integral inequality holds: Latif et al. [
22] for the first time introduced the idea of the
-Hermite-Hadamard inequality and obtained the
-Hermite-Hadamard-type inequalities for coordinated convex functions on the quantum integral. Furthermore, Alp et al. [
23], proved the corrected
-Hermite-Hadamard-type inequalities for coordinated convex functions by using quantum calculus.
Theorem 3. Suppose that is coordinated convex on ; the following inequalities hold for all : This paper is organized in the following way. After this Introduction, in
Section 2, we define some new concepts regarding the
-calculus of the function of two variables and discuss some properties and examples. Moreover, our aim is to describe the Hermite-Hadamard-type inequalities for functions of two variables using
-calculus, which is a generalization of Theorem 3 presented by Alp et al. [
23]. Next, we provide some new estimates for
-analogues of the Hermite-Hadamard-type inequalities of the functions of two variables using convexity and quasi-convexity on coordinates; these results are a generalization of all those results that were presented in [
22].
2. Preliminaries and Main Results
Definition 1. Let be a continuous function in each variable; let , ; the partial -, - and -derivatives at are, respectively, defined as: In a similar way, higher order partial derivatives can be defined.
Definition 2 ([
24])
. For any real number , the -analogue of n is defined as: Example 1. Let and then:and:
which are the
-analogues of the usual partial derivatives:
and
The
-exponential functions are defined as:
In the limiting case, as
and for
, the exponential function defined by (
3) reduces to the exponential function:
Replacing
x by
in (
3), we obtain:
Example 2. Now, we consider the following exponential function in the -calculus of the functions of two variables defined by: We find the partial derivative of this function with respect to s using the -calculus of the functions of two variables is: Again, taking the partial derivative with respect to t on both sides of Equation (5) using the -calculus of the functions of two variables, we have:
which is the
-analogue of the usual partial derivative:
The
-trigonometric functions are defined as:
Replacing
x by
in (
6), we obtain:
Example 3. Now, we consider the following trigonometric function of two variables in the -calculus of the functions of two variables defined by: The partial derivative with respect to s using the -calculus of functions of two variables is: Again, taking the partial derivative with respect to t on both sides of Equation (7) using the -calculus of functions of two variables, we have:which is the -analogue of the usual partial derivative: Definition 3. Let be a continuous function in each variable; let where then the definite -integral on is defined as:for Example 4. Let for , then we have:which is the -analogue of the usual integral: Remark 1. By Definition 3, it directly follows:
for Setting and for , Definitions 1 and 3 reduce to the classical definitions of partial derivatives and the classical integration of double integrals, respectively.
For where , Definitions 1 and 3 reduce to Definitions 1 and 2 in [22], respectively.
Theorem 4. Let be continuous functions in each variable, then we can prove the following easily:
Proof. The proof is directly followed by -calculus. We omit the details. □
Theorem 5. Let be convex on the coordinates on for where , then the following inequalities hold: Proof. Since
is a convex function on the coordinates, it follows that the partial mapping
defined by:
is convex on
for
, and hence, by Theorem 2:
By
-integrating the inequality (
9) on
, we have:
Similarly, consider
defined by
Then, obviously,
is a convex function. Again, the application of Theorem 2 yields the following inequality:
By
-integrating the inequality (
11) on
, we have:
Setting
and
in (9) and (11) respectively:
and hence, by the addition of (14) and (15), we get:
By using Theorem 2 on the right-hand side of (13), we have:
The addition of inequalities (16)–(19) yields the following:
□
Remark 2. Setting and for then Theorem 5 reduces to Theorem 1.
Also, if for in Theorem 5, then Theorem 5 reduces to Theorem 3.
Theorem 6 ([
22])
. Let and be sequences of real (or complex) numbers. If with , then the following inequality holds: Provided that all the sums are finite.
Theorem 7. Let g and h be functions defined on and for . If with , then the following inequality holds: Proof. The proof is directly followed by -calculus. We omit the details. □
Lemma 1. Let be a function such that -derivatives exist on (the interior of ) and for Moreover, if is continuous and integrable on , then: Proof. By the Definition 1 of partial
-derivatives and the Definition 3 of definite
-integrals, we have:
Using (20)–(32) and simplifying, we get:
Multiplying both sides of (
33) by
, we get the desired equality.
□
Remark 3. Setting and for in Lemma 1, then Lemma 1 reduces to Lemma 1 proved in [25]. Furthermore, if for in Lemma 1, then Lemma 1 reduces to Lemma 2 proven in [22].
Now, we can introduce quantum integral inequalities for functions whose partial -derivative on satisfies the convexity on coordinates on .
Theorem 8. Let be a function such that -derivatives exist on (the interior of ) and for . Moreover, if is continuous and integrable on and is a convex function on the coordinates on for , then the given inequality holds: Proof. By applying Lemma 1, using the
-Hölder inequality, and the convexity of
on the coordinates on
, then we have:
By using Definition 3, we get:
Using the values of the above -integrals, we get the desired inequality. □
Theorem 9. Let be a function such that -derivatives exist on (the interior of ) and for Moreover, if is continuous and integrable on and is a convex function on the coordinates on . If and , then the given inequality holds: Proof. By applying Lemma 1, the
-Hölder inequality, and the convexity of
on the coordinates on
, then we have:
Considering the first
-integral from (
34) and making use of the substitution
, we obtain:
Considering the second
-integral from (
34) and making use of the substitution
, we obtain:
After this calculation, we get:
Remark 4. Setting and for in Theorems 8 and 9, then Theorems 8 and 9 reduce to Theorems 4 and 5 proven in [25]. Furthermore, if for in Theorems 8 and 9, then Theorems 8 and 9 reduce to Theorems 6 and 7 proven in [22].
Theorem 10. Let be a function such that -derivatives exist on (the interior of ) and for . Moreover, if is continuous and integrable on and is quasi-convex on the coordinates on for , then the inequality holds:where and are defined in Theorem 8. Proof. By applying Lemma 1, using the
-Hölder inequality, and the quasi-convexity of
on the coordinates on
, then we have:
Now, using the properties of the supremum and Definition 3, we get the desired result from (
35). □
Theorem 11. Let be a function such that -derivatives exist on (the interior of ) and for Moreover, if is continuous and integrable on and is a quasi-convex function on the coordinates on . If and , then the given inequality holds:where and are defined in Theorem 9. Proof. By applying Lemma 1, using the
- Hölder inequality, and the quasi-convexity of
on the coordinates on
, then we have:
Now, using the properties of the supremum and Definition 3, we get the desired result. □