1. Introduction
The study of convex functions is an important field in mathematics which occurs almost everywhere in pure and applied branches of mathematics, but mainly it plays a key role in predicting the approximate solutions for linear and non-linear programming. A well known inequality called the Hermite–Hadamard inequality, introduced first by Hadamard [
1], states:
Let
be a convex function on
, then
inequality holds.
Over the last three decades, a number of researchers concentrated their research on various types of Hadamard’s inequality and its related applications. But we are able to acknowledge with confidence that the actual work on it was diversified by Dragomir in 1992 (see [
2]). Many interested researchers have generalized (
1) and constructed multiple formulations in different forms as a result of this progressive work. Dragomir established a number of Hermite–Hadamard-type inequalities for many kinds of functions using various assumptions, like convex functions defined on a disc in the plane and on a ball in the space. See Ref. [
2] for further details. The Hermite–Hadamard-type inequalities for convex functions on
n-dimensional convex bodies were investigated in 2006 by de la Cal and Cárcamo [
3]. They derived two main results; the first one is related to mappings on polytopes in
, and the second one is related (
1) via symmetric random vectors on arbitrary norms on
. Another extension of (
1) for a function defined on a convex subset of
was introduced by Yang in 2012 [
4]. Furthermore, matrix and operator inequalities of the Hermite–Hadamard type were introduced by Moslehian recently in [
5]. The idea of a coordinated convex function was given by Dragomir [
6]; it is defined as follows:
For any two-dimensional interval , with and . A function is a coordinated convex if the partial mappings , and , are convex for all and .
Using this definition, Dragomir derived the Hermite–Hadamard inequality on a coordinated convex function, which is stated as follows:
Theorem 1 ([
6])
. Let be a convex coordinated function, theninequalities hold. In the current era, the study of quantum calculus is also a center of attention for researchers. Euler (1707–1783) was the first person to propose the concept of quantum calculus after the 17th century. Its role provides a bridge between physics and mathematics. F. H. Jackson and others focused more on quantum calculus in the early 20th century. Quantum calculus (
-calculus), a subfield of time-scale calculus, deals with the study of difference equations and provides solutions to a wide range of dynamical problems. Furthermore, we can state that q-calculus generalizes the derivative and integration of classical calculus and that we retrieve the classical conclusions as
. The Euler notion was first studied by Jackson [
7] in 1910. He used it to define
-integration and
-derivatives for continuous functions over an interval (0, ∞), commonly called calculus without limits. Al-Salam [
8] introduced the concepts of
-fractional and
-Riemann–Liouville fractional inequalities in 1966. The fundamental principles of
-calculus were first discussed by Kac and Cheung in their book [
9]. We recommend reading [
10,
11] for some details on quantum calculus.
The -integral and -derivative of continuous functions over finite intervals were specifically presented in 2013 by Tariboon and Ntouyas.
Definition 1 ([
12])
. For a continuous function , suppose that for Then, is defined as follows: If we put , (
3)
becomes Definition 2 ([
12])
. For a continuous function , the -integral on can be obtained byIf we put = 0, (
4)
becomes Furthermore, Jackson [
7] derived its subsequent form:
Utilizing the above essentials of quantum calculus, Tariboon and Ntouyas developed various well-known inequalities on finite intervals in [
13] and the Hermite–Hadamard inequality is one of them. But there is an error in the first form of the quantum Hermite–Hadamard inequality, which was pointed out by Alp et al. in [
14], where they gave a counter-example to the previously established form of this inequality. Thus, they formulated the corrected form of Hermite–Hadamard inequality in quantum calculus, which can be presented as
Theorem 2 ([
14])
. For being a convex function on , theninequality holds for . Dragomir et al., in [
15], published some work on quantum calculus. They provided some definitions of the partial
-derivative,
-derivative,
-derivative, and
-integral and shifted the notion of quantum calculus on finite intervals with work on bi-intervals (i.e., coordinates in plane).
Definition 3 ([
15])
. Let be a continuous function of two variables from to ℜ and , thenandare called the partial -derivative, -derivative, and -derivative at . Definition 4 ([
15])
. Let be a continuous function from to ℜ, thenis called the -integral on . With the help of these definitions, they derived the Hermite–Hadamard inequality for the convex function of two variables.
Theorem 3. Let be a convex coordinated function on , then for and , the following inequalities hold: In July 2020, N. Alp and M. Z. Sarikaya [
16] published an article in which they gave a counter-example that disproves the (
5) and derived the correct form of the Hermite–Hadamard inequality.
Theorem 4 ([
16])
. Let be a convex coordinated function on , then for and , the following inequalities hold: In the recent past, a number of publications have been presented regarding the improvement and development of different variants of the quantum Hermite–Hadamard and related inequalities (see [
17,
18,
19,
20], and references therein). However, in the present article we are interested in exploring such findings under the new and latest perspective of symmetric quantum calculus.
2. Preliminaries and New Results in Symmetric Quantum Calculus
The symmetric partial derivatives of the function are defined below.
Definition 5 ([
21])
. Let be a function which is defined on , thenare said to be symmetric partial derivatives of with respect to and , respectively. Da Cruz et al. introduced a new concept called symmetric quantum calculus in [
22]. It plays an important role in developing hypergeometric and a class of harmonic functions in complex analysis [
23]. The symmetric quantum analogue of any number
can be defined as [
9]
In addition to this, the
-differential and
-differential for
and
can be defined as
The derivative and integral in symmetric quantum calculus can be derived as
Definition 6 ([
24])
. For a continuous function , then the -derivative or -symmetric derivative on is defined asif , this becomes Definition 7 ([
24])
. For a continuous function , then the -integral or -symmetric integral on is defined ashere, or In this section, with the help of all of the notions above, we will define some new definitions in symmetric quantum calculus that assist in constructing the -symmetric Hölder’s inequality and Hermite–Hadamard inequalities.
We introduce and provide definitions below.
Definition 8. Let be a continuous function of two variables from to ℜ and , thenandare called the partial -, -, and -symmetric derivatives at . If , and exist for all , then is called partially -, -, and -symmetric differentiable on .
Example 1. Let be a function.
- Case 1.
If is a non-convex function. Suppose that , then the partial -symmetric derivative of is
The partial -symmetric derivative of is And also, the partial -symmetric derivative of isand finally, - Case 2.
If is a convex function. Suppose that , then the partial -symmetric derivative of is
Similarly, the partial -symmetric derivative of is And also, the partial -symmetric derivative of is Remark 1. If in both cases of Example 1, then the results will coincide with the symmetric partial derivatives of such functions coming from (
7)
and (
8)
, respectively. Definition 9. Let be a continuous function from to ℜ, then for ,oris called -symmetric integral on . From (
10), we can say that
Moreover, for any point
, we can write
Some results that are given below hold for Definitions 8 and 9.
Theorem 5. Let be a continuous function from to ℜ, thenfor . Proof. (1) Using (
10), and then, (
9),
- (2)
Using (
9), and then, (
10),
- (3)
Using (
11), and then, (2) results in
Hölder’s inequality for the double sum can be stated as
Theorem 6 ([
15])
. For any two real or complex sequences , and , with inequality holds for finite sums. Now, with the help of Theorem 5, we construct Hölder’s inequality for the function of two variables in symmetric quantum calculus.
Theorem 7. (-Symmetric Hölder’s Inequality). Let and be continuous functions from to ℜ, then for , , and = 1, with , theninequality holds. Proof. Using the definition of a
-symmetric integral and (
12), we have
□
In our previous work [
25], we derived some results in symmetric quantum calculus which are given below.
For any function
which is convex differentiable on
and
, then
3. Hermite–Hadamard Inequalities for Coordinated Convex Function in Symmetric Quantum Calculus
In this section, we will construct the Hermite–Hadamard inequality for the convex coordinated function in symmetric quantum calculus and its types. For this, let be a class of continuous functions from a one- or two-dimensional interval to ℜ that satisfies some of the following properties of symmetric quantum integrals:
- (i)
For with and , then for all .
- (ii)
For , then .
It is important to mention here that throughout the rest of the paper all the functions that we will consider belong to the class .
Theorem 8. Let be a convex coordinated function on , then for and ,inequalities hold. Proof. Let
be a convex function from
to
ℜ and defined as
. Then, using (
14), we have
for
and for all
. For any
, taking the symmetric
-integral on
Similarly, let
be a convex function from
to
ℜ and defined as
. Then, again using (
14), we have
for
and for all
, for any
.
Taking the symmetric
-integral on
,
Replacing
by
in (
18) and
by
in (
20), then adding, we have
Applying (
14) on the right-hand side of (
22):
Joining (
22)–(
24), we obtain the desired result. □
Remark 2. If approach 1 in Theorem 8, then it is reconstructed into the classical one (
2).
Now, we derive another new result of symmetric quantum calculus which will assist in proving some related results of Hermite–Hadamard inequalities.
Theorem 9. Let be a convex function on and differentiable as well, then for inequalities hold. Proof. Let and be the line segments joining the points and , respectively. Owing to the convexity of on , is always below the line segments and . The equations of these line segments are and .
From
Figure 1, we can write the equations of
, and
using the two-point form formula of the equation of a line as
Also, from
Figure 1, from
to
,
Similarly, from
to
Taking the symmetric
-integral, the inequalities (
26) and (
27) from
to
and
to
, respectively, then from
Figure 1,
where
.
So, inequality (
28) becomes
From
Figure 1, we can also conclude that
Finally, if we combine the left-hand side of (
14) with (
32) and (
33), we obtain the desired result. □
Remark 3. If approaches 1 in Theorem 9, then it is reconstructed into corollary 2.1 from [
26].
Theorem 10. Let be a convex coordinated function on , then for and ,inequalities hold. Proof. Let
be a convex function from
to
ℜ and defined as
. Then, using (
32), we have
holds for all
and
. Taking the symmetric
-integral from
to
:
Similarly, let
be a convex function from
to
ℜ and defined as
. Then, again using (
32), we have
holds for all
and
. Now, taking the symmetric
-integral from
to
.
Now,
(
35) +
(
36), we have
Now, applying (
32) on the right-hand side of (
37), we have
Adding (
38)–(
43), and then, combining the obtained result with (
37) and the left-hand sides of (
22) and (
23), we obtain the desired result. □
Remark 4. If approach 1 in Theorem 10, then it is reconstructed into the classical one of theorem 2.2 from [
26].
Theorem 11. Let be a convex coordinated function on , then for and ,inequalities hold. Proof. Let
be a convex function from
to
ℜ and defined as
. Then, using (
16), we have
for and for all . For any .
Taking the symmetric
-integral on
and dividing by
, we have
Similarly, let
be a convex function from
to
ℜ and defined as
. Then, again using (
16), we have
for
and for all
, for any
.
Taking the symmetric
-integral on
and dividing by
, we have
Adding (
46) and (
48), we have
Replacing
by
and
by
in (
45) and (
47), respectively, then adding, we have
Finally, by (
22), (
24), (
49), and (
50) we obtain the desired result. □
Theorem 12. Let be a convex coordinated function on , then for and ,inequalities hold. Proof. Using (
15) and same methodology of Theorem 5, we will get the desired result. □
Now, we will investigate our main result (
17) through examples.
Example 2. Let be a convex coordinated function on and defined as , applying it on (
17)
. Let , , , and . - Case 1.
If , put and .
Then, the term on the left-hand side of (
17)
becomes The right-hand part of the extreme left term of (
17)
becomes The middle term of (
17)
becomes The second term from the right-hand side of (
17)
becomes The extreme right-hand term of (
17)
becomes Which endorses our result.
- Case 2.
If , put and .
Then, the term on the left-hand side of (
17)
becomes The right-hand part of the extreme left term of (
17)
becomes The middle term of (
17)
becomes The second term from the right-hand side of (
17)
becomes The extreme right-hand term of (
17)
becomes Which also endorses our result.
- Case 3.
If , put .
Then, the term on the left-hand side of (
17)
becomes The right-hand part of the extreme left term of (
17)
becomes The middle term of (
17)
becomes The second term from the right-hand side of (
17)
becomes The extreme right-hand term of (
17)
becomes Which also endorses our result.