On some new midpoint inequalities for the functions of two variables via quantum calculus

In this paper, first we obtain a new identity for quantum integrals, the result is then used to prove midpoint type inequalities for differentiable coordinated convex mappings. The outcomes provided in this article are an extension of the comparable consequences in the literature on the midpoint inequalities for differentiable coordinated convex mappings.


Introduction
Quantum calculus, which is also named q-calculus, is occasionally mentioned as calculation method without limits. Herewith, one achieves q-analogues of mathematical tools that may be got back as q → 1. There are two techniques in q-addition, one of them is the Nalli-Ward-Al-Salam q-addition (NWA) and the other is Jackson-Hahn-Cigler qaddition (JHC). The first one is commutative and associative, at the same time as the second one is neither. Because of this, there are multiple q-analogs from time to time. These operators constitute the base of the method that combine hypergeometric collection with q-hypergeometric collection and gives many formulations of q-calculus a natural shape. The history of quantum calculus may be traced back to Euler (1707-1783), he first added the expression q in the tracks of Newton's infinite series. Recently, a great number of researchers have shown an eager hobby in studying and investigating quantum calculus and accordingly it emerged as an interdisciplinary subject. The quantum theory has become a cornerstone in theoretical mathematics and applied sciences, due to the fact that quantum analysis is very helpful in several fields and has huge applications in various areas of natural and applied sciences such as computer science and particle physics. Specifically, the theory has been seen as a critical tool for researchers operating with analytic number theory or in theoretical physics. This calculus method is a bridge that provides the connection between mathematics and physics. Owing to a large numbers of applications in quantum group theory, the quantum calculus also has a significant role for physicists. For some recent trends in quantum calculus the reader is referred to [1][2][3][4][5][6].
In recent decades the idea of convex functions has been drastically studied because of its fantastic significance in numerous fields of pure and applied sciences. Theory of inequalities and concept of convex functions are closely related to each other, thus they resemble inequalities that could be obtained inside the literature which are derived for convex and differentiable convex mappings; see [7][8][9][10][11][12][13].
We now consider how the convex functions of two-variables on the coordinates, which may be also called a coordinated convex function, is defined. Dragomir [14] presented the definition of coordinated convexity as follows.
In [14], Hermite-Hadamard type inequalities for convex function of two-variable on the coordinates are established by Dragomir as follows.

Theorem 1
If : → R is coordinated convex, then one has the inequalities The above inequalities are sharp. The inequalities in (1.2) hold in reverse direction if the mapping is a concave mapping on the coordinates.

Some important definitions and theorems with regard to quantum calculus
In this section, we review some valuable definitions, notations and inequalities associated to quantum calculus.
is the familiar q-derivative of at κ ∈ [α, β] defined by the expression (see [5]) We should note that the notation of the quantum numbers (see [5]) which will be used many times in our main results is defined by Moreover, we need to give the following lemma in order to prove our main results readily.

Lemma 1 ([21]) One has the identity
In [7], Alp et al. established the following q α -Hermite-Hadamard inequalities by using convex functions and quantum integral.

Then we have the q-Hermite-Hadamard inequalities
q n q n κ + 1q n β .
q αγ -integral and partial q-derivatives for two variables functions are defined by Latif in [22]. Definition 5 Let : ⊂ R 2 → R be a continuous function. Then, for (κ, η) ∈ , the definite q αγ -integral on is defined by Definition 6 ( [22]) Assume that : ⊆ R 2 → R is a continuous function of two variables. Then the partial q 1 -derivatives, q 2 -derivatives and q 1 q 2 -derivatives at (κ, η) ∈ can be given as follows: For more details related to q-derivatives and integrals for the mappings of two variables, one can refer to [22].
In addition to all these definitions, definitions of q δ α , q β γ and q βδ integrals and related inequalities of Hermite-Hadamard type are presented by Budak et al. in [23]. Definition 7 ([23]) Let : ⊂ R 2 → R be a continuous function. Then, for (κ, η) ∈ , the q δ α , q β γ and q βδ integrals on are defined by respectively.
Budak et al. gave two similar inequalities in addition to the above result. Also, Latif introduced a quantum version of Hölder's inequality for double integrals in [22].
Theorem 5 (q 1 q 2 -Hölder's inequality for two variables functions, [22]) Let x, y > 0, 0 < q 1 , Inspired by these ongoing studies, we establish some new quantum analogues of midpoint type inequalities for q-differentiable coordinated convex functions. Integral inequalities form a crucial branch of analysis and were combined with various types of quantum integrals but we had never seen these before with the integrals that we use here. For this reason, we studied the midpoint type inequalities in quantum calculus.

q-Derivatives for the functions of two variables
In this section, we recall partial q-derivatives for mappings of two variables offered by Ali et al. in [24].

Essential lemmas
In this section, we address three new identities, which are necessary to obtain our crucial results.
Let us start with the following lemma.
Also, it is easily observed that Now by the definition of definite q 1 q 2 -integrals and properties of q 1 q 2 -integrals, we obtain By using the similar operations, one can obtain F(a, c), Using the calculated integrals (I 1 )-(I 9 ) in (4.2), then we obtain the desired identity (4.1) which ends the proof.
Remark 1 Under the given conditions of Lemma 2 with q 1 , q 2 → 1 -, then we have the following identity: where 0 < q 1 , q 2 < 1 and is defined as in Lemma 2.
Proof If the strategy which was used in the proof of Lemma 2 are applied by taking into account the definition of where 0 < q 1 , q 2 < 1 and is defined as in Lemma 2.
Proof If the strategy which was used in the proof of Lemma 2 is applied by taking into

Some new q 1 q 2 -Hermite-Hadamard like inequalities
For brevity, we give some calculated integrals before giving new estimates: Now we give some new quantum estimates by using the identities given in last section. Let us start to find some new quantum estimates by using Lemma 2. We first examine a new result for functions whose partially q 1 q 2 -derivatives in modulus are convex in the following theorem.
Proof On taking the modulus of the identity of Lemma (4.1), because of the properties of the modulus, we find that This completes the proof.   which shows that the proved inequality is valid for convex functions.
Remark 4 Under the given conditions of Theorem 6 with q 1 , q 2 → 1 -, then we obtain the following inequality: which is given by Latif and Dragomir in [25, Theorem 2].
Proof Applying the well-known Hölder inequality for q 1 q 2 -integrals to the integrals on the right side of (5.19), it is found that b,d I q 1 ,q 2 (a, b, c, d)(F) (5.23) ≤ q 1 q 2 (ba)(dc) . Now, if we apply the concept of Lemma 1 for a = 0 to the above quantum integrals, we obtain Remark 5 Under the given conditions of Theorem 7 with q 1 , q 2 → 1 -, then we obtain the following inequality: