Inclusions for Co-Ordinated Interval-Valued Functions via Post-Quantum Calculus

: In this paper, the notions of post-quantum integrals for two-variable interval-valued functions are presented. The newly described integrals are then used to prove some new Hermite– Hadamard inclusions for co-ordinated convex interval-valued functions. Many of the ﬁndings in this paper are important extensions of previous ﬁndings in the literature. Finally, we present a few examples of our new ﬁndings. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role.

J. Tariboon and S. K. Ntouyas [21] described and proved some of the properties of the q-derivative and q-integral of a continuous functions on finite intervals in 2013. Moreover, they proved Hermite-Hadamard-type inequalities and many others for convex functions in the setting of quantum calculus; for more information, see [22].
M. Tunç and E. Göv [23] presented the (p, q)-derivative and (p, q)-integral on finite intervals in 2016, proved some of their properties, and proved a number of integral inequalities using the (p, q)-calculus. Many researchers have recently begun working in this direction, based on the works of M. Tunç and E. Göv, and some further findings on the analysis of (p, q)-calculus can be found in [24][25][26][27].
In [28], S. S. Dragomir proved the following inequalities, which are Hermite-Hadamard type inequalities for co-ordinated convex functions on the rectangle from the plane R 2 . Theorem 1. Suppose that f : [a, b] × [c, d] → R is co-ordinated convex; then we have the following inequalities: The above inequalities are sharp. The inequalities in (1) hold in the reverse direction if the mapping f is a co-ordinated concave mapping.
The quantum variant of above inequality (1) was given by M. Kunt et al. in [29], S. Bermudo et al. [30] recently used q-calculus to describe new q b -derivative and q b -integral, as well as to give the Hermite-Hadamard inequality. H. Budak et al. [31] defined some new q b -integrals for co-ordinates and Hermite-Hadamard inequalities for co-ordinated convex functions as a result of this. F. Wannalookkhee et al. [32] in 2021 gave some new definitions of (p, q) b -integrals and used them to prove the following Hermite-Hadamard inequalities:

Interval Calculus
In this section, we provide notation and background information on interval analysis. The space of all closed intervals of R is denoted by I c , and ∆ is a bounded element of I c . We have the representation where Θ 1 , Θ 1 ∈ R and Θ 1 ≤ Θ 1 . L(∆) = Θ 1 − Θ 1 can be used to express the length of the interval ∆ = Θ 1 , Θ 1 . The left and right endpoints of interval ∆ are denoted by the numbers Θ 1 and Θ 1 , respectively. The interval ∆ is said to be degenerate when Θ 1 = Θ 1 , and the form ∆ = Θ 1 = [Θ 1 , Θ 1 ] is used. In addition, if Θ 1 > 0, we can say ∆ is positive, and if Θ 1 < 0, we can say ∆ is negative. I + c and I − c denote the sets of all closed positive intervals and closed negative intervals of R, respectively. Between the intervals ∆ and Λ, the Pompeiu-Hausdorff distance is defined by (I c , d) is a complete metric space, as far as we know (see, [33]). |∆| denotes the absolute value of ∆, which is the maximum of the absolute values of its endpoints: |∆| = max Θ 1 , Θ 1 .
The following are the concepts for fundamental interval arithmetic operations for the intervals ∆ and Λ: The interval ∆'s scalar multiplication is defined by where µ ∈ R. The opposite of the interval ∆ is where µ = −1.
Aside from any of these characteristics, the distributive law does not always apply to intervals. As an example, Another distinct feature is the inclusion ⊆, which is described by In [37], Zhao et al. gave the notions about the co-ordinated convex interval-valued functions and inclusions of Hermite-Hadamard type.
c is said to be co-ordinated convex interval-valued function if the following inclusion holds: c is said to be co-ordinated convex interval-valued function if and only if F and F are co-ordinated convex and concave, respectively.
It is easy to prove that an interval-valued convex function is an interval-valued co-ordinated convex, but the converse may not be true. For this, we can see the following example.
is an interval-valued convex on co-ordinates, but it is not an interval-valued convex on [0, 1] 2 .
For more recent inclusions of Hermite-Hadamard type for co-ordinated convex interval-valued functions one can read [37,38].

Basics of Quantum and Post-Quantum Calculus
In this section, we review some necessary definitions about q and (p, q)-calculus for real-valued and interval-valued functions. Moreover, here and further, we use the following notations with 0 < q < p ≤ 1:

Definition 4 ([24]).
For a function f : The (p, q) ac integral of f is given as: x a y c f (t, s) c d p 2 ,q 2 s a d p 1 , Recently, in [39], the authors gave the notions of quantum integral for the intervalvalued functions and stated the following: for all x ∈ [a, b].
In [41], Ali et al. gave the post-quantum version of Definition 7 and defined it as: for all In The I(p, q) b c integral of F is given as: The I(p, q) ac integral of F is given as: x a y c F(t, s) c d I p 2 ,q 2 s a d I p 1 , The I(p, q) bd integral of F is given as:
From I(p, q) d a -integral:

Some New (p, q)-Hermite-Hadamard Inclusions
In this section, we deal with the Hermite-Hadamard-type inclusions for co-ordinated convex interval-valued functions using the newly defined interval-valued (p, q)-integrals in the last section.
is a co-ordinated convex interval-valued function on co-ordinates [a, b] × [c, d], F and F are co-ordinated convex and concave on co-ordinates [a, b] × [c, d], respectively. Hence, from co-ordinated convexity of F and using (2), we have F(x, c) a d p 1 ,q 1 x + q 2 From co-ordinated concavity of F and again using (2), we have F(x, c) a d p 1 ,q 1 x + q 2 Now, from the inequalities (12) and (13), we have following inclusions: F(x, y) d d p 2 ,q 2 y a d p 1 ,q 1 x, We obtain the required result (11) by combining the inclusions (14)- (17).
The following inclusions of Hermite-Hadamard type hold for I(p, q) b cintegral: Proof. Following arguments similar to those in the proof of Theorem 2 by taking into account the inclusions (4), the desired inclusion (19) can be attained. Proof. Following arguments similar to those in the proof of Theorem 2 and the concepts of inequalities (2)-(4), by taking into account the I(p, q) ac -integral, the desired inclusion can be attained.

Remark 10.
In Theorem 5, if F = F, then we have the following inequality: This can be found as a special case in [26].