New Post Quantum Analogues of Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions

Abstract

The main objective of this paper is to introduce I p , q ϱ -derivative and I p , q ϱ -integral for interval-valued functions and discuss their key properties. Also, we prove the I p , q ϱ -Hermite–Hadamard inequalities for interval-valued functions is the development of p , q ϱ -Hermite–Hadamard inequalities by using new defined I p , q ϱ -integral. Moreover, we prove some results for midpoint- and trapezoidal-type inequalities by using the concept of Pompeiu–Hausdorff distance between the intervals. It is also shown that the results presented in this paper are extensions of some of the results already shown in earlier works. The proposed studies produce variants that would be useful for performing in-depth investigations on fractal theory, optimization, and research problems in different applied fields, such as computer science, quantum mechanics, and quantum physics.

On Some New AB-Fractional Inclusion Relations

“…In [24], they discussed trigonometric convexity and related integral containment, implementing the fuzzy ordered relation. In [25], Kalsoom et al used the idea of I.V. convexity and (p, q) calculus to establish some new refinements of existing results.…”

Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators

“…As further refinements, An et al [14] introduced interval (h 1 , h 2 )− convex function, Nwaeze et al [15] proved H − H inequality for n-polynomial convex interval-valued function, Zhao et al [16] introduced the notion of interval-valued coordinated convex function, and Ali et al [17], Kalsoom et al [18], and Tariboon et al [19] refined this concept via quantum calculus. Recently, this concept is also generalized to convex fuzzy interval-valued functions by Khan et al [20].…”

Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings

“…The inequalities of Minkowski and Beckenbach for interval valued functions were established by Roman-Flores et al in [10]. We direct readers to [11][12][13][14][15][16] for further results that are related to generalization of convex and interval-valued convex functions As additional references, Zhao et al [17] introduced the idea of interval-valued coordinated convex functions; An et al [18] introduced interval (h1, h2) convex functions; Nwaeze et al [19] proved H-H inequality for n-polynomial convex interval-valued functions; and Tariboon et al [20], Kalsoom et al [21] and Ali et al [22] refined this idea using quantum calculus. Recently, this concept was also generalized to convex fuzzy interval-valued functions by Khan et al [23].…”