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

Abstract

The main objective of this paper is to introduce $I{\left(p,q\right)}_{\mathrm{\varrho }}$ -derivative and $I{\left(p,q\right)}_{\mathrm{\varrho }}$ -integral for interval-valued functions and discuss their key properties. Also, we prove the $I{\left(p,q\right)}_{\mathrm{\varrho }}$ -Hermite–Hadamard inequalities for interval-valued functions is the development of ${\left(p,q\right)}_{\mathrm{\varrho }}$ -Hermite–Hadamard inequalities by using new defined $I{\left(p,q\right)}_{\mathrm{\varrho }}$ -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.