Some inequalities based on a general quantum difference operator

Abstract

In this paper, some integral inequalities based on the general quantum difference operatorwhere β is a strictly increasing continuous function, defined on an interval I ⊆ R, that has one fixed point s 0 ∈ I. The β-Hölder and β-Minkowski inequalities are proved. Also, the β-Gronwall, β-Bernoulli, and some related inequalities are shown. Finally, the β-Lyapunov inequality is established.

A $β$-Sturm Liouville problem associated with the general quantum operator

“…In 2015, Hamza et al [33] introduced a general quantum difference operator, the β−derivative, generalizing the Hahn's quantum operator (for certain functions β), and its inverse operator, the β−integral. Also in 2015 [34], β−Hölder, β−Minkowski, β−Gronwall, β−Bernoulli and β−Lyapunov inequalities were exhibited. In 2016, it was proved the existence and uniqueness of solutions of general quantum difference equations [35].…”

Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order linear quantum difference equations

“…Quantum difference operators allows us to deal with non-differentiable functions in the usual sense. They have an essential role due to their applications in several mathematical areas such as orthogonal polynomials, basic hypergeometric function, combinatorics, the calculus of variations and the theory of relativity (see [1,4,9]). New results in quantum calculus can be found in [8] and the references cited therein.…”

The Directional Derivative in General Quantum Calculus

“…when the series on the right hand side converges. Some integral inequalities based on D β were introduced in [23]. Moreover, the β-Laplace transform and the β-convolution theorem were given in [24,25].…”