A Unified Representation of q- and h-Integrals and Consequences in Inequalities

Abstract

This paper aims to unify q-derivative/q-integrals and h-derivative/h-integrals into a single definition, called q−h-derivative/q−h-integral. These notions are further extended on the finite interval [a,b] in the form of left and right q−h-derivatives and q−h-integrals. Some inequalities for q−h-integrals are studied and directly connected with well known results in diverse fields of science and engineering. The theory based on q-derivatives/q-integrals and h-derivatives/h-integrals can be unified using the q−h-derivative/q−h-integral concept.