General Opial type inequality

Abstract

In this paper our goal is to give a general Opial type inequality. We consider two functions, convex and concave and prove a new general inequality on a measure space (Ω,Σ,μ). The obtained inequalities are not direct generalizations of the Opial inequality but are of Opial type because the integrals contain a function and its integral representation. We apply our result to numerous symmetric functions and obtain new results that involve Green’s functions, Lidstone series and the Hermite’s interpolating polynomials.