Different Bounds for an Original Double Integral Involving Concave Functions

Abstract

Establishing bounds for integrals involving concave functions is an important topic in mathematical analysis. In this article, we investigate a class of double integrals whose integrands are given by the ratio of the sum of two concave functions. This form is particularly interesting because of its connection with well-known inequalities of Hilbert integral type. We derive a series of upper and lower bounds using various valuable techniques, leading to elegant and new results. Where appropriate, multiple proofs of the same result are given. Numerical examples are also used to illustrate and support the findings.