The Bushell-Okrasi\'nski inequality

TL;DR

This paper provides a comprehensive overview of the Bushell-Okrasiński inequality, including its origins, history, generalizations, and various extensions, highlighting its significance in nonlinear Volterra equations and related fields.

Contribution

It offers an extensive expository account of the inequality, covering its motivation, historical development, and numerous generalizations and variations.

Findings

Generalizations strengthen the inequality

Extensions include different kernels and nonlinearities

Conditions for equality are identified

Abstract

We present an expository account of the Bushell-Okrasi\'nski inequality, the motivation behind it, its history, and several generalizations. This inequality originally appeared in studies of nonlinear Volterra equations but very soon gained interest of its own. The basic result has quickly been generalized and extended in different directions strengthening the assertion, generalizing the kernel and nonlinearity, providing the optimal prefactor, finding conditions under which it becomes an equality, and formulating variations valid for other than Lebesgue integrals. We review all of these aspects.