New Jensen-type inequalities and their applications

TL;DR

This paper introduces new Jensen-type inequalities for highly convex functions, providing simple tools to extend existing results and explore applications across economics, probability, and mathematical analysis.

Contribution

It presents novel Jensen-type inequalities for very convex functions, enabling broader applications and generalizations in various fields.

Findings

New inequalities for very convex functions introduced

Applications demonstrated in risk measures and probability theory

Extensions of classical inequalities like Hermite-Hadamard

Abstract

Convex analysis is fundamental to proving inequalities that have a wide variety of applications in economics and mathematics. In this paper we provide Jensen-type inequalities for functions that are, intuitively, "very" convex. These inequalities are simple to apply and can be used to generalize and extend previous results or to derive new results. We apply our inequalities to quantify the notion "more risk averse" provided in \cite{pratt1978risk}. We also apply our results in other applications from different fields, including risk measures, Poisson approximation, moment generating functions, log-likelihood functions, and Hermite-Hadamard type inequalities.