On Some Integral Means

TL;DR

This paper introduces the concepts of mean functions and integral means, establishing bounds and inequalities among them, expanding the mathematical understanding of classical means through a geometric and analytical approach.

Contribution

It defines mean functions and integral means, providing new bounds and inequalities, and extends classical results with a unified framework.

Findings

Established bounds on mean functions and integral means.

Derived new inequalities among various classical means.

Unified geometric and analytical approach to means.

Abstract

Harmonic, Geometric, Arithmetic, Heronian and Contraharmonic means have been studied by many mathematicians. In 2003, H. Evens studied these means from geometrical point of view and established some of the inequalities between them in using a circle and its radius. In 1961, E. Beckenback and R. Bellman introduced several inequalities corresponding to means. In this paper, we will introduce the concept of mean functions and integral means and give bounds on some of these mean functions and integral means.