Exploring nine different proofs of a famous logarithmic inequality

Authors

  • Christophe Chesneau Department of Mathematics, LMNO, University of Caen-Normandie, France

DOI:

https://doi.org/10.56947/amcs.v23.294

Keywords:

Inequalities, convex functions, Integral inequality

Abstract

This article explores and analyzes nine different proofs of a fundamental logarithmic inequality, say "log(1+x)/x  1/ (1+x/2) for x>-1". Some proofs are already published; others are new or consider new approaches or new angles of research. They are based on various techniques, such as differentiation, series expansion, and the application of well-known inequalities such as the primitive, Cauchy-Schwarz, logarithmic mean, hyperbolic tangent function, Jensen, and Hermite-Hadamard inequalities. A graphical work illustrates some results. Therefore, our study clarifies the different mathematical foundations of this seemingly straightforward but important inequality and goes beyond it with some new material.

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Published

2024-04-26

Issue

Section

Articles