Inequalities for the exponential remainder of the Taylor series

TL;DR

This paper investigates inequalities related to the exponential function's Taylor series sections, providing new proofs for optimal inequalities, exploring their generalizations, and discussing related conjectures.

Contribution

It offers new proofs for the main inequalities with best constants and extends these results to multiple generalizations, advancing understanding of exponential series inequalities.

Findings

Established new proofs for key inequalities with optimal constants

Formulated and proved several conjectures related to exponential series inequalities

Extended inequalities to broader classes of functions and series sections

Abstract

This is a preprint of 1992 with some updates. We study sections of the exponential function Taylor series. Interesting inequalities for these sections were considered by G.Hardy, Kesava Menon, W. Gautschi, H.Alzer and others. The main aim of this preprint is to investigate new proofs for the main inequality with best constants and its multiple generalizations. Some conjectures are formulated (and some of them were proved recently, see comments of 2016).