Inequalities for the exponential remainder of the Taylor series

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).