Some Tight Polynomial-Exponential Lower Bounds for an Exponential Function

Abstract

This note is devoted to new sharp lower bounds for exp(x²) over the whole real line. We first introduce and study a new lower bound defined with polynomial of degree 2 and exponential (or hyperbolic) functions. Then we propose two improvements of this lower bound by using two different approaches; the first approach consists in adding well-chosen polynomial term to it, whereas the second approach aims to transform it for large values of |x|. We show that they are better to well-known lower bounds. The analytic results are supported by some numerical studies and graphics. A part of the study is devoted to some integral methods having the ability to generate new lower bounds for exp(x²).

Key words and phrases. Algebraic bounds, exponential function.

2000 Mathematics Subject Classification. 33B10, 26D07