Relations between growth of entire functions and behavior of its Taylor coefficients

TL;DR

This paper establishes precise, non-asymptotic relations between the growth of entire functions and the decay of their Taylor coefficients, using convex analysis techniques for functions of one and several complex variables.

Contribution

It provides the first exact bilateral relations linking entire function growth and Taylor coefficient decay, applicable to multiple complex variables.

Findings

Derived exact relations between growth and coefficients

Applied convex analysis methods to complex functions

Extended results to multivariable entire functions

Abstract

We derive in the closed and unimprovable form the bilateral non-asymptotic relations between growth of entire functions and decay rate at infinity of its Taylor coefficients. We investigate the functions of one as well as of several complex variables. We will apply the convex analysis: Young-Fenchel (Legendre) transform, Young inequality, saddle-point method etc.