Hardy-Petrovitch-Hutchinson's problem and partial theta function

TL;DR

This paper investigates conditions on the coefficients of entire functions with real-rooted finite segments, establishing sharp bounds related to the partial theta function's membership in the Laguerre-Polya class.

Contribution

It provides sharp lower bounds on coefficient ratios for functions with real-rooted finite segments, linking these bounds to the partial theta function's properties.

Findings

Sharp lower bounds on coefficient ratios a_i^2/a_{i-1}a_{i+1} are established.

The limit of these bounds relates to the inverse of the maximal parameter for the partial theta function in the Laguerre-Polya class.

The results connect classical inequalities with the analytic properties of special functions.

Abstract

In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminantal inequalities one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J.I.Hutchinson has shown that an entire function p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the property that any its finite segment a_ix^i+...+a_jx^j has all real roots if and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or equal to 4. In the present paper we give sharp lower bounds on the ratios a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular, we show that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.