Hardy–Petrovitch–Hutchinson’s problem and partial theta function

Abstract

In 1907, M. Petrovitch initiated the study of a class of entire functions all whose finite sections (i.e., truncations) are real-rooted polynomials. He was motivated by previous studies of E. Laguerre on uniform limits of sequences of real-rooted polynomials and by an interesting result of G. H. Hardy. 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 discriminant 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 $p\left(x\right)=a_0+a_{1}x+\cdots +a_n{x}^n+\cdots$ with strictly positive coefficients has the property that all of its finite segments $a_i{x}^i+a_{i+1}{x}^{i+1}+\cdots +a_j{x}^j$ have only real roots if and only if $a_i^2/a_{i-1}{a}_{i+1}\ge 4$ for $i=1,2,\dots$ . In the present paper, we give sharp lower bounds on the ratios $a_i^2/a_{i-1}{a}_{i+1}$ ($i=1,2,\dots$) for the class considered by M. Petrovitch. In particular, we show that the limit of these minima when $i\to \infty$ equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre–Pólya class $\mathcal{L-P}I$. We also explain the relation between Newton’s and Hutchinson’s inequalities and the logarithmic image of the set of all real-rooted polynomials with positive coefficients.