HARDY-PETROVITCH-HUTCHINSON'S PROBLEM AND PARTIAL THETA FUNCTION

By Vladimir Petrov Kostov, Boris Z. Shapiro. 2011

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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(x) = a(0) + a(1)x + ... + a(n)x(n) + ... with strictly positive coefficients has the property that all of its finite segments a(i) x(i) + a(i+1)x(i+1) + ... + a(j)x(j) have only real roots if and only if a(i)(2)/a(i-1)a(i+1) >= 4 for i = 1, 2,.... In the present paper, we give sharp lower bounds on the ratios a(i)(2)/a(i-1)a(i+1) (i = 1, 2,...) for the class considered by M. Petrovitch. In particular, we show that the limit of these minima when i -> 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 L - PI. 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5e21b5206761f582137b4c61a1d046a1aa9152b0