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1 April 2013 Hardy–Petrovitch–Hutchinson’s problem and partial theta function
Vladimir Petrov Kostov, Boris Shapiro
Duke Math. J. 162(5): 825-861 (1 April 2013). DOI: 10.1215/00127094-2087264


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)=a0+a1x++anxn+ with strictly positive coefficients has the property that all of its finite segments aixi+ai+1xi+1++ajxj have only real roots if and only if ai2/ai1ai+14 for i=1,2, . In the present paper, we give sharp lower bounds on the ratios ai2/ai1ai+1 (i=1,2,) for the class considered by M. Petrovitch. In particular, we show that the limit of these minima when i equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre–Pólya 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.


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Vladimir Petrov Kostov. Boris Shapiro. "Hardy–Petrovitch–Hutchinson’s problem and partial theta function." Duke Math. J. 162 (5) 825 - 861, 1 April 2013.


Published: 1 April 2013
First available in Project Euclid: 29 March 2013

zbMATH: 1302.30008
MathSciNet: MR3047467
Digital Object Identifier: 10.1215/00127094-2087264

Primary: 30C15
Secondary: 12D10 , 26C05

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 5 • 1 April 2013
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