## Duke Mathematical Journal

### 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 $p(x)=a_{0}+a_{1}x+\cdots+a_{n}x^{n}+\cdots$ 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 ${a_{i}^{2}}/{a_{i-1}a_{i+1}}\ge4$ for $i=1,2,\ldots$ . In the present paper, we give sharp lower bounds on the ratios ${a_{i}^{2}}/{a_{i-1}a_{i+1}}$ ($i=1,2,\ldots$) 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.

#### Article information

Source
Duke Math. J., Volume 162, Number 5 (2013), 825-861.

Dates
First available in Project Euclid: 29 March 2013

https://projecteuclid.org/euclid.dmj/1364562912

Digital Object Identifier
doi:10.1215/00127094-2087264

Mathematical Reviews number (MathSciNet)
MR3047467

Zentralblatt MATH identifier
1302.30008

#### Citation

Kostov, Vladimir Petrov; Shapiro, Boris. Hardy–Petrovitch–Hutchinson’s problem and partial theta function. Duke Math. J. 162 (2013), no. 5, 825--861. doi:10.1215/00127094-2087264. https://projecteuclid.org/euclid.dmj/1364562912

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