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On a converse of Laguerre’s theorem. (English) Zbl 0911.30020

Authors’ summary: The problem of characterizing all real sequences \(\{\gamma_k \}_{k=0}^\infty\) with the property that if \(p(x)= \sum_{k=0}^n a_kx^k\) is any real polynomial, then \(\sum_{k=0}^n \gamma_k a_kx^k\) has no more nonreal zeros than \(p(x)\), remains open. Recently, the authors solved this problem under the additional assumption that the sequences \(\{\gamma_k \}_{k=0}^\infty\), with the aforementioned property, can be interpolated by polynomials. The purpose of this paper is to extend this result to certain transcendental functions. In particular, the main result establishes a converse of a classical theorem of Laguerre for these transcendental entire functions.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30D10 Representations of entire functions of one complex variable by series and integrals
26C10 Real polynomials: location of zeros
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