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.


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
Full Text: EuDML EMIS