## 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
Full Text: