Interpolation and the Laguerre-Pólya class

Acta Mathematica Hungarica, 2012

ABSTRACT The principal goal of this paper is to investigate and report results concerning the following problem. Determine the family of all real entire functions of positive order, φ, in the Laguerre–Pólya class, such that if p is an arbitrary, non-constant real polynomial which has no zeros in common with φ, then the entire function f=φ+p possesses some non-real zeros. Ramifications of the results obtained are also considered in relation to the Hermite–Poulain Theorem and the theory of multiplier sequences.

Geometric And Functional Analysis, 2003

Journal of Mathematical Analysis and Applications, 1980

Let P(z) be a polynomial of degree n with real or complex coefficients, In this paper we obtain a ring shaped region containing all the zeros of P( z), Our results include, as special cases, several known extensions of Enestrom-Kakeya theorem on the zeros of a polynomiaL We shall also obtain zero free regions for certain class of analytic functions. n THEOREMB. LetP(z) = L aki ¥=-°be apolynomial with complex coefficients k=O such that I arg ak -131 :::; a :::;~, k = 0, l, ... , n for some 13, and Mathematics subject classification (1991): 30CIO,30CI5.

1998

Let P(z) be a polynomial of degree n with real or complex coefficients, In this paper we obtain a ring shaped region containing all the zeros of P( z), Our results include, as special cases, several known extensions of Enestrom-Kakeya theorem on the zeros of a polynomiaL We shall also obtain zero free regions for certain class of analytic functions. n THEOREMB. LetP(z) = L aki ¥=-°be apolynomial with complex coefficients k=O such that I arg ak -131 :::; a :::;~, k = 0, l, ... , n for some 13, and Mathematics subject classification (1991): 30CIO,30CI5.

Journal of Combinatorial Theory, Series A, 2005

Let f (x) and g(x) be two real polynomials whose leading coefficients have the same sign. Suppose that f (x) and g(x) have only real zeros and that g interlaces f or g alternates left of f . We show that if ad ≥ bc then the polynomial

Acta Mathematica, 2007

We prove Pólya's conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the n-th derivative tends to infinity as n → ∞. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane. ∞ k=1 |z k | −s−1 converges, then f has the Weierstrass representation f (z) = z m e P (z) ∞ k=1

Pacific Journal of Mathematics, 1981

An inequality is established which provides a unifying principle for the distribution of zeros of real polynomials and certain entire functions. This inequality extends the applicability of multiplier sequences to the class of all real polynomials. The various consequences obtained generalize and supplement several results due to Hermite-Poulain, Laguerre, Marden, Obreschkoff, Polya and Schur.

Illinois Journal of Mathematics, 1997

Annales UMCS, Mathematica, 2011

For a polynomial of degree n, we have obtained some results, which generalize and improve upon the earlier well known results (under certain conditions). A similar result is also obtained for analytic function.

1997

The problem of characterizing all real sequences 0 with the property that if 0 is any real polynomial, then 0 has no more nonreal zeros than , remains open. Recently, the authors solved this problem under the additional assumption that the sequences 0 , with the aforementioned property, can be interpolated by polynomials. The purpose of this paper is to