Conjectures and theorems in the theory of entire functions

Motivated by the recent solution of Karlin's conjecture, properties of functions in the Laguerre-Polya class are investigated. The main result of this paper establishes new moment inequalities fur a class of entire functions represented by Fourier transforms. The paper concludes with several conjectures and open problems involving the Laguerre-Polya class and the Riemann xi -function.

Keywords

Laguerre-Polya class, Karlin's conjecture, Turan inequalities, double Turan inequalities, Fourier transform, admissible kernel, Riemann hypothesis, de Bruijn-Newman constant