The Laguerre inequality and the distribution of zeros of entire functions

Abstract

The purpose of this paper is fourfold: (1) to survey some classical and recent results in the theory of distribution of zeros of entire functions, (2) to generalize the Laguerre inequality, (3) to establish several special cases of the Hawai‘i Conjecture, and (4) to present some new results dealing with the polar derivatives of polynomials. In addition, applications of the extended Laguerre inequalities are given. The paper concludes with several open problems.