Deriving inequalities in the laguerre-pólya class from properties of half-plane mappings

Abstract

Newton, Euler and many after them gave inequalities for real polynomials with only real zeros. We show how to extend classical inequalities ensuring a guaranteed minimal improvement. Our key is the construction of mappings with bounded image domains such that existing coefficient criteria from complex analysis are applicable. Our method carries over to the Laguerre-Pólya class 𝓛–𝓟 which contains real polynomials with exclusively real zeros and their uniform limits. The class 𝓛–𝓟 covers quasi-polynomials describing delay-differential inequalities as well as infinite convergent products representing entire functions, while it is at present not known whether the Riemann ξ-function belongs to this class. For the class 𝓛–𝓟 we obtain a new infinite family of inequalities which contains and generalizes the Laguerre-Turán inequalities.