Deriving Inequalities in the Laguerre-Pólya Class from Properties of Half-Plane Mappings

Conference paper

DOI: 10.1007/978-3-0348-0249-9_5

Volume 161 of the book series International Series of Numerical Mathematics (ISNM)
Cite this paper as:
Batra P. (2012) Deriving Inequalities in the Laguerre-Pólya Class from Properties of Half-Plane Mappings. In: Bandle C., Gilányi A., Losonczi L., Plum M. (eds) Inequalities and Applications 2010. International Series of Numerical Mathematics, vol 161. Birkhäuser, Basel

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 \(\mathcal{L}\)\(\mathcal{P}\) which contains real polynomials with exclusively real zeros and their uniform limits. The class \(\mathcal{L}\)\(\mathcal{P}\) 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 \(\mathcal{L}\)\(\mathcal{P}\) we obtain a new infinite family of inequalities which contains and generalizes the Laguerre-Turán inequalities.

Keywords

Coefficient inequalities Reality of zeros Moment problem Hankel determinants Logarithmic derivative 

Mathematics Subject Classification

26D05 30D20 33C45 

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Institute f. Computer Technology (E-13)Hamburg University of TechnologyHamburgGermany