Turán-type inequalities and the distribution of zeros of entire functions

Abstract

The purpose of this note is to survey a select group of classical and new results which highlight the role that the Turán-type inequalities play in the study of the distribution of zeros of polynomials and transcendental entire functions.

Turán inequalities from Chebyshev to Laguerre polynomials

“…Proofs include transcendental functions and Jensen polynomials. This initiated research in several directions [KS60], including higher Turán inequalities capturing the conjectured hyperbolic property of Jensen polynomials associated with the Riemann zeta function [Cs14,CJW19,GORZ19]. We also refer to the open session report by van Assche [VA10], where Nevai emphasized also the importance of studying Turán inequalities.…”

New Formulas for the Euler-Mascheroni Constant and other Consequences derived from the Acceptance of Hyperbolicity of Jensen Polynomials and the Analysis of the Turán Moments for the ξ-Function

“…( 90), that lets consider an expansion on the Bernoulli numbers. Of course, other infinite sets of relevant numbers like the Gregory coefficients [23] and the Cauchy numbers of the second kind [24] can describe special summations on those ones which converge to the Euler-Mascheroni constant. Now, the evidence shows that three new sets of coefficients a 2n , b n and C n , can contribute to represent this important constant as it has been inferred in the current article.…”

Monotonicity of ratios of q-Kummer confluent hypergeometric and q-hypergeometric functions and associated Turán types inequalities

“…[9]- [10]. A survey of recent results on Turán type inequalities [7] is published in the proceedings of the conference [8] dedicated to Paul Turán's achievements in different areas of mathematics and applications.…”