On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities

2019/12/10 by Mikhail Tyaglov, Tyaglov, Mikhail, Mohamed Jalel Atia +1
Mathematics · #Mathematical functions and polynomials #Meromorphic and Entire Functions #Mathematics and Applications

paper · pdf · doi:10.48550/arxiv.1912.04951

Abstract

Given a real polynomial p with only real zeroes, we find upper and lower bounds for the number of non-real zeroes of the differential polynomial F_\varkappa[p](z):= p(z)p''(z)-\varkappa[p'(z)]², where \varkappa is a real number. We also construct a counterexample to a conjecture by B. Shapiro on the number of real zeroes of the polynomial F_\tfracn-1n[p](z) in the case when the real polynomial p of degree n has non-real zeroes. We formulate some new conjectures generalising the Hawaii conjecture.

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