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

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 $$F_{\varkappa }[p](z):=p(z)p^{\prime \prime }(z)-\varkappa [p^{\prime }(z){]}^2,$$ 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 $F_{\frac{n-1}{n}}[p](z)$ in the case when the real polynomial $p$ of degree $n$ has non-real zeroes. We formulate some new conjectures generalising the Hawaii conjecture.

30 pages, 2 figures

References in corpus (1)

• On the number of real critical points of logarithmic derivatives and the Hawaii conjecture