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

Abstract

For a given real entire function $\varphi$ with finitely many nonreal zeros, we establish a connection between the number of real zeros of the functions $Q=({\varphi }^{\prime }/\varphi {)}^{\prime }$ and $Q_1=({\varphi }^{\prime \prime }/{\varphi }^{\prime }{)}^{\prime }$. This connection leads to a proof of the Hawaii conjecture [T.Craven, G.Csordas, and W.Smith, The zeros of derivatives of entire functions and the Pólya-Wiman conjecture, Ann. of Math. (2) 125 (1987), 405--431] stating that the number of real zeros of $Q$ does not exceed the number of nonreal zeros of $\varphi$.

52 pages, 3 figures

References in corpus (1)

• Proof of a conjecture of Polya on the zeros of successive derivatives of real entire functions

Cited by in corpus (2)

• On the number of non-real zeroes of a homogeneous differential polynomial and a generalization of the Laguerre inequalities
• Problems around polynomials - the good, the bad and the ugly ...