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

TL;DR

This paper proves the Hawaii conjecture by establishing a relationship between the zeros of certain derivatives of a real entire function with finitely many nonreal zeros, confirming the conjecture's bound on real zeros.

Contribution

It introduces a new connection between the zeros of derivatives of entire functions, leading to a proof of the longstanding Hawaii conjecture.

Findings

Proves the Hawaii conjecture for functions with finitely many nonreal zeros.

Establishes a relationship between zeros of $Q$ and $Q_1$ functions.

Confirms the bound on the number of real zeros of $Q$.

Abstract

For a given real entire function $\phi$ with finitely many nonreal zeros, we establish a connection between the number of real zeros of the functions $Q=(\phi'/\phi)'$ and $Q_1=(\phi''/\phi')'$. 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 P\'olya-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 $\phi$.