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

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 $\phi$.