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

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

2009-02-03

arxiv.org/abs/0902.0413v3

J. Anal. Math. 114 (2011), pp. 1-62

Mathematics

Classical Analysis and ODEs

52 pages, 3 figures

Scientific paper

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 $\phi$.

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