Derivatives of entire functions and a question of Pólya. II

Abstract

It is shown that if f is an entire function of infinite order, which is real on the real axis and has, along with f ′ f’ , only real zeros, then f f has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions f for which f , f ′ f,f’ , and f f have only real zeros are those in the Laguerre-Pólya class, i.e. \[ f ( z ) = z m exp ⁡ { − a z 2 + b z + c } ∏ n ( 1 − z z n ) e z / z n , f(z) = {z^m}\exp \{ - a{z^2} + bz + c\} \prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}}, \] a ⩾ 0 , b , c a \geqslant 0,b,c and the z n {z_n} real, and Σ z n − 2 > ∞ \Sigma z_n^{ - 2} > \infty . This gives a strong affirmative version of an old conjecture of Pólya.