Derivatives of entire functions and a question of Pólya

Abstract

The purpose of this note is to announce a partial solution to an old question of Pólya.To state concisely his question and our results we introduce the following notation: For each integer p > 0, denote by V 2 the class of entire functions of the form f{z) = exp (-az 2p + 2 )g(z) where a > 0 and g(z) is a constant multiple of a real entire function of genus < 2p + 1 with only real zeros. Recall that a real entire function is one which assumes real values on the real axis. Now set U 0 = V 0 , and for p > 1, set In

On a problem of Hellerstein, Shen and Williamson

“…But since B > 0, Gei/2 (for the defrmtion see e.g. [3]) and so by Lemma 8 in [3], G' has exactly two additional zeros in all of ÈrThis, together with (i), (ii) and (6), shows that / has only real zeros if and only if C is admissible. The proof of Theorem 1 is now complete.…”

“…We would like to thank Hellerstein and Williamson for bringing Lemma 8 in [3] to our attention. This considerably shortened our original proof.…”

Derivatives of meromorphic functions of finite order

“…In a series of papers [3,4,6] Hellerstein, Shen and Williamson settled an old conjecture of Pólya by characterizing those entire functions / for which /, /' and /" have only real zeros. The first two of these papers characterized the real entire functions with this property while the third characterized the strictly nonreal entire functions with this property.…”

Real Zeros of Derivatives of Meromorphic Functions and Solutions of Second Order Differential Equations

“…In a series of papers [3,4,6] the authors recently settled an old conjecture of Pólya by characterizing those entire functions / for which /, /' and /" have only real zeros. These results may be summarized by THEOREM A.…”