The Zeros of Derivatives of Entire Functions and the Pólya-Wiman Conjecture

Abstract

An entire function f(x) which assumes only real values on the real axis is said to be a real entire function. Thus, if f is a real entire function, then its Maclaurin coefficients are all real, and consequently the zeros of f are symmetrically located with respect to the real axis. One of the intuitive principles underlying the theory of distribution of zeros of successive derivatives of real entire functions was formulated by Polya [P4] in his celebrated survey article entitled On the zeros of the derivatives of a function and its analytic character. In this work [P4, p. 181] Polya states, The real axis seems to exert an influence on the complex zeros of f(n)(z); it seems to attract these zeros when the order is less than 2, and it seems to repel them when the order is greater than 2. Indeed, the first confirmation of this principle was made by Alander [A2] (see also P6lya [P2]) who in 1930 showed that the Polya-Wiman conjecture is true if the order of f(x) is less than 2/3. In [Wil] and [Wi2] Wiman established the validity of this conjecture for functions f(x) of order at most 1, while in 1937 Polya [P3] proved it for functions of order less than 4/3. A general survey of these results and related conjectures may be found in [P2] and [P4]. (For more recent surveys of this and related areas of research see Boas [B2] and Prather [Pr].) It seems that no progress has been made on the P6lya-Wiman conjecture since the publication of Polya's 4/3 theorem in 1937. Our proof of the conjecture is self-contained and does not rely on the results cited above. In Section 2 we will (1) introduce some definitions and notations, (2) state some of the properties of functions in the Laguerre-P6lya class, and (3) prove the Polya-Wiman conjecture (Theorem 3). In the proof of Theorem 3 we