A POLYA 'SHIRE' THEOREM FOR ENTIRE FUNCTIONS.

摘要

George Polya first studied the limiting distribution of the zeros of the successive derivatives of an entire or meromorphic function of f. He defined the final set of f to be the set of points z in the complex plane such that every neighborhood of z contains zeros of infinitely many derivatives of f.;Theorem (Polya): If f is meromorphic in the plane and has at least one pole, the final set of f is the union of the boundaries of all the shires.;In effect, the poles of f "repel" the zeros of f('(k)).;Polya (and subsequently R. M. McLeod and A. Edrei) also determined the final sets of certain entire functions. In each of the cases examined by those authors, the function f has large growth on certain rays, and the zeros of f('(k)) (large k) avoid those rays. In others words, the rays "repel" zeros. Proceeding from that observation, we introduce for the first time a notion of "shire" for entire functions--with "pole" replaced by "ray of maximal growth".;For each pole A of f, let the shire of f be the set of points closer to A than to any other pole.;With a suitable definition of "shire" we show that, for a certain class of entire functions f, the final set of f equals the union of the boundaries of all the shires. This class includes many of the examples previously considered, as well as some new ones, such as the Mittag-Leffler Functions (their final sets are void) and certain sums of generalized Mittag-Leffler Functions.;To prove that the final set is contained in the boundaries of the shires, we adapt a saddle point method employed by Edrei and McLeod, which in turn is based on a technique of Hayman.;To prove that every point on the boundaries is in the final set, we determine asymptotically the number of zeros of the k('th) derivative in small disks centered at points on the boundaries, using a slight modification of an argument due to Edrei.