Accession Number : AD0255904

Title :   MEROMORPHIC FUNCTIONS WITH SECTORS FREE OF ZEROS AND POLES

Corporate Author : SYRACUSE UNIV N Y

Personal Author(s) : HELLERSTEIN,SIMON

Report Date : JAN 1961

Pagination or Media Count : 1

Abstract : Let f(z) be a meromorphic function which is not a polynomial. Assume that all the zeros and poles of f lie on the real axis. Let (>0) be given and denote by n (r,k) the number of zeros of fe U., N. Y. MEROMORPHIC FUNCTIONS WITH SECTORS FREE OF ZEROS AND POLES, by Simon Hellerstein. Jan 61, 15p. (Contract AF 49(638)571) (AFOSR-222)Unclassified report DESCRIPTORS: *Mathematics, *Functions. Let f(z) be a meromorphic function which is not a polynomial. Assume that all the zeros and poles of f lie on the real axis. Let (>0) be given and denote by n (r,k) the number of zeros of f(k)(z) (taking multiplicities into account) which lie in the disk z r and outside the angles - < arg z < ; - < arg z < + . Then for functions of finite order and r sufficiently large, n (r,k) < (K/ )r, where K depends only on f and k. For functions of infinite order n (r,k) < (K/ )r2log r log2T(r,f), provided r avoids the values of an exceptional set of finite measure, (T(r,f) is the Nevanlinna characteristic of f). The theorem is a consequence of a more general one for meromorphic functions with one sector S free of zeros and poles. Here again it is possible to prove that a sector interior to S contains few zeros of the successive derivatives of f. (Author)

Descriptors :   *FUNCTIONS(MATHEMATICS), *MATHEMATICS

Distribution Statement : APPROVED FOR PUBLIC RELEASE