# Accession Number:

## AD0255904

# Title:

## MEROMORPHIC FUNCTIONS WITH SECTORS FREE OF ZEROS AND POLES

# Descriptive Note:

# Corporate Author:

## SYRACUSE UNIV N Y

# Personal Author(s):

# Report Date:

## 1961-01-01

# Pagination or Media Count:

## 1.0

# Abstract:

Let fz 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 49638571 AFOSR-222Unclassified report DESCRIPTORS Mathematics, Functions. Let fz 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 fkz 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 log2Tr,f, provided r avoids the values of an exceptional set of finite measure, Tr,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