Accession Number : AD0716559

Title :   A Note on the Tumura-Clunie Theorem (Zamechanie k Teoreme Tumura-Kluni),

Corporate Author : NAVAL RESEARCH LAB WASHINGTON D C

Personal Author(s) : Goldberg,A. A.

Report Date : 11 DEC 1970

Pagination or Media Count : 4

Abstract : In 1937, Tumura expressed the following statement: If the entire function f(z) and (f sup k)(z), k = or > 2, have no zeros, then f(z) = exp(az + b); however, its proof had important gaps. A strict proof of Tumura's theorem was recently given by Clunie, as reported in a survey by Hayman; Clunie also proved the following theorem: If f(z) is meromorphic in z not equal to infinity, and f(z) and (f sup k)(z), k = or > 2, have a finite number of zeroes and poles, then f(z) has finite order rho. In the present note the author proves a statement not contained in Clunie's result, but which, with the use of this result, will permit one to give an exact expression for rho within the assumptions of Clunie's theorem. (Author)

Descriptors :   (*ANALYTIC FUNCTIONS, THEOREMS), (*MEROMORPHIC FUNCTIONS, THEOREMS), USSR

Subject Categories : Numerical Mathematics

Distribution Statement : APPROVED FOR PUBLIC RELEASE