# Fourier series and {delta}-subharmonic functions of finite {gamma}-type in a half-plane

Let {gamma}(r) be a growth function and let v(z) be a proper {delta}-subharmonic function in the sense of Grishin in a complex half-plane, that is v=v{sub 1}-v{sub 2}, where v{sub 1} and v{sub 2} are proper subharmonic functions (limsup{sub z{yields}}{sub t}v{sub i}(z){<=}0, for each real t, i=1,2), let {lambda}={lambda}{sub +}-{lambda}{sub -} be the full measure corresponding to v and let T(r,v) be its Nevanlinna characteristic. The class J{delta}({gamma}) of functions of finite {gamma}-type is defined as follows: v element of J{delta}({gamma}) if T(r,v){<=}A{gamma}(Br)/r for some positive constants A and B. The Fourier coefficients of v are defined in the standard way. The central result of the paper is the equivalence of the following properties: (1) v element of J{delta}({gamma}); (2) N(r){<=}A{sub 1}{gamma}(B{sub 1}r)/r, where N(r)=N(r,{lambda}{sub +}) or N(r)=N(r,{lambda}{sub -}), and |c{sub k}(r,v)|{<=}A{sub 2}{gamma}(B{sub 2}r). It is proved in addition that J{delta}({gamma})=JS({gamma})-JS({gamma}), where JS({gamma}) is the class of proper subharmonic functions of finite {gamma}-type.
Authors:
[1]
1. Ukrainian Academy of Banking, Sumy (Ukraine)
Publication Date:
OSTI Identifier:
21205612
Resource Type:
Journal Article
Resource Relation:
Journal Name: Sbornik. Mathematics; Journal Volume: 192; Journal Issue: 6; Other Information: DOI: 10.1070/SM2001v192n06ABEH000572; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPLEX MANIFOLDS; FUNCTIONS; MATHEMATICAL LOGIC