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Some Examples of the Dependence of the Nevanlinna Deficiency Upon the Choice of Origin
Author(s) -
Miles Joseph
Publication year - 1983
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-47.1.145
Subject(s) - meromorphic function , mathematics , complement (music) , order (exchange) , function (biology) , pure mathematics , nevanlinna theory , entire function , chemistry , biochemistry , finance , evolutionary biology , complementation , biology , economics , gene , phenotype
Two examples are given of entire functions f for which the Nevanlinna deficiency d (0, f ( z )) of 0 depends on the choice of the origin. The first is a function of infinite order for which d (0, f ( z ))=0 and d (0, f ( z + a ))=1 for all a ≠ 0. The second is a function of finite order for which d (0, f ( z ))= 0 and d (0, f ( z + a )) ⩾ ε > 0 for all a ≠ 0. These examples complement earlier examples given by Hayman for entire functions and by Gol'dberg for meromorphic functions.