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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An Abelian theorem for a class of subharmonic functions

Author: Faruk F. Abi-Khuzam
Journal: Proc. Amer. Math. Soc. 67 (1977), 253-259
MSC: Primary 31A05; Secondary 30A64
MathSciNet review: 0460667
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Abstract: We show that if the Riesz-mass of a subharmonic function u, of finite order $ \lambda $, is distributed along a ray, then regular variation (with exponent $ \lambda $) of the mean value of $ u(r{e^{i\theta }})$ implies regular variation (with exponent $ \lambda $) of each of the $ {L_s}( - \pi ,\pi )$ means of $ u(r{e^{i\theta }})$. This result extends a known theorem of Edrei and Fuchs, but our method differs from theirs. In particular, for the case of integral orders we obtain the theorem for a much more general distribution of the Riesz-mass. A corollary, which appears to be new, on the deficiency of the value zero of entire functions with positive integral order, follows.

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Keywords: Riesz-mass, order of a subharmonic function, regular variation
Article copyright: © Copyright 1977 American Mathematical Society