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 , is distributed along a ray, then regular variation (with exponent ) of the mean value of implies regular variation (with exponent ) of each of the means of . 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.

**[1]**Albert Edrei and W. H. J. Fuchs,*Tauberian theorems for a class of meromorphic functions with negative zeros and positive poles*, Contemporary Problems in Theory Anal. Functions (Internat. Conf., Erevann, 1965) Izdat. “Nauka”, Moscow, 1966, pp. 339–358. MR**0213561****[2]**William Feller,*An introduction to probability theory and its applications. Vol. II*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0210154****[3]**Joseph Miles and Daniel F. Shea,*An extremal problem in value-distribution theory*, Quart. J. Math. Oxford Ser. (2)**24**(1973), 377–383. MR**0324041****[4]**L. A. Rubel,*A Fourier series method for entire functions*, Duke Math. J.**30**(1963), 437–442. MR**0152651**

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DOI:
https://doi.org/10.1090/S0002-9939-1977-0460667-6

Keywords:
Riesz-mass,
order of a subharmonic function,
regular variation

Article copyright:
© Copyright 1977
American Mathematical Society