An Abelian theorem for a class of subharmonic functions
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- by Faruk F. Abi-Khuzam PDF
- Proc. Amer. Math. Soc. 67 (1977), 253-259 Request permission
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.References
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Additional Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 67 (1977), 253-259
- MSC: Primary 31A05; Secondary 30A64
- DOI: https://doi.org/10.1090/S0002-9939-1977-0460667-6
- MathSciNet review: 0460667