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Proceedings of the American Mathematical Society

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

 

 

A spread relation for entire functions with negative zeros


Author: Faruk F. Abi-Khuzam
Journal: Proc. Amer. Math. Soc. 110 (1990), 951-960
MSC: Primary 30D35
MathSciNet review: 1028282
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Abstract: Let $ g$ be a canonical product having only real negative zeros and nonintegral order $ \lambda $, and let $ \phi $ be the set function defined by $ 2\pi \phi (E) = {\smallint _E}\pi \lambda \csc \pi \lambda \cos \lambda \theta d\theta $. It is shown that if $ E(r)$ is the set of values of $ \theta \in ( - \pi ,\pi ]$ where $ \vert g(r{e^{i\theta }})\vert \geq 1,{r_n}$ is a sequence of Polya peaks of $ g$ and $ \delta $ is the deficiency of the value zero of $ g$ then $ \phi (E({r_n})) \geq 2{(1 - \delta )^{ - 1}}$. This inequality leads to a sharp spread relation for $ g$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1990-1028282-X
Article copyright: © Copyright 1990 American Mathematical Society