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Maximum modulus convexity and the location of zeros of an entire function


Author: Faruk F. Abi-Khuzam
Journal: Proc. Amer. Math. Soc. 106 (1989), 1063-1068
MSC: Primary 30D20
DOI: https://doi.org/10.1090/S0002-9939-1989-0972225-3
MathSciNet review: 972225
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Abstract: Let $ f$ be an entire function with non-negative Maclaurin coefficients and let $ b\left( r \right) = r{\left( {rf'\left( r \right)/f\left( r \right)} \right)' }$. It is shown that if all the zeros of $ f$ lie in the angle $ \left\vert {\arg z} \right\vert \leq \delta $, where $ 0 < \delta \leq \pi $, then $ \lim {\sup _{r \to \infty }}b\left( r \right) \geq \frac{1}{4}{\text{cose}}{{\text{c}}^2}\frac{1}{2}\delta $. In particular, we always have $ \lim {\sup _{r \to \infty }}b\left( r \right) > \frac{1}{4}$ for such functions.


References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/S0002-9939-1989-0972225-3
Article copyright: © Copyright 1989 American Mathematical Society

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