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Proceedings of the American Mathematical Society

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

 

 

An inequality for functions of exponential type not vanishing in a half-plane


Author: N. K. Govil
Journal: Proc. Amer. Math. Soc. 65 (1977), 225-229
MSC: Primary 30A64
DOI: https://doi.org/10.1090/S0002-9939-1977-0454010-6
MathSciNet review: 0454010
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Abstract: Let $ f(z)$ be an entire function of order 1, type $ \tau $ having no zero in $ \operatorname{Im} \;z < 0$. If $ {h_f}( - \pi /2) = \tau, {h_f}(\pi /2) \leqslant 0$ then it is known that $ {\sup _{ - \infty < x < \infty }}\vert f'(x)\vert \geqslant (\tau /2){\sup _{ - \infty < x < \infty }}\vert f(x)\vert$. In this paper we consider the case when $ f(z)$ has no zero in $ \operatorname{Im} \;z < k, k \leqslant 0$ and obtain a sharp result.


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DOI: https://doi.org/10.1090/S0002-9939-1977-0454010-6
Keywords: Special classes of entire functions and growth estimates, inequalities in the complex domain, extremal problems
Article copyright: © Copyright 1977 American Mathematical Society