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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some inequalities for entire functions of exponential type


Authors: Robert B. Gardner and N. K. Govil
Journal: Proc. Amer. Math. Soc. 123 (1995), 2757-2761
MSC: Primary 30D15; Secondary 30A10
MathSciNet review: 1257107
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Abstract: If $ f(z)$ is an asymmetric entire function of exponential type $ \tau $,

$\displaystyle \left\Vert f \right\Vert = \mathop {\sup }\limits_{ - \infty < x < \infty } \vert f(x)\vert,$

then according to a well-known result of R. P. Boas,

$\displaystyle \left\Vert {f'} \right\Vert \leq \frac{\tau }{2}\left\Vert f \right\Vert$

and

$\displaystyle \vert f(x + iy)\vert \leq \frac{{({e^{\tau \vert y\vert}} + 1)}}{2}\left\Vert f \right\Vert,\quad - \infty < x < \infty , - \infty < y \leq 0.$

Both of these inequalities are sharp. In this paper we generalize the above two inequalities of Boas by proving a sharp inequality which, besides giving as special cases the above two inequalities of Boas, yields some other results as well.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1995-1257107-7
PII: S 0002-9939(1995)1257107-7
Keywords: Special classes of entire functions and growth estimates, inequalities in the complex domain, approximation in the complex domain
Article copyright: © Copyright 1995 American Mathematical Society