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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.


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

  • [1] R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954. MR 0068627 (16:914f)
  • [2] -, Inequalities for asymmetric entire functions, Illinois J. Math. 1 (1957), 94-97. MR 0084577 (18:884c)
  • [3] N. K. Govil, On the derivative of a polynomial, Proc. Amer. Math. Soc. 41 (1973), 543-546. MR 0325932 (48:4278)
  • [4] P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509-513. MR 0010731 (6:61f)
  • [5] Q. I. Rahman, On asymmetric entire functions, Proc. Amer. Math. Soc. 14 (1963), 507-508. MR 0148916 (26:6412)
  • [6] P. Turán, Über die Ableitung von polynomen, Compositio Math. 7 (1939), 89-95. MR 0000228 (1:37b)

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Additional Information

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

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