Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
MathSciNet review: 972225
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] V. S. Boĭčuk and A. A. Gol′dberg, On the three lines theorem, Mat. Zametki 15 (1974), 45–53 (Russian). MR 0344465
  • [2] W. K. Hayman, Note on Hadamard’s convexity theorem, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 210–213. MR 0252639
  • [3] B. Kjellberg, The convexity theorem of Hadamard-Hayman, Proc. of the Sympos. in Math., Royal Institute of Technology, Stockholm (June 1973), 87-114.
  • [4] R. R. London, A note on Hadamard’s three circles theorem, Bull. London Math. Soc. 9 (1977), no. 2, 182–185. MR 0444948,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30D20

Retrieve articles in all journals with MSC: 30D20

Additional Information

Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society