The minimum of small entire functions
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- by P. C. Fenton PDF
- Proc. Amer. Math. Soc. 81 (1981), 557-561 Request permission
Abstract:
It is shown that if $f(z)$ is entire and satisfies $\overline {\lim } \log M(r,f)/(\log r)^2 = \sigma < \infty$ then for a sequence of $r \to \infty$ \[ m(r,f)/M(r,f) > \prod \limits _1^\infty \left ( \frac {1 - \exp (-(2k - 1)/4\sigma )} {1 + \exp (-(2k - 1)/4\sigma )} \right )^2 + o(1).\] This proves a long-standing conjecture of P. D. Barry.References
- J. M. Anderson, K. F. Barth, and D. A. Brannan, Research problems in complex analysis, Bull. London Math. Soc. 9 (1977), no. 2, 129–162. MR 440018, DOI 10.1112/blms/9.2.129
- P. D. Barry, The minimum modulus of small integral and subharmonic functions, Proc. London Math. Soc. (3) 12 (1962), 445–495. MR 139741, DOI 10.1112/plms/s3-12.1.445 E. C. Titchmarsh, Theory of functions, Oxford Univ. Press., New York, 1939.
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 81 (1981), 557-561
- MSC: Primary 30D15
- DOI: https://doi.org/10.1090/S0002-9939-1981-0601729-0
- MathSciNet review: 601729