The minimum modulus of certain small entire functions
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- by P. C. Fenton PDF
- Trans. Amer. Math. Soc. 271 (1982), 183-195 Request permission
Abstract:
Suppose that $f(z)$ is an entire function satisfying \[ \min \limits _{|z| = r} |f(z)| \leqslant C(\sigma )\max \limits _{|z| = r} |f(z)|,\] for $r \geqslant {\rho _0} > 0$, where $\sigma > 0$ and \[ C(\sigma ) = \prod \limits _{k = 1}^\infty {{{\left \{ {\frac {{1 - \exp ( - (2k - 1) / 4\sigma )}} {{1 + \exp ( - (2k - 1) / 4\sigma )}}} \right \}}^2}.} \] It is shown that \[ \underline {\lim } \limits _{r \to \infty } \frac {{{{\max }_{|z| = r}}|f(z)| - \sigma {{(\log r)}^2}}} {{\log r}} \geqslant - 2\sigma \log (\max ({\rho _0},|{a_1}|)),\] where ${a_1}$ is the first nonzero zero of $f$.References
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Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 271 (1982), 183-195
- MSC: Primary 30D15
- DOI: https://doi.org/10.1090/S0002-9947-1982-0648085-5
- MathSciNet review: 648085