On the area of the region where an entire function is greater than one
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- by Li-Chien Shen PDF
- Proc. Amer. Math. Soc. 102 (1988), 68-70 Request permission
Abstract:
Using Carleman’s inequality, we prove that if $f$ is entire and of finite order $\rho \geq 1$, then \[ \lim sup\limits _{r \to \infty } \frac {{A(r)}}{{{r^2}}} \geq \frac {\pi }{{2\rho }},\] where $A(r)$ is the area of the region $\{ z:|f(z)| \geq 1\;{\text {and}}\;|z| \leq r\}$.References
- Kihachiro Arima, On maximum modulus of integral functions, J. Math. Soc. Japan 4 (1952), 62–66. MR 49320, DOI 10.2969/jmsj/00410062
- A. Edrei and P. Erdős, Entire functions bounded outside a finite area, Acta Math. Hungar. 45 (1985), no. 3-4, 367–376. MR 791456, DOI 10.1007/BF01957033
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 68-70
- MSC: Primary 30D15,; Secondary 30D20
- DOI: https://doi.org/10.1090/S0002-9939-1988-0915718-6
- MathSciNet review: 915718