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Is the following statement true?

Suppose, $f:\mathbb C\to \mathbb C $ be an entire function. $ |f(z)| $ is bounded in a region where $ \alpha\le \arg(z)\le \beta $ with $|\beta-\alpha|>\pi $. Then $f(z) $ is constant.

Chandan
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1 Answers1

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This is not true. The Mittag Leffler function is a counterexample. For example, see Hayman W.K. Meromorphic functions, Clarendon Press, Oxford, 1975. p. 19.

user64494
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  • Can you please give a specific example? I mean some specific values of $\alpha, \beta $ in $ E_{\alpha,\beta}$. In that note its written that the function is bounded when $ \frac{\pi}{2k}<|arg z|<\pi $, but that doesn't satisfy the condition I wrote. – Chandan Mar 26 '14 at 12:59
  • Put $k=2$. In fact, the ray $\arg z =\pi$ also lie in that set. – user64494 Mar 26 '14 at 13:05
  • But putting $k=2 $ gives $ \frac {\pi}{4}<|arg z|<\pi $. But I wrote that $ \alpha \le arg z \le \beta $ where $|\beta-\alpha|>\pi $ which is not true in this case. – Chandan Mar 26 '14 at 13:12
  • W. K. Hayman treats $-\pi <\arg z \le \pi$, cutting along the negative ray. You treat $0<\arg z \le 2\pi$, cutting along the positive ray. – user64494 Mar 26 '14 at 13:17
  • Ok. Thanks. Got it. – Chandan Mar 26 '14 at 13:19