Prove that entire function satisfying this is constant

Asked Sep 18 '20 at 06:26

Active Sep 18 '20 at 06:26

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This question was part of an assignment which I am trying to solve but couldn't.

Let f be an entire function such that $\frac{f(z) } {z} \in 0$ as $ |z|\to \infty$ . Show that f is constant.

Entire function means that f is holomorohic on all of $\mathbb{C} $ but what result can I use? One can also proceed by assuming that f is not constant. But in this way also I was unable to find a contradiction.

Kindly help!!

asked Sep 18 '20 at 06:26

I think this question has already an answer here : https://math.stackexchange.com/questions/1745437/if-f-is-entire-and-lim-limitsz-to-infty-fracfzz-0-show-that-f – jacopoburelli Sep 18 '20 at 06:29
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Does this answer your question? If $f$ is entire and $\lim_\limits{z\to\infty} \frac{f(z)}{z} = 0$ show that $f$ is constant – jacopoburelli Sep 18 '20 at 06:29

Prove that entire function satisfying this is constant

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