$\lim_{z\to \infty} \frac{f(z)}{z}=0$ what can we conclude about $f$?

Question

Let $f$ be a entire function such that $$\lim_{z\to \infty} \frac{f(z)}{z}=0$$

What can we conclude about $f$.

Using the change of variable $w=1/z$ we can conclude $\lim_{w\to 0} f(1/w)w=0$ so at $w=0$ we have removable singularity so we conclude that $f$ has removable singularity at $z=\infty$

Now what should be my next step?

Another one: Find all entire function $f$ such that $\lim_{z\to \infty}\left|\frac{f(z)}{z}\right|=0$. — Martin R, Jun 11 '19 at 14:16
@MartinR Thanks a lot. Did you search them using Approach.xyz? Somehow it did not show any results to me. — Shweta Aggrawal, Jun 11 '19 at 14:18
Yes: https://approach0.xyz/search/?q=%24%5Clim_%7Bz%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bf(z)%7D%7Bz%7D%3D0%24&p=1. – The second one is from the “Related” section. — Martin R, Jun 11 '19 at 14:19

score 1 · Accepted Answer · answered Jun 11 '19 at 14:13

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You can deduce from Cauchy's inequalities that $f$ is constant. Indeed, if the Taylor series of $f$ centered at $0$ is $\sum_{n=0}^\infty a_nz^n$, the$$(\forall n\in\mathbb N)(\forall r>0):\lvert a_n\rvert\leqslant\sup_{\lvert z\rvert=n}\frac{\bigl\lvert f(z)\bigr\rvert}{r^n}\to_{r\to\infty}0.$$

answered Jun 11 '19 at 14:13

José Carlos Santos

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Obrigado pela ajuda :) – Shweta Aggrawal Jun 11 '19 at 14:21
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$\ddot\smile{}$ – José Carlos Santos Jun 11 '19 at 14:23

$\lim_{z\to \infty} \frac{f(z)}{z}=0$ what can we conclude about $f$?

1 Answers1