Prove that a complex function is linear

Asked Jun 05 '20 at 07:20

Active Jun 05 '20 at 07:20

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I am dealing with this problem:

Let $f: \Bbb C \rightarrow \Bbb C$ entire function such that for each $z\in\Bbb C$ :

$|f(z)| \leq A +B|z|^\frac{3}{2}$

Prove $f$ is linear.

I thought using Liouville's theorem, but it is about completely bounded function.I thought somehow to conclude that because the function is bounded by specific $z$ I can deduce the function is linear. I am struggling to formalize it properly.

asked Jun 05 '20 at 07:20

Igor

4

This has been asked many times. Let $g(z)=\frac {f(z)-f(0)} z$ for $z \neq 0$ and $g(0)=f'(0)$. Check that $g$ is a bounded entire function. – Kavi Rama Murthy Jun 05 '20 at 07:25
And how does it show that $f$ is linear? – Igor Jun 05 '20 at 07:27
1

If $g$ is constant what is $f$? Can you write $f$ in terms of $g$? – Kavi Rama Murthy Jun 05 '20 at 07:36
OK, I get that now. Thanks you! – Igor Jun 05 '20 at 07:41
Need to solve it without Taylor series – Igor Jun 05 '20 at 10:51
2

Here is another straight forward solution: https://math.stackexchange.com/q/227226/42969 – Martin R Jun 05 '20 at 11:02

Prove that a complex function is linear

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