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From an old examination paper:

Let $f(z)$ be entire and $|f(w)-f(z)|\leq R|w-z|$ for arbitrary $w, z$ in $\mathbb C$ and $R>0$. Prove that $f(z)$ is a polynomial of degree less than 2.

I have absolutely no idea where to start so any help would be much appreciated!

2 Answers 2

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If you rewrite the inequality as

$ \frac{|f(w) - f(z)|}{|w - z|} \leq R $

Then what does this tell you about the derivative of $f$?

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    @katherinebarry I'm glad that it helped you =)2012-11-04
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Since $\dfrac{f(w) - f(0)}{w}$ is a bounded entire function (set $z = 0$, and continue analytically to $w=0$), by Liouville's theorem it's constant.

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    @CameronBuie Yes, of course.2012-11-03