# Describe all holomorphic functions.

Problem: Describe the class of all holomorphic functions on $\mathbb{C}-\{0\}$ such that

$$\sup_{(x,y)\neq (0,0)}\frac{|f(x+iy)|}{|\log(x^2+y^2)|}<\infty.$$

Attempt at a solution:

Let $z=x+iy$, then we have:

$$\frac{|f(z)|}{|\log|z|^2|}\leq c$$. So,

$|f(z)|\leq c|\log|z|^2|.$

For large enough $z$, we have $|\log|z|^2|\leq |z|^2$ so we get:

$$|f(z)|\leq c|z|^2.$$ Now by extented Liouville's Theorem, $f(z)$ must reduce to a polynomial of degree at most two.

Is this correct?

Thanks!

• Why stop at $|z|^2$? And even if you stop at $|z|$, you still might not get the right answer ;) – Evan Aug 14 '13 at 23:43
• Do polynomials of degree at most two satisfy your constraint? – Patrick Da Silva Aug 14 '13 at 23:43
• And Liouville's theorem applies on functions that are holomorphic over the whole complex plane, not over functions who are missing a point of holomorphicness. (The proof relies on a Taylor expansion around $0$ and then using Cauchy's theorem, so you expect the pole at $0$ to play a role here...) :P – Patrick Da Silva Aug 14 '13 at 23:45

## 1 Answer

We also need to consider the fact that $f$ might have a singularity at $z=0$. However, we can handle that because \begin{align} \lim_{z\to0}|zf(z)| &\le\lim_{z\to0}|z|2c\log(|z|)\\ &=0 \end{align} Riemann's Theorem says that the singularity at $z=0$ is removable. Now, using Cauchy's Integral Formula we get that $$f^{(n)}(z)=\frac{n!}{2\pi i}\oint\frac{f(w)\,\mathrm{d}w}{(w-z)^{n+1}}$$ where the integral is over a counter-clockwise circle of radius $R$. This means that $$\left|\,f^{(n)}(z)\,\right|\le\frac{n!}{R^n}2c\log(R)$$ Just as with the proof of Liouville's Theorem, we show that $f^{(n)}(z)=0$ and $f$ must be constant, and since $f(z)=0$ when $|z|=1$, we know that $f(z)=0$.

Note added

I mentioned above that your conditions imply that $f(z)=0$ when $|z|=1$. This in itself implies that $f(z)=0$ everywhere since $\{z:f(z)=0\}$ has an accumulation point (see this section).

• I already showed that $f$ had a removable singularity, but I should have included it in my attempt I guess. Thanks for the solution, i like it! – V-B Aug 14 '13 at 23:59
• In this case as well, it suffices to take $n=1$. – V-B Aug 15 '13 at 0:01
• @V-B: Yes, it does, but I got carried away :-) – robjohn Aug 15 '13 at 0:02
• @V-B: I apologize for the accusatory tone of my first sentence. It has been toned down. – robjohn Aug 15 '13 at 0:06
• Don't worry about the accusatory tone :) To your note: That is actually fantastic! I can't believe I missed that. That follows directly from the Identity Theorem (or the fact that holomorphic functions, which do not vanish identically, have isolated zeros). – V-B Aug 15 '13 at 0:21