# Finding entire entire functions

Hi I was trying to solve the following problems:

1) Let $f$ be a function analytic inside and on the unit circle. Suppose that $|f(z)-z|<|z|$ on the unit circle.

$a) \text{ Show that } |f'(1/2)|\le 8.$

$b) \text{ Show that$f$has precisely one zero inside that unit circle.}$

2) Find all entire functions $f$ which satisfy $\text{Re}(f(z))\le\frac{2}{|z|}$ for $|z|>1$.

Finally I could solve the first problem but I couldn't do the second one. Any help would be highly appreciated. Thanks in advance.

• Does the condition $|f(z)-z|<|z|$ apply for all $|z|\le 1$ or just for $|z|=1$? – idok Jan 6 '18 at 16:10
• @idok I should have made more clear. No that condition is not for problem 2. – mint Jan 6 '18 at 16:18

Hint for problem 2: consider $e^{f(z)}$.

• Aguirre Since $|e^{f(z)}|\le e^2$, $f(z)$ is constant. Is it right? – mint Jan 6 '18 at 17:21
• Yes, but the inequality holds for $|z|\ge1$, – Julián Aguirre Jan 6 '18 at 17:47