3

Let $f:\mathbb{D}\to \mathbb{C},f\in Hol(\mathbb{D})$ such that $|f'(0)|=1$ and $|f'(z)|\leq 2$ for all $z\in \mathbb{D}$. I need to show that $f(\mathbb{D}$) contains a disc of radius $3-\sqrt 8$.

I`ve read about Bloch and Landau's theorems, that give a disk with a constant radius of $1/14$ in-case $|f'(0)|=1 $.

However, I have no idea how to get a larger radius of $3-\sqrt8$ given that $|f'(z)|\leq2$ .

I feel like I might be looking at this problem the wrong way, so I`d be happy if someone can give me a hand and point me to the right direction.

Sar
  • 907
  • 1
    I'm pretty sure it should be $|f'(z)|\leq 2$. – Mark Mar 09 '19 at 11:18
  • Yes it was a typo. – Sar Mar 09 '19 at 11:53
  • 2
    Maybe you might find this helpful. Look at the answer t.b wrote. In the end he proves a lemma which looks related to your problem. https://math.stackexchange.com/questions/63401/picards-little-theorem-proofs/63562#63562 – Mark Mar 09 '19 at 12:01

0 Answers0