$f:\mathbb D\rightarrow \mathbb D$ be holomorphic with $f(1/2)=0$ and $f(0)=1/2$

Question

Let $f:\mathbb D\rightarrow \mathbb D$ be holomorphic with $f(1/2)=0$ and $f(0)=1/2,$ where $\mathbb D=\{z:|z|<1\}.$ Then which of the following statements are correct?

(a) $|f'(0)|\leq 3/4,$

(b) $|f'(1/2)|\leq 4/3,$

(c) $|f'(1/2)|\leq 4/3$ and $|f'(0)|\leq 3/4,$

(d) $f(z)=z$ for $z\in \mathbb D.$

by Schwarz-Pick Lemma $|f'(z)|\le {1-|f(z)|^2\over 1-|z|^2}$ we get $|f'(0)|\le {3\over 4}$ and $|f'(1/2)|\le {4\over 3}$ so so $a,b,c$ are true, but I don't know how to prove or disprove $d$ could anyone tell me? here $f(0)\ne 0$ so I am not able to use Schwarz-Pick Lemma directly.

Is there a typo in (d)? 'cause otherwise, since $f(0) \neq 0$ and $0 \in \mathbb{D}$, (d) is obviously false, no? — fgp, May 27 '13 at 14:01
ooops sorry by the given condition $f(z)=z$ is automatically false as $f(z)=z\Rightarrow f(0)=0$, — Myshkin, May 27 '13 at 14:02
Possible duplicate of Multiple Choice Question: Let f be holomorphic on D with $ f(0) = \frac{1}{2}$ and $ f(\frac{1}{2}) = 0 $, where $ D = { z : |z|\leq 1 }$. — , Nov 15 '17 at 13:53
Is d) asking for the existence of a fixed point, $f(z)=z$ for some (but not all) $z\in \Bbb D$? — Lutz Lehmann, Mar 19 '18 at 11:43

Easy · Answer 1 · 2013-05-27T14:06:16.013

1

How about $f(z)=\frac{2z-1}{z-2}$?

edited May 27 '13 at 14:06

answered May 27 '13 at 14:00

Easy

4,485

score 0 · Answer 2 · answered Mar 19 '18 at 09:03

0

Let $f(z)=1/2 - z$ then $f(1/2)=0$ and $f(0)=1/2$. $|f'(0)|=1$ option 1 is false, now $|f'(1/2)|=1<4/3$ so optn b is correct, check similarly

answered Mar 19 '18 at 09:03

user532616

45

Your $f(-\frac34)=\frac54$ is far outside $\Bbb D$. – Lutz Lehmann Mar 19 '18 at 11:45
Yaa bcoz no 1 is not correct, thts why, i have given a way to know which is correct one – user532616 Mar 19 '18 at 13:46
No, your function does not satisfy the basic condition $f(\Bbb D)\subset \Bbb D$, $\Bbb D={z:|z|< 1}$.. Thus all conclusions are invalid. – Lutz Lehmann Mar 19 '18 at 14:26

$f:\mathbb D\rightarrow \mathbb D$ be holomorphic with $f(1/2)=0$ and $f(0)=1/2$

2 Answers2