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1).Let $f: \mathbb{C}\to \mathbb{C}$ be a holomorphic function. Prove that if $f(0) = f(1) =0 $ and $f:\lvert f'(z) \rvert\leq 1 $ for all $z\in \mathbb{C}$, then $f(z)=0$ for all $z\in \mathbb{C}$.

2).True or false: If $f:\mathbb{C}\to\mathbb{C}$ is holomorphic, then f must be either a constant function or must be surjective.

I know that Liouville's theorem states that Every bounded entire function is constant. if $f'(z) \leq 1$ then $F(z) = cz+D.$ Where D is a constant. by the condition $f(0) = f(1) =0 $. substituting 0 and 1 for z. I found that c and D is equal to 0. So would it be true for $f(z)=0$ for all $z\in \mathbb{C}$.

Please correct me if im heading in the wrong direction *Not sure of Part 2*.By Liouville's theorem i think its true for a constant but cannot identify a surjective function.

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You are more likely to get an answer if you show your progress towards a solution and addressing the parts which you are specifically confused about, rather than simply stating the problem. –  Alexander Gruber May 24 '13 at 19:49
    
This question is phrased in the kind of language appropriate for posing homework questions: "Prove this!" That makes it look as if you've passed on to us a question written by someone else, and you might not even understand the question, so you're not really asking a question. You should demonstrate that you've understood something. –  Michael Hardy May 24 '13 at 19:52
    
@ Alexander Gruber: True - sorry for late editing of what i understood about the question. Iwas still editing when this comments came in. –  Avinesh May 24 '13 at 20:05
    
Please suggest for changes if there is problem in the question. Voting down a question is not helping. While i was still editing my question was voted down. –  Avinesh May 24 '13 at 20:41
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2 Answers 2

up vote 2 down vote accepted

If $f'$ is bounded, then Liouville says $f'$ is constant, and the fact that $f(0)=f(1)$ tells you WHICH constant it is.

For you second question, consider that $\exp$ is not surjective since there is one number that is not in its image.


What I wrote above should tell you the answers to these questions. But if you write them verbatim and hand them in as homework solutions, you won't get much if any credit, if the person grading them has any sense. Take them as "hints".

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I have shown that the constants C and D is equal to 0. –  Avinesh May 24 '13 at 20:07
    
The hints are good enough for me. Thanks for your help –  Avinesh May 24 '13 at 20:38
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For 1. $f'(x)$ is holomorphic, and if it is bounded it is constant.

For 2. Think of the exponentialfunction.

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