# Thread: complex analysis and Liouville's theorem problems

1. ## complex analysis and Liouville's theorem problems

I had these two problems on a problem set and my solutions were incorrect and I haven't been able to figure them out.

Let f(z) be analytic on the complex plane. Prove f is a constant in the following cases:

$\displaystyle |f(z)|\geq7$

and

$\displaystyle Re f(z)\geq 7$, hint: consider $\displaystyle \exp(f(z))$
both for all z in the complex plane.

Thanks.

2. A non-constant entire function achieves all values in $\displaystyle \mathbb{C}$ at least once with at most one exception: If it's a non-constant polynomial, it attains all values n times, and if it's non-polynomial, it reaches all values infinitely often with at most one exception.

3. Since $\displaystyle \vert f(z)\vert \geq 7$ we have that $\displaystyle g(z)=\frac{1}{f(z)}$ is an entire bounded function. For the second one notice that $\displaystyle \vert e^{f(z)} \vert =\vert e^{Re(f(z))} \vert \geq e^7$ now argue as in the first problem.

4. Thanks shawsend. Do you or does anyone have something more mathematical? I was half-expecting a Cauchy integral to be used or something. I have some other problems like this and I'm sure I'd be asked something like this for the final where I'd be expected to be more rigorous.

5. Originally Posted by Jose27
Since $\displaystyle \vert f(z)\vert \geq 7$ we have that $\displaystyle g(z)=\frac{1}{f(z)}$ is an entire bounded function. For the second one notice that $\displaystyle \vert e^{f(z)} \vert =\vert e^{Re(f(z))} \vert \geq e^7$ now argue as in the first problem.
Yeah, I inverted the function like you did but apparently it's not complex analytic according to the person who graded my problem set so it was deemed incorrect.

6. Can anyone verify that $\displaystyle g(z)=\frac{1}{f(z)}$ is indeed analytic based on the above?
Thanks.

7. I felt Jose did a better job than me showing that under those constraints, the function must be constant. Everything he did is completely analytic and entire. I don't see what the problem was.

8. Okay great, thanks to the both of you.

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### liouville's theorem solved question

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