Analytic function with 3 or less repeats

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SUMMARY

An analytic function defined over the entire complex plane cannot take any value more than three times. This conclusion stems from the properties of non-constant analytic functions, which are open mappings. It is established that such functions cannot be polynomials of degree four or higher due to the fundamental theorem of algebra, which guarantees at least as many roots as the degree of the polynomial. Therefore, the only candidates for such functions are polynomials of degree three or less.

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Grothard
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Homework Statement

A function f that is analytic in the whole plane does not take any value more than 3 times. What can it be?

Relevant equations

Cauchy-Riemann equations

The attempt at a solution

I am aware that any non-constant analytic function is open, but I'm not sure where to go from there.
The function cannot be a polynomial of degree 4 or higher since it would have 4 or more roots. It could be a polynomial of degree 3 or less I think.
I'm not sure how I'd go about proving that nothing other than a polynomial can work (or if that's even true).
 
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