Can an Analytic Function in the Whole Plane Take Any Value More Than 3 Times?

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

Please PM me the solution so that this can stay posted as a puzzle. Otherwise, the HH rules would apply.

Thanks.
 
Grothard said:
A function f that is analytic in the whole plane does not take any value more than 3 times. What can it be?
How about a cubic polynomial?
 

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