How can we show the complex function f is an affine mapping?

I'm a bit lost at this, couldn't give a formal proof: Let f be an entire function. Show that f is Lipschitz if, and only if, f is an affine mapping. The if part is immediate, but I'm stuck at the only if part. Is this true if, instead of entire, f is holomorphic in a proper subset of the complex... show more I'm a bit lost at this, couldn't give a formal proof:

Let f be an entire function. Show that f is Lipschitz if, and only if, f is an affine mapping. The if part is immediate, but I'm stuck at the only if part.

Is this true if, instead of entire, f is holomorphic in a proper subset of the complex plane?

Thank you.
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