Entire function dominated by another entire function is a constant multiple These two questions I didn't even find the way to solve
So please if you can help


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*Suppose $f (z)$ is entire with $|f(z)| \le |\exp(z)|$ for every $z$
I want to prove that $f(z) = k\exp(z)$  for some $|k| \le 1$

*Can a non constant entire function be bounded in half a plane? Prove if yes , example if not.

Hint:
1) Liouville + $\,\displaystyle{\frac{f(z)}{e^z}}\,$ is analytic and bounded...
2) Develop the function in power series and extend analytically into the "other" halpf plane.
