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.
Best Answer
(1) can actually be generalized to show that no entire function $f$ can dominate another entire function $g$ unless $g = \lambda f$ for some constant $\lambda$. Let $Z(f)$ be the set of zeros of $f$. If $|g| \le |f|$, then: $$ \left|\frac{g(z)}{f(z)}\right| \le 1 \quad \forall z \in \Bbb C - Z(f) $$
But $g/f$ is bounded in a deleted neighborhood of each $a \in Z(f)$. Thus each $a \in Z(f)$ is a removable singularity of $g/f$. Therefore $g/f$ is entire and bounded, hence constant by Liouville's theorem. It follows that $g = \lambda f$.
In your particular question, $f(z) = \exp(z)$ and this function doesn't have any zeros, so it isn't necessary to discuss the zeros of $f$.