I have just been introduced to (generalised) Liouville's theorem which tell us that if an entire function is bounded by some monomial $m |z|^n$, then we know that the function is in fact a polynomial of at most degree $n$. Now when I got to the exercises, this was the first exercise:
Find an entire function such that $\forall z \in \mathbb C$, we have that $|f(z)| \leq |\exp(z)|$ and $f(\pi i)=i$.
Of course, we can express the exponential as a power series, as how it is often originally defined, but this is not a polynomial. My assumption would be that in fact $f(z)= \exp(z/2)$, this would satisfy the second property. I do however run into trouble because take for instance $\exp(-5) < \exp(-5/2)$, this does not satisfy the first property. I am still struggling with seeing a good approach how one goes about and finds these entire functions satisfying such properties. Does anyone have any good pointers as to a general approach I could try?
You can simply take $f(z)=-ie^z$. That will work.