I have to find a non-polynomial entire function $f(z)$ such that $|f(z)| \leq e^{A|z|^{\alpha}}$ where $A \gt 0$ and $\alpha \lt \frac{1}{2}$.
I tried to insert $z = 0$ and I get $|f(0)| \leq e^{A|0|^{\alpha}} = 1$, so I would look at a function which maps $0$ inside the closed unit disc. Looking at the form of the inequality, I would assume that a function of the form $e^z$ would be a candidate, firstly because it is not a polynomial function. But I don't know how to exactly determine the function $f(z)$