Entire function Problem I am new to complex analysis and I am facing difficulty in the following problem:
Let $f(z)$ be an entire function satisfying $|f(z)| \leq k|z|^2$ for some +ve constant k and all z. Show that $ f(z) = az^2$ for some constant a.
Which concepts will be used here?

This is an application of Liouville's theorem.
$|f(z)|\leq|(\sqrt{k}z)^2|\implies|\frac{f(z)}{(\sqrt{k}z)^2}|\leq1$ Notice that $f(z)$ and $(\sqrt{k}z)^2$ are both entire functions thus $\frac{f(z)}{(\sqrt{k}z)^2}$ is entire and it's also bounded therefore it's constant and the result follows.
