Hardy’s theorem

Theorem.

Let $f$ be a holomorphic function on $B\mathit{}\mathrm{(}\mathrm{0}\mathrm{,}R\mathrm{)}$ (the open ball of radius $R$)and $f$ is not a constant function, then

$$I(r):=\frac{1}{2\pi }{\int }_0^{2\pi }|f(r{e}^{i\theta })|𝑑\theta$$

is strictly increasing and $\log \mathit{}I\mathit{}\mathrm{(}r\mathrm{)}$is a convex function of $\log \mathit{}r$.