Hardy's theorem

In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions.

Let ${\displaystyle f}$ be a holomorphic function on the open ball centered at zero and radius ${\displaystyle R}$ in the complex plane, and assume that ${\displaystyle f}$ is not a constant function. If one defines

${\displaystyle I(r)={\frac {1}{2\pi }}\int _{0}^{2\pi }\!\left|f(re^{i\theta })\right|\,d\theta }$

for ${\displaystyle 0<r<R,}$ then this function is strictly increasing and is a convex function of ${\displaystyle \log r}$.