Hardy's theorem

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

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


 * $$I(r) = \frac{1}{2\pi} \int_0^{2\pi}\! \left| f(r e^{i\theta}) \right| \,d\theta$$

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