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

:$I(r)\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_0^\{2\backslash pi\}\backslash !\; \backslash left|\; f(r\; e^\{i\backslash theta\})\; \backslash right|\; \backslash ,d\backslash theta$

for then this function is strictly increasing and is a convex function of $\backslash log\; r$.