# Hardy’s theorem

###### Theorem.

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

 $I(r):=\frac{1}{2\pi}\int_{0}^{2\pi}\lvert f(re^{i\theta})\rvert d\theta$

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

## References

• 1 John B. Conway. . Springer-Verlag, New York, New York, 1978.
Title Hardy’s theorem HardysTheorem 2013-03-22 14:19:44 2013-03-22 14:19:44 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 30C80 msc 30E20 HadamardThreeCircleTheorem