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

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

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

See also

• Maximum principle
• Hadamard three-circle theorem

References

• John B. Conway. (1978) Functions of One Complex Variable I. Springer-Verlag, New York, New York.

This article incorporates material from Hardy's theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.