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 ${\displaystyle \log r}$.

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.

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