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

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

for then this function is strictly increasing and logarithmically convex.

ee also

* maximum principle
* Hadamard three-circle theorem

References

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