Hardy's theorem explained

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)=

	1
	2\pi

	2\pi
\int
	0

\left|f(re^i\theta)\right|d\theta

for

0<r<R,

then this function is strictly increasing and is a convex function of

logr

.

See also

References

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