Hardy's theorem

In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let [math]\displaystyle{ f }[/math] be a holomorphic function on the open ball centered at zero and radius [math]\displaystyle{ R }[/math] in the complex plane, and assume that [math]\displaystyle{ f }[/math] is not a constant function. If one defines

[math]\displaystyle{ I(r) = \frac{1}{2\pi} \int_0^{2\pi}\! \left| f(r e^{i\theta}) \right| \,d\theta }[/math]

for [math]\displaystyle{ 0\lt r \lt R, }[/math] then this function is strictly increasing and is a convex function of [math]\displaystyle{ \log r }[/math].

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.