Hardy's theorem

In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions.

Let <math>f</math> be a holomorphic function on the open ball centered at zero and radius <math>R</math> in the complex plane, and assume that <math>f</math> is not a constant function. If one defines

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

for <math>0< r < R,</math> then this function is strictly increasing and is a convex function of <math>\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.