Hardy's theorem

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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}{2\pi }}\int _{0}^{{2\pi }}\!\left|f(re^{{i\theta }})\right|\,d\theta

for 0<r<R, then this function is strictly increasing and logarithmically convex.

See also

References

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

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