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Hardy's theorem (Theorem)
Theorem 1   Let $f$ be a holomorphic function on $B(0,R)$ (the open ball of radius $R$ and $f$ is not a constant function, then \begin{equation*} I(r) := \frac{1}{2\pi} \int_0^{2\pi} \lvert f(r e^{i\theta}) \rvert d\theta \end{equation*}is strictly increasing and $\log I(r)$ is a convex function of $\log r$

Bibliography

1
John B. Conway. Functions of One Complex Variable I. Springer-Verlag, New York, New York, 1978.




"Hardy's theorem" is owned by jirka.
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See Also: Hadamard three-circle theorem

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Cross-references: convex function, strictly increasing, constant function, radius, open ball, holomorphic function

This is version 3 of Hardy's theorem, born on 2004-04-23, modified 2005-06-10.
Object id is 5798, canonical name is HardysTheorem.
Accessed 2344 times total.

Classification:
AMS MSC30C80 (Functions of a complex variable :: Geometric function theory :: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination)
 30E20 (Functions of a complex variable :: Miscellaneous topics of analysis in the complex domain :: Integration, integrals of Cauchy type, integral representations of analytic functions)

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