Hadamard three-circle theorem

In complex analysis, a branch of mathematics, the Hadamard three-circle theorem is a result about the behavior of holomorphic functions.

Let $f(z)$ be a holomorphic function on the annulus

:$r\_1leqleft|\; z\; ight|\; leq\; r\_3.$

Let $M(r)$ be the maximum of $|f(z)|$ on the circle $|z|=r.$ Then, $log\; M(r)$ is a convex function of the logarithm $log\; (r).$ Moreover, if $f(z)$ is not of the form $cz^lambda$ for some constants $lambda$ and $c$, then $log\; M(r)$ is strictly convex as a function of $log\; (r).$

The conclusion of the theorem can be restated as

:$logleft(frac\{r\_3\}\{r\_1\}\; ight)log\; M(r\_2)leq\; logleft(frac\{r\_3\}\{r\_2\}\; ight)log\; M(r\_1)+logleft(frac\{r\_2\}\{r\_1\}\; ight)log\; M(r\_3)$ for any three concentric circles of radii

History

A statement and proof for the theorem was given by J.E. Littlewood in 1912, but he attributes it to no one in particular, stating it as a known theorem. H. Bohr and E. Landau claim the theorem was first given by Jacques Hadamard in 1896, although Hadamard had published no proof.ref|Ed74

ee also

*maximum principle
*logarithmically convex function
*Hardy's theorem

References

* H.M. Edwards, "Riemann's Zeta Function", (1974) Dover Publications, ISBN 0-486-41740-9 "(See section 9.3.)"
* E. C. Titchmarsh, "The theory of the Riemann Zeta-Function", (1951) Oxford at the Clarendon Press, Oxford. "(See chapter 14)"
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