Hadamard three-circle theorem

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 r_1

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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