Hadamard three-circle theorem explained

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

Statement

Hadamard three-circle theorem: Let

f(z)

be a holomorphic function on the annulus

r_1\leq\left|z\right|\leqr₃

. Let

M(r)

be the maximum of

|f(z)|

on the circle

|z|=r.

Then,

logM(r)

is a convex function of the logarithm

log(r).

Moreover, if

f(z)

is not of the form

cz^λ

for some constants

λ

and

c

, then

logM(r)

is strictly convex as a function of

log(r).

The conclusion of the theorem can be restated as

for any three concentric circles of radii

Proof

The three circles theorem follows from the fact that for any real a, the function Re log(z^af(z)) is harmonic between two circles, and therefore takes its maximum value on one of the circles. The theorem follows by choosing the constant a so that this harmonic function has the same maximum value on both circles.

The theorem can also be deduced directly from Hadamard's three-line theorem.

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. Harald Bohr and Edmund Landau attribute the theorem to Jacques Hadamard, writing in 1896; Hadamard published no proof.

See also

• Maximum principle
• Logarithmically convex function
• Hardy's theorem
• Hadamard three-line theorem
• Borel–Carathéodory theorem
• Phragmén–Lindelöf principle

References

• • • E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 14)

External links

• "proof of Hadamard three-circle theorem"