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.

Let

f(z) be a holomorphic function on the annulusr _{1\leq\left|} z\right| \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 ^{λ} for some constants λ and c , then log M(r) is strictly convex as a function of log (r).

The conclusion of the theorem can be restated as

log \left( (r _{3})/(r _{1}) \right) log M(r _{2 )\leq log \left( (r _{3})/(r _{2})} \right) log M(r _{1 ) +log \left( (r _{2})/(r _{1})} \right) log M(r _{3)} for any three concentric circles of radii r _{1<r} _{2<r} _{3.}

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.

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.

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"