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.
Statement
Hadamard three-circle theorem: Let be a holomorphic function on the annulus. Let be the maximum of on the circle Then, is a convex function of the logarithm Moreover, if is not of the form for some constants and, then is strictly convex as a function of
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 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.