In mathematics, the Borel–Carathéodory theorem in complex analysis shows that an analytic function may be bounded by its real part. It is an application of the maximum modulus principle. It is named for Émile Borel and Constantin Carathéodory.
Statement of the theorem
Here, the norm on the left-hand side denotes the maximum value of f in the closed disc:
(where the last equality is due to the maximum modulus principle).
Define A by
First let f(0) = 0. Since Re f is harmonic, we may take A>0. f maps into the half-plane P to the left of the x=A line. Roughly, our goal is to map this half-plane to a disk, apply Schwarz's lemma there, and tease out the stated inequality.
sends P to the standard left half-plane. sends the left half-plane to the circle of radius R centered at the origin. The composite, which maps 0 to 0, is the desired map:
From Schwarz's lemma applied to the composite of this map and f, we have
Take |z| ≤ r. The above becomes
as claimed. In the general case, we may apply the above to f(z)-f(0):
which, when rearranged, gives the claim.
- Lang, Serge (1999). Complex Analysis (4th ed.). New York: Springer-Verlag, Inc. ISBN 0-387-98592-1.
- Titchmarsh, E. C. (1938). The theory of functions. Oxford University Press.