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.
be analytic on a closed disc of radius R centered at the origin.
Suppose that r < R. Then, we have the following inequality: 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 If f is constant c, the inequality follows from
Since Re f is harmonic, Re f(0) is equal to the average of its values around any circle centered at 0.
Now 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 make 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 so as claimed.
In the general case, we may apply the above to f(z)-f(0): which, when rearranged, gives the claim.
WLOG, subtract a constant away, to get
using Cauchy integral formula:
Then by linearly combining the three integrals, we get
) d t ( 1 + cos ( 2 π n t + θ ) ) =
) d t ( 1 + cos ( 2 π n t + θ ) ) =
d t ( 1 + cos ( 2 π n t + θ ) ) =
It suffices to prove the case of
By previous result, using the Taylor expansion,
Corollary 2 (Titchmarsh, 5.51, improved) — With the same assumptions, for all
It suffices to prove the case of
}{(R-r)^{n+1}}}R(M-u(0))\end{aligned}}}
Borel–Carathéodory is often used to bound the logarithm of derivatives, such as in the proof of Hadamard factorization theorem.
is an entire function, such that there exists a sequence
is a constant function, and it converges to zero, so it is zero.
Corollary — If an entire function
has no roots and is of finite order
Apply the improved Liouville theorem to