In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Borel–Carathéodory theorem in complex analysis

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

 shows that an analytic function

Analytic function

In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

 may be bounded by its real part. It is an application of the maximum modulus principle

Maximum modulus principle

In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f| cannot exhibit a true local maximum that is properly within the domain of f....

. It is named for Émile Borel

Émile Borel

Félix Édouard Justin Émile Borel was a French mathematician and politician.Borel was born in Saint-Affrique, Aveyron. Along with René-Louis Baire and Henri Lebesgue, he was among the pioneers of measure theory and its application to probability theory. The concept of a Borel set is named in his...

 and Constantin Carathéodory

Constantin Carathéodory

Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...

.

Statement of the theorem

Let a function be analytic on a closed disc of radius

Radius

In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

 R centered at the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

. 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).

Proof

Define A by

First, assume that . If f is identically zero, then the theorem is trivially true. If f is not identically zero, then A > 0. Define the function g by


The function g has a removable singularity at z = 0. Furthermore, the factor is nonzero when |z| ≤ R, because


Therefore, g is analytic in the disc { z ∈ C : |z| ≤ R }. If z is on the boundary of this disc, then


because


Since g is analytic, we can apply the maximum modulus principle to find that |g(z)| ≤ 1/R for all z with |z| ≤ R.

Now, suppose that ω is a complex number with . Then


So


whence


Thus,


In the general case, where f(0) does not necessarily vanish, let . Then, by the triangle inequality

Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side ....

,


Because , we can say that


if |z| ≤ r. Furthermore,


so


Therefore,


This completes the proof.