In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Borel–Carathéodory theorem in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
shows that an
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. Functions of each type are infinitely differentiable, but complex an ...
may be
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
by its
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. It is an application of the
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 strict local maximum that is properly within the domain of ''f''.
In other words, eit ...
. It is named for
Émile Borel
Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Math ...
and
Constantin Carathéodory
Constantin Carathéodory ( el, Κωνσταντίνος Καραθεοδωρή, Konstantinos Karatheodori; 13 September 1873 – 2 February 1950) was a Greek mathematician who spent most of his professional career in Germany. He made significant ...
.
Statement of the theorem
Let a function
be analytic on a
closed disc
In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not.
For a radius, r, an open disk is usu ...
of
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
''R'' centered at the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
. 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
:
If ''f'' is constant, the inequality is trivial since
, so we may assume ''f'' is nonconstant. First let ''f''(0) = 0. Since Re ''f'' is harmonic, Re ''f''(0) is equal to the average of its values around any circle centered at 0. That is,
:
Since ''f'' is regular and nonconstant, we have that Re ''f'' is also nonconstant. Since Re ''f''(0) = 0, we must have Re
for some ''z'' on the circle
, so we may take
. 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.
References
* Lang, Serge (1999). ''Complex Analysis'' (4th ed.). New York: Springer-Verlag, Inc. .
* Titchmarsh, E. C. (1938). ''The theory of functions.'' Oxford University Press.
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Theorems in complex analysis
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