Borel–Carathéodory Theorem
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
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 ...
may be bounded 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 ...
. 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 maximum that is strictly within the domain of f. In other words, either f is locally ...
. It is named for
Émile Borel Félix Édouard Justin Émile Borel (; 7 January 1871 – 3 February 1956) was a French people, French mathematician and politician. As a mathematician, he was known for his founding work in the areas of measure theory and probability. Biograp ...
and
Constantin Carathéodory Constantin Carathéodory (; 13 September 1873 – 2 February 1950) was a Greeks, Greek mathematician who spent most of his professional career in Germany. He made significant contributions to real and complex analysis, the calculus of variations, ...
.


Statement of the theorem

Let a function f 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 usua ...
of
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
''R'' centered at the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
. Suppose that ''r'' < ''R''. Then, we have the following inequality: : \, f\, _r \le \frac \sup_ \operatorname f(z) + \frac , f(0), . Here, the norm on the left-hand side denotes the maximum value of ''f'' in the closed disc: : \, f\, _r = \max_ , f(z), = \max_ , f(z), (where the last equality is due to the maximum modulus principle).


Proof

Define ''A'' by : A = \sup_ \operatorname f(z). If ''f'' is constant ''c'', the inequality follows from (R+r), c, +2r\operatornamec\ge(R-r), c, , 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, : \operatorname f(0) = \int_0^1 \operatorname f(R^)s. Since ''f'' is regular and nonconstant, we have that Re ''f'' is also nonconstant. Since Re ''f''(0) = 0, we must have Re f(z) > 0 for some ''z'' on the circle , z, =R, so we may take A>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 In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex differential geometry that estimates the (squared) pointwise norm , \partial f , ^2 of a holomorphic map f:(X,g_X) \to (Y,g_Y) between Hermitian manifo ...
there, and make out the stated inequality. w \mapsto w/A - 1 sends ''P'' to the standard left half-plane. w \mapsto R(w+1)/(w-1) 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: :w \mapsto \frac. From Schwarz's lemma applied to the composite of this map and ''f'', we have :\frac \leq , z, . Take , ''z'', ≤ ''r''. The above becomes :R, f(z), \leq r, f(z) - 2A, \leq r, f(z), + 2Ar so :, f(z), \leq \frac, as claimed. In the general case, we may apply the above to ''f''(''z'')-''f''(0): : \begin , f(z), -, f(0), &\leq , f(z)-f(0), \leq \frac \sup_ \operatorname(f(w) - f(0)) \\ &\leq \frac \left(\sup_ \operatorname f(w) + , f(0), \right), \end which, when rearranged, gives the claim.


Alternative result and proof

We start with the following result:


Applications

Borel–Carathéodory is often used to bound the logarithm of derivatives, such as in the proof of
Hadamard factorization theorem In mathematics, and particularly in the field of complex analysis, the Hadamard factorization theorem asserts that every entire function with finite order can be represented as a product involving its zeroes and an exponential of a polynomial. It i ...
. The following example is a strengthening of Liouville's theorem.


References


Sources

* Lang, Serge (1999). ''Complex Analysis'' (4th ed.). New York: Springer-Verlag, Inc. . * Titchmarsh, E. C. (1938). ''The theory of functions.'' Oxford University Press. {{DEFAULTSORT:Borel-Caratheodory theorem Theorems in complex analysis Articles containing proofs