Borel–Carathéodory theorem

Borel–Carathéodory theorem 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 Let a function {estilo de exibição f} be analytic on a closed disc of radius R centered at the origin. Suppose that r < R. Then, we have the following inequality: {displaystyle |f|_{r}leq {frac {2r}{R-r}}sup _{|z|leq R}operatorname {Re} f(z)+{frac {R+r}{R-r}}|f(0)|.} Here, the norm on the left-hand side denotes the maximum value of f in the closed disc: {displaystyle |f|_{r}=max _{|z|leq r}|f(z)|=max _{|z|=r}|f(z)|} (where the last equality is due to the maximum modulus principle). Proof Define A by {displaystyle A=sup _{|z|leq R}operatorname {Re} f(z).} If f is constant, the inequality is trivial since {displaystyle (R+r)/(R-r)>1} , 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. Aquilo é, {nome do operador de estilo de exibição {Re} f(0)={fratura {1}{2pi }}int_{|z|=R}nome do operador {Re} f(z)dz.} Since f is regular and nonconstant, we have that Re f is also nonconstant. Since Re f(0) = 0, we must have Re {estilo de exibição f(z)>0} for some z on the circle {estilo de exibição |z|=R} , so we may take {displaystyle 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 there, and make out the stated inequality.

{displaystyle wmapsto w/A-1} sends P to the standard left half-plane. {displaystyle wmapsto 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 para 0, is the desired map: {displaystyle wmapsto {fratura {Rw}{w-2A}}.} From Schwarz's lemma applied to the composite of this map and f, temos {estilo de exibição {fratura {|Rf(z)|}{|f(z)-2UMA|}}leq |z|.} Take |z| ≤ r. The above becomes {estilo de exibição R|f(z)|ler|f(z)-2UMA|ler|f(z)|+2Ar} assim {estilo de exibição |f(z)|leq {fratura {2Ar}{R-r}}} , as claimed. In the general case, we may apply the above to f(z)-f(0): {estilo de exibição {começar{alinhado}|f(z)|-|f(0)|&leq |f(z)-f(0)|leq {fratura {2r}{R-r}}e aí _{|W|leq R}nome do operador {Re} (f(W)-f(0))\&leq {fratura {2r}{R-r}}deixei(e aí _{|W|leq R}nome do operador {Re} f(W)+|f(0)|certo),fim{alinhado}}} que, when rearranged, gives the claim.

References Lang, Sarja (1999). Complex Analysis (4ª edição). Nova york: Springer-Verlag, Inc. ISBN 0-387-98592-1. Titchmarsh, E. C. (1938). The theory of functions. imprensa da Universidade de Oxford. Categorias: Theorems in complex analysis