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 {stile di visualizzazione 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. Questo è, {nome dell'operatore dello stile di visualizzazione {Re} f(0)={frac {1}{2pi }}int _{|z|=R}nome operatore {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 {stile di visualizzazione f(z)>0} for some z on the circle {stile di visualizzazione |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 a 0, is the desired map: {displaystyle wmapsto {frac {Rw}{w-2A}}.} From Schwarz's lemma applied to the composite of this map and f, noi abbiamo {stile di visualizzazione {frac {|Rf(z)|}{|f(z)-2UN|}}leq |z|.} Take |z| ≤ R. The above becomes {stile di visualizzazione R|f(z)|leq r|f(z)-2UN|leq r|f(z)|+2Ar} Così {stile di visualizzazione |f(z)|leq {frac {2Ar}{R-r}}} , as claimed. In the general case, we may apply the above to f(z)-f(0): {stile di visualizzazione {inizio{allineato}|f(z)|-|f(0)|&leq |f(z)-f(0)|leq {frac {2r}{R-r}}sup _{|w|leq R}nome operatore {Re} (f(w)-f(0))\&leq {frac {2r}{R-r}}sinistra(sup _{|w|leq R}nome operatore {Re} f(w)+|f(0)|Giusto),fine{allineato}}} quale, when rearranged, gives the claim.

References Lang, Serge (1999). Complex Analysis (4th ed.). New York: Springer-Verlag, Inc. ISBN 0-387-98592-1. Titchmarsh, e. C. (1938). The theory of functions. la stampa dell'università di Oxford. Categorie: Teoremi in analisi complessa