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By J. Topuszanski

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Therefore, using what we have already proved, we again obtain the result that f(z) = c in B. This completes the proof of the lemma. Accessible boundary points. Let us now examine the behavior of the function J= f(z), which maps a given simply connected domain B univalently onto the disk Ill < 1 close to the boundary of the domain B. Since such a domain can be mapped univalently by means of elementary functions onto a bounded domain, it will be sufficient in what follows to consider the case of a bounded domain B and it is for such a domain that we shall prove all the theorems..

Consequently (ki (0)/¢2 (0)I _< 1 with equality holding if and only if EC, where 1Ej = 1, that is, if and only if ¢1(0 = 02(EO. One immediate result of Lindelof's principle is the fact that the conformal radius of a domain increases when the domain is increased c) Minimization of area. Let B denote a simply connected domain that includes the point z = 0 and has more than one boundary point. Let R denote the set of all functions F W with F (O) = 0 and F '(0) = 1 that are regular in B. Then the same function F as in a) is the unique function minimizing the quantity S(F)=551F'(z)I9da B (1) where do is the element of area.

9(0)1, with equality holding if and only if ¢1(0 = 02(E0' where IEI = 1. Proof. Let t = f2(z) denote the function inverse to z = 02(x"). Then, the function z = f2(01(0) maps the disk 1C1 < 1 onto a domain contained in the disk I zl < 1. Also, it maps the point C = 0 into the point z = 0. Consequently (ki (0)/¢2 (0)I _< 1 with equality holding if and only if EC, where 1Ej = 1, that is, if and only if ¢1(0 = 02(EO. One immediate result of Lindelof's principle is the fact that the conformal radius of a domain increases when the domain is increased c) Minimization of area.

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An Intro to Symmetry and Supersym. in Quantum Field Theory by J. Topuszanski


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