By Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)

ISBN-10: 3642855903

ISBN-13: 9783642855900

ISBN-10: 364285592X

ISBN-13: 9783642855924

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Y; is actually the maxi- § 3. 1. Let IX be a point set consisting of finitely many boundary arcs of the region G. According to the results of § 2 of this chapter, there exists one and only one bounded harmonic function in G that takes the value 1 at every interior point of IX, while vanishing at every point of the complementary portion {3 of the boundary T =IX + {3. 5, we call the value taken by this function at an interior point z of G the harmonic measure of the arc IX with respect toG at the point z; we denote it by w(z, ex, G) or by w(z, ex), if the rest is clear.

Applying the minimum principle in the region G. punctured at the poles of the difference gives the desired result. ) = A, g(w(z), w0 , Gw) > A, and equality holds for every z and each A > 0 provided it holds at one point z and for one value of A. ) + iJ-L, where -h is the function conjugate to g and f-1- is a real parameter. 3. Of course, the extremal functions w(z, f-1-) in general will not be single-valued for multiply connected regions. In these cases a sharp estimate can be obtained by mapping the universal covering surfaces G';" and G';;' conformally onto the unit disk, whereby the extremal problem is reduced to simply connected regions, for which Lindelof's principle does give the best possible estimate.

And Gw have boundaries so "weak" that their Green's functions do not exist; this is the case, for example, when the boundaries consist of isolated points 2 • E. DELOF [2]. A schlicht region G possesses a Green's function if and only if the set of its boundary points is not "of harmonic measure zero". This question will be investigated more thoroughly in Chapter V. 1 2 § 3. , that these surfaces be of hyperbolic type, which is the case if and only if the boundaries each contain at least three boundary points (cf.

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Analytic Functions by Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)


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