Analytic Functions by Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B.

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

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K. , then the argument of the arc differential drp increases by k• . , the argument of dz, on the other hand, by 2n or -2n, according as v = 1 or v > 1. The total growth of arg ']! 1 is hence 2n(l:' k. + p - 2), and using the argument principle we conclude: The function rp'(z) has altogether n + p - 2 zero points in the region G, where n is the number of arcs in

5. On the Level Lines of the Harmonic Measure 33 The harmonic measure of a boundary arc at a point z is equal to 21n times the angle enclosed by the h-lines running from the end points of the arc to z. § 5. 1. : IX is a proper subarc of the boundary, w(z, IX, G) = A are circular arcs which join the end points of IX. It follows from this by conformal mapping that the system of level lines exhibits similar behavior when IX is a boundary arc of a region G bounded by the Jordan curve r: The lines K;.

Fig. A) (in the shaded region of the figure). From here one deduces: If a function which is bounded in the upper half plane converges to a limit as z-+ oo on the positive real axis, then it converges uniformly to the 1 For more results of this kind, cf. F. and R . NEVANLINNA [1]. § 3. The Principle of Hyperbolic Measure 45 same value in every angle 0 < argz< n - 'I} ('I}> 0). We observe that all of the harmonic measures used here are associated with regions so simple that finding the corresponding measures is an elementary problem that can be solved immediately without bringing to bear general existence theorems from the theory of conformal mapping.

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