Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B.'s Analytic Functions PDF

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

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L. Now it is immediately clear that the harmonic measure w is equal to 1/n times the magnitude of the angle formed by the line through z and ! 0 and that part of the real axis lying to the right of t0 as seen from z (nw = n - arg (z- t0 )), and the region A< w < 1 is thus an angle whose vertex lies at z = t0 and which is bounded by the positive real axis and the half ray arg (z - t0 ) = n(l -A). 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.

Y. r.. yy y;, - u(x) 1 =- 2n 1J u(eifJ ) -~~. -·-----~~ ~ · ·--d{}. 1+r 2r cos ({} - rp) 2" - 0 r2 2 - To be sure, the integrand u(eifJ) in general exhibits discontinuities of an involved kind; but the integral nevertheless makes sense (even as a § 2. Solution of the General Boundary Value Problem 25 Cauchy integral), since it can be written as a convergent series whose terms are integrals along the individual segments y!. As can be seen at once, this series is absolutely and uniformly convergent in every subregion of the interior of K, and according to the theorem of ·weierstrass, the sum is therefore a harmonic function inK.

On the level line w(z, eX, G) =A of a function w(z) defined in G. 2 with respect to this region is obviously equal to ~ ~~, and the two-constants theorem yields the rela- tion logM;. (,l- A1} log M;. A) log M;. ::::;; }. 2 , the arc of the curve y = log M;. ,) and (1. ,) does not run above the chord joining these end points, whose ordinates are given by the right hand side of the above inequality. From this one concludes: The logarithm of the maximum modulus log M;. is a convex function of the parameter A.

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