By Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)
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Extra info for Analytic Functions
We fix a number A. > 0 and consider the values of w(z) in the region g(z, z0 , G,):? ; if one letsp, (p > 0) stand for the minimum ofg(w(z), w0 , Gw) in this region, then the point w = w(z) lies in the region g(w, w0 , Gw) > p for these same values of z. fe can now apply our principle to the regions G~: 0 < g(z, z0 ) i g(z, z 0, G,).
The same result also holds when t; is the end point of an arc cx. , suppose that a point on the boundary moves in the positive direction when it moves along the arc IX starting from the point (. *(x) then takes the value 1 on the segment 0 < x - ~ < e of the real axis, and the value zero on the segment 0 > x - ; > -e (for sufficiently small (! > 0). This is also a property of the function 1 1 If 1 = in log~-__:_:~:' 35 § 5. On the Level Lines of the Harmonic Measure which maps the upper half plane onto the parallel strip P in such a way that the line w = 1 corresponds to the part of the real axis x < ~.
_ 1 < t < t. (r = 1, ... , p; t0 = 0, tp = 1), and for each r = 1, ... , p suppose that the rth subinterval is mapped continuously and one-to-one onto the Jordan curve 1~. ·· + Tp) in such a way that the point Wl on the boundary of G runs over Tin the positive direction with respect toG as t increases continuously from 0 to 1. Let cx(t) be the part of the boundary T that corresponds to the interval (0, t). We now prove that for a fixed z, w(z, cx(t)) is a monotonically increasing, contimtous function oft.