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
This can also be established directly using the maximum principle in the following way. ::Tu~ defines a function that is regular for jzf < 1 and whose absolute value has the upper bound one on the circle fzl = 1; it follows from this that its absolute value is also at most equal to one at every interior point of the unit disk, and equality holds only if the quotient is constantly equal to a 48 III. Function Theoretic Majorant Principles number of absolute value 1 so that the function w = w(z) maps this disk conformally onto itself.
Having the points z0 and w0 as poles. 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,).
If t; is a point of continuity of w, then on a segment that contains ~ in its interior, the harmonic measure either takes the value 0 or the value 1, and according to the reflection principle w is therefore harmonic in a certain neighborhood Ix - ~I < r of ~; the function
Analytic Functions by Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)