By Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)
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Lrx- DELOF. 24 II. Solution of the Dirichlet Problem for a Schlicht Region at every boundary point t;, except perhaps at the points t;1 , ••. , t;,. where the boundary value function u(t;) is discontinuous, but in any case also bounded. An application of the maximum and minimum principles now shows that the difference v(z) vanishes identically, proving the uniqueness of the solution to the boundary value problem. We shall later see (Chapter VII) that the Poisson integral leads to the solution of the Dirichlet problem under far more general hypotheses.
1). Lindelof's principle can be given an especially suggestive form if like PICK  and CARATHEODORY  one introduces a la Poincare a noneuclidean metric on these hyperbolic surfaces, as has already been set forth in I, § 1, for the disk. 4. In order to attain this general formulation we must discuss somewhat more thoroughly the simplest case, where G, as well as G11• is the unit disk. If we then set z0 = w0 = 0, the principle of Lindelof becomes Schwarz's lemma, which can be proved most simply in a direct way by applying the maximum principle to the function wjz, which is regular for jzj < 1: If w(z) is regular and bounded for Jzj < 1: and w(O) = 0, then [w(z)[ < 1 [w(z) [ < [z[ for every [z\ < 1.
Zl -+ oo so fast that the lower limit lim ~g M(j_ I' 1 E . PHRAGMEN and E. LI N D E LOF . A still sharper form of this theorem was given by F. and R. NEVANLINNA . 44 III. Function Theoretic Majorant Principles is positive; or the function is bounded so that fw(z)f < 1 at every point of the half plane. As a further result we have this corollaryl: If under the hypotheses of the above theorem w(z) tends to zero on a sequence of half circles so that . - logM(r) - = -oo r ' hm - ·1'~00 then w(z) vanishes identically.
Analytic Functions by Professor Dr. Rolf Nevanlinna (auth.), Professor Dr. B. Eckmann, Professor Dr. B. L. van der Waerden (eds.)