By Hille E.
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Extra info for An Integral Equality and its Applications
The limit f either being constant or a schlicht function (see [Ding6l], p. 256) satisfies I f'(zo)I = C > 0. Hence, f is schlicht in D. It remains to show f [D] = D. Assume this is not true. Then there exists a d E D\ f [D] # 0 satisfying d 54 0 because f(zo) = 0. Then any branch of the function fi(z) f(z) d , fI(zo) = V 1 - df(z) is single-valued and schlicht in D different from 0 and oo. fe(z) - fi(zo) 1 - fi(zo)fi(z)' z E D , 21 Function theoretical tools which is in So as f1 E S. Now fi(zo) fo(zo) = 1- fi(zo) - I - Idl , so that fo(zo) = 1 + Idlf,(zo), i7-=d lfo(zo)I = 2f'(zo) , 2V---d I - Ifi(zo)I2 1 + IdlC 2 >C.
The CAUCHY-RIEMANN system for the function (g, h) leads to On , as an The integral in the above representation of h can be taken along the straight line from a to z. h is determined up to an arbitrary additive constant. It is a multi-valued function. The so-called complex GREEN function [Mikh35], see [Gakh66], p. 209, M(z, zo) = g(z, zo) + ih(z, zo) in the neighborhood of zo behaves like - log(z - zo)+ analytic function. Remark. For simply connected domains the function f(z, zo) := exp(-M(z, zo)) is a single-valued analytic function mapping D conformally onto D with f(zo, zo) = 0.
Remark. The existence of the GREEN function for a given domain D can be proved if the DIRICHLET problem for harmonic functions can be solved for D. The DIRICHLET problem demands us to find a harmonic function in I) attaining prescribed boundary values on the boundary 8D. In case of continuous boundary values this problem can be shown to be (uniquely) solvable for a wide class of domains by the method of PERRON-RADO-RIESZ. If then u(z, zo) is the harmonic function in D satisfying lim u(z, zo) = log ]r; - zoj, =-C (E 8D , for zo E D fixed then g(z,zo) log I Iz - zo[ + u(z,zo) is the GREEN function for D.