A Fixed Point Theorem for Analytic Functions
نویسنده
چکیده
We denote that set by A(z,w,k) and (following [1]) call it the Apollonius circle of constant k associated to the points z and w. The set A(z,w,k) is a circle for all values of k other than 1 when it is a line. In this paper, we consider z,w ∈ U, show that if z = w, then necessarily A(z,w, √ (1−|w|2)/(1−|z|2)) meets the unit circle twice, consider the arc on the unit circle with those endpoints, situated in the same connected component of C \ A(z,w, √ (1−|w|2)/(1−|z|2)) as z, and denote it by Γz,w. We prove that if Z = (z1, . . . ,zN ) and W = (w1, . . . ,wN ) are N-tuples with entries in U such that zj = wj for all j = 1, . . . ,N and
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