A fixed point theorem for analytic functions
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We prove that each analytic self-map of the open unit disk which interpolates between certain n-tuples must have a fixed point. 1. Introduction Let U denote the open unit disk centered at the origin and T its boundary. For any pair of distinct complex numbers z and w and any positive constant k, we consider the locus of all points ζ in the complex plane C having the ratio of the distances to w and z equal to k, that is, we consider the solution set of the equation |ζ − w | = k. |ζ − z|
(1.1)
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 z j = w j for all j = 1,...,N and T=
N j =1
Γz j ,w j ,
(1.2)
then each analytic self-map of U interpolating between Z and W must have a fixed point. The next section contains the announced fixed point theorem (Theorem 2.2). Copyright © 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:1 (2005) 87–91 DOI: 10.1155/FPTA.2005.87
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A fixed point theorem for analytic functions
2. The fixed point theorem For each eiθ ∈ T and k > 0, the set
2
HD eiθ ,k := z ∈ U : eiθ − z < k 1 − |z|2
(2.1)
called the horodisk with constant k tangent at eiθ is an open disk internally tangent to T at eiθ whose boundary HC(eiθ ,k) := {z ∈ U : |eiθ − z|2 = k(1 − |z|2 )} is called the horocycle with constant k tangent at eiθ . The center and radius of HC(eiθ ,k) are given by C=
eiθ , 1+k
R=
k , 1+k
(2.2)
respectively. One should note that HD(eiθ ,k) extends to exhaust U as k → ∞. Let ϕ be a self-map of U. For each positive integer n, ϕ[n] = ϕ ◦ ϕ ◦ · · · ◦ ϕ, n times. The following is a combination of results due to Denjoy, Julia, and Wolff. Theorem 2.1. Let ϕ be an analytic self-map of U. If ϕ has no fixed point, then there is a remarkable point w on the unit circle such that the sequence {ϕ[n] } converges to w uniformly on compact subsets of U and
ϕ HD(w,k) ⊆ HD(w,k)
k > 0.
(2.3)
The remarkable point w is called the Denjoy-Wolff point of ϕ. Relation (2.3) is a consequence of a geometric function-theoretic result known as Julia’s lemma. In case ϕ has a fixed point, but is not the identity or an elliptic disk automorphism, one can use Schwarz’s lemma in classical complex analysis to show that {ϕ[n] } tends to that fixed point, (which is also regarded as a constant function), uniformly on compact subsets of U. These facts show that if ϕ is not the identity, then it may have at most a fixed point in U. Good accounts on all the results summarized above can be found in [2, Section
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