Harmonic Mappings of Multiply Connected Domains

Theorem 1. Let D be a bounded finitely connected Jordan domain, and let Ωbe a bounded domain of equal connectivity whose outer boundary is a Jordan curve and whose inner boundary components are locally connected. Let f ∗be an orientation-preserving weak homeomorphism of ∂D onto ∂Ω. Let f be the solution of the Dirichlet problem, the harmonic extension of f ∗ to D. If f(D) ⊂Ω, then f maps D univalently onto Ω. Conversely, if f is univalent in D, then f(D) = Ω. Since the harmonic extension f is in

Theorem 2. Let D be a finitely connected domain bounded by Jordan curves C0, C1, . . . , Cn, where C0 is the outer boundary component. Let Ωbe a bounded convex domain. Suppose that f ∗is an orientation-preserving weak homeomorphism of C0 onto ∂Ω. Then there is a function f harmonic in D and continuous in D, mapping D univalently onto Ωwith n points removed, with the prescribed boundary values: f(z) = f ∗(z) on C0. It must be emphasized that the locations of the punctures are not pre- scribed inde

Corollary 1. Let D be a finitely connected domain bounded by Jordan curves C0, C1, . . . , Cn, where C0 is the outer boundary component. Let Ωbe a bounded convex domain. Suppose that f ∗is an orientation-preserving weak homeomorphism of C0 onto ∂Ω. Let f = h + g be any function harmonic in D and continuous in D, with h and g (globally) analytic in D, such that f(z) = f ∗(z) on C0 and f(z) is constant on each of the inner boundary components C1, . . . , Cn. Then f is a univalent mapping of D onto

Corollary 2. Let f = h + g be harmonic in a finitely connected domain D and continuous in D. Then f is a univalent mapping of D onto a punctured convex domain if and only if for each α ∈R the function φα = eiαh −e−iαg is a conformal mapping of D onto a slit domain convex in the horizontal direction. This last result will be further generalized in Section 4. We now conclude the present section with some remarks and two examples. First we note

Theorem 3. Every domain D ⊂ˆC containing the point at infinity admits a canonical harmonic punctured-plane mapping F = H +G, where H and G are analytic (and single-valued) in D. If ∂D has countably many components, such a mapping is unique.

Theorem 4. Let D be a domain containing infinity, not a punctured plane, and let F = H + G be the canonical harmonic punctured-plane mapping of D. Let G have the expansion G(z) = ∞ X n=1 Bnz−n near infinity. Then B1 < 0 and D(f, f) ≥D(G/B1, G/B1) for every func- tion f analytic in D with the form f(z) = 1 z + ∞ X n=2

Theorem 5. Let D ⊂ˆC be a finitely connected domain containing the point at infinity, let F = H + G be the canonical harmonic punctured-plane mapping of D, and let A = G′/H′ be its dilatation function. Let f ∈P H(D) be a solution of fz = Afz in D. Then f = F.