Lemma 1.1 Assume ϕ is a holomorphic mapping from D into itself.
Theorem 3.1 Let 0 < p < ∞, 0 ≤ s ≤ 1, and 0 < α ≤ 1. Assume that ϕ is a holomorphic mapping from D into itself.
Proposition 3.1 Let 0 < p < ∞, 0 ≤ s ≤ 1 and 0 < α ≤ 1. Assume that ϕ is a holomorphic mapping from D into itself.
Assume that (f:OmegatoOmega) is a holomorphic mapping such that (operatorname{dist}(f(x), partialOmega) geepsilon) for some (epsilon> 0) and all (x inOmega).
We are interested in the set T a [ Y 0 ] of marked once-holed tori X for which there is a holomorphic mapping of X into Y 0. It possesses an interesting quantitative property.
We compare it with the subset T ∞ [ Y 0 ] of marked once-holed tori X such that there is a holomorphic mapping f : X → Y 0 for which the cardinal numbers of f − 1 ( p ), p ∈ Y 0, are bounded.
Then, for some l ∈ [ 0, 1 ), there is a holomorphic mapping f of X τ ( l ) into Y 0. Recall that T τ ( l ) is the horizontal slit domain T τ ∖ π τ ( [ 0, l ] ) of the torus T τ.
Moreover, the proofs of Theorems 3.1 and 3.2 yield the following result: Theorem 3.3 Let 0 < p < ∞, − 1 < s ≤ 1, and 0 < α ≤ 1. Assume that ϕ is a holomorphic mapping from D into itself.
If f : R → R ′ is holomorphic and maps a and b onto loops freely homotopic to a ′ and b ′ on R ′, respectively, then we say that f is a holomorphic mapping of Y into Y ′ and use the notation f : Y → Y ′.
Since the inclusion mapping S n → T τ ( l ) is a conformal mapping of the marked once-holed torus W n : = ( S n, χ τ ( l ) ) into X τ ( l ), the restriction f n of f to S n is a holomorphic mapping of W n into Y 0. As S n is relatively compact in T n ( l ), we know that d ( f n ) < ∞.
