Quasihomeomorphisms and univalent harmonic mappings onto punctured bounded convex domains

Proposition 2.1. Let f be a function of T into a Jordan curve C, and let F be a sense-preserving homeomorphism of T onto C. (i) If f is a sense-preserving quasihomeomorphism of T onto C, then there is a real-valued nondecreasing function ϕ on R such that ϕ(t + 2π) = ϕ(t) + 2π and f(eit) = F(eiϕ(t)). (ii) If f(eit) = F(eiϕ(t)), where ϕ is a real-valued nondecreasing function on R such that ϕ(t + 2π) = ϕ(t) + 2π, and if E is the countable set of points eiϕ(t) where ϕ is discontinuous, then f coinc

Theorem 2.1. Let f ∈Hu(ρ, G). Then there is a countable set E ⊂T such that the following hold: (i) For each eiθ ∈T \ E, the unrestricted limit f(eiθ) exists and belongs to ∂G. Furthermore, f is continuous in A(ρ, 1) \ E. (ii) For each eiθ0 ∈E, the side-limits limθ↑θ0 f(eiθ) and limθ↓θ0 f(eiθ) exist in ∂G and are distinct. (iii) For each eiθ0 ∈E, the cluster set C(f, eiθ0) lies in ∂G and is the straight-line segment joining the side-limits limθ↑θ0 f(eiθ) and limθ↓θ0 f(eiθ). (iv) co(f(T \ E)) = G.

Theorem 2.2. Suppose that the following are true: (i) f∗is a sense-preserving quasihomeomorphism of T into ∂G, and the constant ζ0 defined by (1.3) on Tρ. (ii) cof∗( eE(f∗)) = G. (iii) f is the Dirichlet solution of f∗in A(ρ, 1).

Lemma 2.1. fn →f locally uniformly in A(ρ, 1).

Lemma 2.2. (a) hn →h locally uniformly in A(ρ2, 1). (b) For z ∈A(ρ, 1), f(z) = h(z) −h(ρ2/z) + ζ0 = ζ0 + X k̸=0 ck(f∗)r2k −ρ2k 1 −ρ2k r−keikθ (2.5) where ck(f∗), k = ±1, ±2, . . . , is the k-th Fourier coefficient of f∗.

Theorem 2.3. Define T : Q(G) →H0(ρ, G) by T(f∗) = f, where f is the Dirichlet solution in A(ρ, 1) of the boundary function which is f∗on T and the average of f∗on Tρ. Then T is bijective. Furthermore, for a sequence {f∗ n} in Q(G), the following statements are equivalent: (a) f∗ n →f∗a.e.. (b) f∗ n →f∗in L1. (c) fn →f locally uniformly in A(ρ, 1).

Proposition 3.1. Suppose that the following are true: (i) f∗is a sense-preserving quasihomeomorphism of T into ∂G such that co(f∗( bE(f∗))) = G. (ii) f is the Dirichlet solution in A(ρ, 1) of the function defined by f∗on T and a constant ζ ∈G on Tρ. (iii) f0 is the Dirichlet solution of the function defined by f∗on T and the average ζ0 of f∗on T. Then there is an analytic function h in A(ρ2, 1) such that f(z) = h(z) −h(ρ2/z) + ζ + 2cζ log(|z|/ρ) (3.1) = f0(z) + 2cζ log |z|, (3.2)

Corollary 3.1. Let f0 be the average associate of f ∈Hu(ρ, G) with f(Tρ) = ζ and f0(Tρ) = ζ0. Then there is an analytic function h in A(ρ2, 1) such that (3.1) and (3.2) hold simultaneously. Suppose now that f ∈Hu(ρ, G) has form (1.2). According to Propo- sition 3.1, f and its average associate f0 have the same analytic and co- analytic part h. Since our interest in this section is exclusively in h, we restrict ourselves to functions f ∈H0(ρ, G) of form (2.2). We shall need the notion of the modu

Theorem 3.1. Suppose f ∈H0(ρ, G) has form (2.2). Then (a) h′ is nonvanishing on Tρ and h maps Tρ homeomorphically onto a convex curve whose diameter is bounded above by D = (4d/π) tanh−1 µ−1(log(1/ρ))
. (b) If h(z) = P∞ −∞anzn, z ∈A(ρ2, 1), then ∞ X n=1 n|a−n|2ρ−2n < ∞ X n=1

Lemma 3.1. Suppose f ∈Hu(ρ, G) has form (2.2), and let Φα(z) = eiαh(z) + e−iαh(ρ2/z), (z ∈A(ρ2, 1)). (3.5) Then Φα is univalent in A(ρ, 1) and it maps A(ρ, 1) onto a slit domain convex in the direction of the real axis. Our second lemma is intuitive and geometric in nature, and it needs some basic notions. A closed curve is a continuous image of T; we use the same notation for the curve and its defining function. Let γ be a closed curve, and let ℓbe a straight line. A point w ∈γ ∩ℓis called a mee

Lemma 3.2. If every straight-line through the origin meets a closed curve γ exactly twice, counting multiplicity, and at crossing points only, then γ is a Jordan curve whose inner domain is starlike with respect to the origin.

Theorem 3.2. Suppose f ∈H0(ρ, G) has form (2.2). Then there is a univa- lent close-to-convex function H of D and a homeomorphism φ of A(ρ, 1)∪T into D with φ(T) = T such that h = H ◦φ.

Lemma 3.3. Fix p, p = 2, 3, . . . . Suppose f ∈H0(ρ, G) has form (2.2) and an unrestricted limit function that satisfies the following properties: (i) f is a sense-preserving local homeomorphism of T onto ∂G. (ii) f(p) exists and is absolutely continuous on T. (iii) f′ is nonvanishing on T. Then (a) h extends to A(ρ2, 1) such that h(eiθ) and h(ρ2eiθ) are continuously (p −1)-differentiable with lim z→reiθ h(k)(z) = h(k)(reiθ), (z ∈A(ρ2, 1)), (3.6) where r is either 1 or ρ2. (b) h′(eiθ) ̸= 0 and h′(

Lemma 3.4. Suppose f ∈H0(ρ, G) has form (2.2), ∂G an analytic curve, and f(eiθ) an infinite-differentiable function with a nonvanishing derivative. Let Γ be the convex curve defined by Γ(θ) = h(ρeiθ), 0 ≤θ ≤2π (see

Theorem 3.1(a)). Then h is a sense-preserving homeomorphism of Tρ onto

Lemma 3.4, each hn is a sense-preserving homeomorphism of Tρ onto Γn with ℜ  1 + ρeiθ h′′ n(ρeiθ) h′n(ρeiθ)  ≥0 for all θ; see (3.22). Using this, with Lemma 2.2(a) and Theorem 3.1(a), we conclude that h satisfies (3.22) and, consequently, h is also a sense- preserving homeomorphism of Tρ onto Γ. Also, by Lemma 3.4, we have each hn univalent in A(ρ, 1). Hence, by Hurwitz’s theorem, h is also a univalent function on A(ρ, 1) or else f is a constant. Define W as above,

Proposition 3.1, the class H(ρ, f∗) yields an analytic function h in A(ρ2, 1), unique up to an additive constant, such that every f ∈Hu(ρ, f∗) is of the forms (3.1) and (3.2). In our second result, we characterize in terms of h and f∗the boundary points of K(ρ, f∗) in a manner leading to a univalence criterion for functions f ∈H(ρ, f∗). Finally, we provide sufficient conditions on ρ, G, and f∗that warrant a nonempty interior for K(ρ, f∗).

Theorem 4.1. K(ρ, f∗) is a nonempty compact subset of G.

Theorem 4.2. Let f ∈Hu(ρ, f∗) be of form (3.1), where f∗: T →∂G is a twice-differentiable function with nonvanishing derivative and absolutely continuous second derivative. Then the dilatation of f and zh′(z)+cζ extend continuously to A(ρ, 1) ∪T such that eiθh′(eiθ) + cζ ̸= 0 for all θ. Moreover, we have: (a) If ζ ∈∂K(ρ, f∗), then either ρeiθ1h′(ρeiθ1) + cζ = 0 for some θ1, or |ω(eiθ2)| = 1 for some θ2. (b) If |ω(eiθ)| = 1 for some θ, then ζ ∈∂K(ρ, f∗). (c) If in (a) and (b) the function |ω(eiθ)|

Theorem 3.1(a), and |ω(eiθ)| = 1 for some θ if and only if ρ2|h′(ρ2eiθ)| = |h′(eiθ)|. We conclude the following Corollary 4.1.

Corollary 4.1. Let f ∈Hu(ρ, f∗) be of form (2.2), where f∗is as in The- orem 4.2. Then the following statements are equivalent: (a) ζ0 ∈∂K(ρ, f∗).

Theorem 4.3. Let f ∈H(ρ, f∗) be of form (3.1), where f∗be smooth as in

Theorem 4.2. Then f ∈Hu(ρ, f∗) if zh′(z) + cζ ̸= 0 for z ∈A(ρ, 1), and if one of the following two inequalities holds for all θ: (a) |ω(eiθ)| ≤1. (b) 2ℜ eiθh′(eiθ) + cζ eiθf′(eiθ)  ≥1. We remark that f∗as defined in Theorem 4.2 yields, by Lemma 3.3, zh′(z) ̸= 0 for z ∈A(ρ, 1). This makes the above sufficiency condition, zh′(z) + cζ ̸= 0 for z ∈A(ρ, 1), easily achievable for functions f ∈H(ρ, f∗) with appropriately small cζ. Finally, we prove the existence of a large family of triplets, 0 < ρ < 1,

Theorem 4.4. Let Ωbe a bounded convex domain, and let h be a homeo- morphism of D onto Ωthat maps D conformally onto Ω. Suppose that h′′ is continuous on D, h′′(eiθ) is absolutely continuous, and ℜ  1 + eiθ h′′(eiθ) h′(eiθ)  > 0 (4.9) for all θ. Then there exists δ > 0 such that for each 0 < ρ < δ we can find a bounded convex domain Gρ such that the harmonic mapping fρ(z) = h(z) −h(ρ2/z), (z ∈A(ρ, 1)), (4.10)

Theorem 5.1. Suppose f ∈Hu(ρ, G) has form (1.2) with ζ0 the average of f on T. If h is analytic in D, then f(z) = ∞ X n=1 λnbn 1 −ρ2n [zn −(ρ2/z)n] + ζ + 2cζ log(|z|/ρ) (5.1) = ∞ X n=1 λnbn

Corollary 5.1. Suppose f ∈Hu(ρ, D) has form (1.2) with ζ0 the average of f on T and h analytic in D. Then there is a unimodular constant λ such that f(z) = λ(1 −|ζ0|2) ( z −ρ2/z 1 −ρ2 + ∞ X n=2 (−λζ0)n−1 1 −ρ2n