The boundary behaviour of harmonic univalent maps

LEMMA 1. Let f = h +~g e SH. Then log/z' is a Block function, that h"{z) h'{z) < JΓ7 = re iΘ), for some absolute constant c.

LEMMA 2. Let h be an analytic and locally univalent function in D satisfying inequality (2). If a > 0, then lim (1 - r) ah'(re ιθ) = 0 for almost all θ.

THEOREM 1. Let f = h + ~g eSH. Then the integrals •1 rl /•! df , J dr ί \h'{re iΘ)\dr, f \g'(re iθ)\dr, and ί Jo Jo Jo converge for almost all θ, and the boundary function f(e ιθ) exists al-

THEOREM 2. Let f = h+J e SH. Then h,g e H λ and f e h λ for every λy 0 < λ < 1/c 2, where c is the absolute constant of Lemma 1.

COROLLARY 1. Let h be analytic in D. If 'log h! is a Block function, then h G H λ for some λ. 3. / constant on some arc of dD. The purpose of this section is to study the boundary behaviour of a function / G SH along a closed subarc of d D on which / extends continuously to a constant function there. First, we need the following: DEFINITION 1. Let D be a simply connected domain in C with at least two boundary points, P an accessible prime end, and (Cn) a null-chain of P. We say that D satisfies

THEOREM 3. Let f = h+~g G SH- Suppose that f extends continu- ously to a nondegenerate subinterval, J, ofdD such that f{z) — w0for all z G /. Then φ{J) = λ, \λ\ = 1; and ifP is the prime end of f{D) that corresponds to λ, under F, then f(D) satisfies the wedge condition at P. For a definition of a prime-end see [8, pp. 271-277]. The proof of the theorem makes use of the following interesting lemma.

LEMMA 3. Let f = h + ~g be continuous on D, where h and g are analytic in D. Suppose that f(z) = w0 for all z belonging to a non- degenerate subinterval J = {e ίθ: θx < θ < θ2} of 3D. Then (a) fθ(z) ^Oas z->ζfor every ζ e Int(/). (b) h and g extend analytically to Int(/), and (c) zh'{z) = zg'(z) for all z e Int(/).

COROLLARY 2. Let f — u + v e SH satisfy u,v e h p, p > 1, and a: z(t), 0 < t < 1, be a Jordan arc in D with \z{t)\ -+ 1 (t -+ 1). // f(z(t))-+wo(t-+l)and (10) exists, then z(t) -* ζ(t -+ I) for some ζ edD so that a is an asymptotic path. For the definition of an asymptotic path see [8, p. 267].

COROLLARY 3. Let f be as in Corollary 2, w0 an accessible point off(D), and aj: zj(t), 0 < t < 1 (7 = 1,2) be Jordan arcs in D with \zj(t)\ -> 1 (ί -> 1). Iff(zj(t)) ->wo(t-> 1), (11) limarg(f(zJ(t))-w0) = Θ (7 = 1,2) and the zero cusp at w0 with sides /(αy) lies in f(D), then Zj(t) —> C (/-> l y = 1,2) for some ζedD.

THEOREM 4. Let λ e G. Then the prime-end P of f(D) corre- sponding to λ, under Fy is accessible and satisfies the wedge condition at P. The next lemma is purely topological and is true for all automor- phisms of D with radial limits almost everywhere.

LEMMA 4. Let ζ, ζ' e dD. (a) If C(φ,ζ) = dD, then φ extends continuously to a constant λ, \λ\ = 1, on dD\ζ, and ζ is unique. (b) IfC(φ,ζ), C(φ,ζ') are not dD, then Int C(φ,ζ) n C(φ,ζ f) = C(φ,ζ)Γ\lntC(φ,ζ') = φ. (c) IfC{φ, ζ) Π C{φ, ζ') = λ, \λ\ = 1, then φ extends continuously to an open circular arc between ζ and ζ 1, and φ = λ there.

LEMMA 5. Let ζ9 ζ 1 e dD. (a) IfC(φ9 ζ) = dD, then f extends continuously to dD\ζ and f is constant there. (b) IfC(φ9 ζ) n C(φ9 C) = λ, \λ\ = 1, then f extends continuously to an open circular arc between ζ and ζ', and f = F(λ) there.

COROLLARY 4. Suppose that f(D) does not satisfy the wedge condi- tion at the prime-end P corresponding to λ G dD; then there exists a unique ζ edD such that λ e C(φ, ζ). In what follows, we assume that f = u + iv satisfies u, v G hP, p > 1. Let [a, b], a Φ b G C u {00}, denote the line segment from a to b. We call [α, b] a side of a domain H if no point of the interior, (a, b), of [α, b] is a point of accumulation of ΘH\[a, b] and H lies on the left of [a,b]. For λ\9 λ2 G <9Z>, let X ^ be the ci

THEOREM 5. Let f = u + iv, u,v e h p, p > 1, Co = e iθo G E and C(φ, Co) = ffi where [α, έ] w a side ofdf(D), and I\— a (I2 = b) iff is constant on the right (left) ofζ0, otherwise Ix = I(PX) (I2 = I(P2))> where P{ are P2 are the prime ends off(D) corresponding to λ\ and λ2, respectively. In this case, P{ (P2) is accessible at a (b)} and I(P{) (I(P2)) has at most one continuum of subsidiary points given by CR(F,λ\) (Cι(F,λ2)).

COROLLARY 5. Under the assumptions of Theorem 5, iff(D) has no sides, then E = 0.

COROLLARY 6. Under the assumptions of Theorem 5, ifP is a prime end of f{D) which is either non-accessible, or accessible but has two subsidairy continua without the wedge condition, then there exists a unique ζ e dD, with ζ e dD\E such that C(φ, ζ) = λ, where λ is the corresponding point to P under F. In the final part of this section, we give conclusive statements of correspondence between various points of dD and the prime-ends of f(D), where / is assumed univalent and in h p, p > 1. First we

Theorem 5 yield the following:

THEOREM 6. There is a one-to-one correspondence between sf and 38. In view of this result, special attention need be given to the case when / is constant on some non-degenerate open subinterval of dD. Let J — ζ\ζι be such a subinterval. Since Schwarz's Theorem [9, p. 131] offers an exact description of / at ζ\ (fo) if / is also constant on the right of ζ\ (left of ζι), we restrict ourselves to the complementary case when / is constant neither on the right of ζ\ nor the left of ζι-

THEOREM 7. Suppose that f(ζ) = w for all ζ e Int£iC2 and f is constant neither on the right ofζ\ nor the left ofζi Then the following is true: (a) Ifζ\,ζ2 € dD\E, then there is a prime end, λ, off(D) satisfying the wedge condition at WQ such that ζ\ £2 ctndλ correspond to each other with C(φ, Ci) = CR(F9λ) and C(φ9 ζ2) = CL(F,λ) (see Figure 3.a).