LOEWNER THEORY FOR QUASICONFORMAL EXTENSIONS:

Lemma 1.1 (Weyl’s lemma (see e.g. [IT92, p.84])). Let f be a continuous function on G whose distributional derivative f¯z is locally integrable on G. If f¯z = 0 in the sense of distributions on G, then f is holomorphic on G. Let B(G) be the open unit ball {µ ∈L∞(G) : ||µ||∞< 1} of L∞(G), where L∞(G) is the complex Banach space of all bounded measurable functions on G, and ||µ||∞:= ess supz∈G |µ(z)| for a µ ∈L∞(G). An element µ ∈B(G) is called the Beltrami coefficient. If f is a k- quasiconformal m

Theorem 1.2 (The measurable Riemann mapping theorem). For a given measurable function µ ∈B(C), there exists a unique solution f of the equation f¯z = µfz (1.2) for which f : C →C is a quasiconformal mapping fixing the points 0 and 1. The equation (1.2) is called the Beltrami equation. Here we give some fundamental properties of quasiconformal mappings we will use later. For the general theory of quasiconformal mappings in the plane, the reader is referred to [Ahl06], [LV73], [AIM09], [Hub06] and

Theorem 1.3 (see e.g. [Leh87, Theorem III-1.2]). The complex dilatations µ and ν are equiv- alent if and only if fµ|H−≡fν|H−. By the above theorem, the universal Teichm¨uller space T can be understood as the set of the normalized conformal mappings fµ|H−which can be extended quasiconformally to the upper half-plane H+. Recall that for a M¨obius transformation f we have Sf◦g = Sg. Therefore, it is natural to consider the mapping T ∋[f] 7→Sfµ|H−∈Q, (1.4) between T and Q, where Q is the space of fu

Theorem 1.4. The mapping (1.4) is a homeomorphism of the universal Teichm¨uller space T onto its image in Q. The mapping (1.4) is called the Bers embedding of Teichm¨uller space. We denote the image of T under (1.4) by T1. It is known that T1 is a bounded, connected and open subset of Q ([Ahl63]). From the viewpoint of the theory of univalent functions, T1 is characterized as follows. Let A be the family of functions f holomorphic in D with f(0) = 0 and f′(0) = 1 and S be the subfamily of A whos

Theorem 2.1 (Gronwall’s area theorem). For a g ∈Σ, we have m(C −g(D∗)) = π

Theorem 2.2 ([Bie16]). If f ∈S, then |a2| ≤2. Equality holds if and only if f(z) is a rotation of the Koebe function defined by K(z) := z (1 −z)2 = 1 4 1 + z 1 −z 2 −1 ! = z + ∞ X n=2

Theorem 2.3 (The Koebe 1/4-theorem). If f ∈S, then f(D) contains the disk centered at the origin with radius 1/4. Since the class S is closed with respect to the Koebe transform fK(z) := f( z+ζ 1+¯ζz) −f(ζ) (1 −|ζ|2)f′(ζ) = z + 1 2(1 −|ζ|2)f′′(ζ) f′(ζ) −¯ζ  z2 + · · · , (2.2) applying Theorem 2.2 to (2.2) we have the inequality (1 −|z|2)f′′(z)

Theorem 2.4. If f ∈S, then

Theorem 2.10. S(k), S∗(k) and Σ(k) are compact families. K¨uhnau gave a fundamental contribution to the coefficient problem with the variational method.

Theorem 2.11 ([K¨uh69]). Let f(z) = z+P∞ n=2 anzn ∈S(k) and g(ζ) = ζ+P∞ n=0 bnz−n ∈Σ(k). Then the followings hold; |b0| ≤2k, |b1| ≤k and |a3 −a2 2| ≤k, in particular |a2| ≤2k.

Theorem 2.12. Let F : (Q(E)) →C be bounded. Then we have ||F||k ≤k||F||1. Some applications of the theorem are demonstrated in [Kru05b, Chapter 3.4]. One of them is the distortion theorem for the class S(k) (see also [Gut73, Corollary 7]); 1 −|z| 1 + |z| k ≤

Theorem 2.13 ([SS76]). For all f ∈S∗(k), we have the sharp estimate |a2| ≤2 −4 arccos k π 2 . For the sharp function, see [SS76, Eq. (4.2)] Since the class S∗(k) is closed with respect to the Koebe transform (2.2), we have the funda- mental estimate for S∗(k)

Theorem 2.14 ([Kru88, Kru95]). For a function f(z) = z + a2z2 + · · · ∈S(k), we have the sharp estimate |an| ≤ 2k n −1 (2.3) for k ≤1/(n2 + 1). The extremal function of the estimate (2.3) is given by f2(z) := z (1 −kz)2 (k ∈[0, 1)), fn(z) := (f2(zn−1))1/(n−1) = z + 2k n −1zn + · · ·

Theorem 2.15 ([AW62]). Let f be a non-constant meromorphic function defined on D and k ∈[0, 1) be a constant. If f satisfies ||Sf|| ≤2k, then f can be extended to a quasiconformal mapping F to bC. In this case the dilatation µF is given by µF (z) :=    −1 2(|z|2 −1)2SF 1 ¯z  1 ¯z4 , |z| > 1 0,

Theorem 2.16 ([Ahl74]). Let f ∈A. If there exists a k ∈[0, 1) such that for a constant c ∈C the inequality c|z|2 + (1 −|z|2)f′′(z) f′(z) ≤k (2.4) holds for all z ∈D, then f ∈S(k) . The case when c = 0 is due to Becker [Bec72]. Remark that the condition |c| ≤k which was stated in the original form is embedded in the inequality (2.4) (see [Hot10]). It is known that many univalence criteria are refined to quasiconformal extension criteria. For instance, Fait, Krzy˙z and Zygmunt proved the following

Theorem 2.17 ([FKZ76]). Every strongly starlike functions of order α has a sin(πα/2)-quasiconformal extension to C. This is generalized to strongly spiral-like functions [Sug12]. Some more results are obtained in [Bro84, Hot09] with explicit quasiconformal extensions which correspond to each subclass of S. In particular, in [Hot09] the research relies on the (classical) Loewner theory, which will be mentioned in the next section. Sugawa approached this problem by means of the holomorphic motions

Theorem 2.18 ([Sug99]). Let k ∈[0, 1) be a constant. For a given f ∈A, let p denote one of the quantities zf′(z)/f(z), 1 + zf′′(z)/f′(z) and f′(z). If

Lemma 3.1. For each fixed z ∈D, a Loewner chain ft satisfies |ft(z) −fs(z)| ≤ 8|z| (1 −|z|)4 |et −es| for all 0 ≤s ≤t < ∞. Hence ft is absolutely continuous on t ∈[0, ∞) for all fixed z ∈D. A necessary and sufficient condition for a Loewner chain is shown by Pommerenke.

Theorem 3.2 ([Pom65, Pom75]). Let 0 < r0 ≤1. Let ft(z) = etz +P∞ n=2 an(t)zn be a function defined on D × [0, ∞). Then ft is a Loewner chain if and only if the following two conditions are satisfied; (i) ft is holomorphic in z ∈Dr0 for each t ∈[0, ∞), absolutely continuous in t ∈[0, ∞) for each z ∈Dr0 and satisfies |ft| ≤K0et (z ∈Dr0, t ∈[0, ∞)) (3.1) for some positive constant K0.

Theorem 3.5. For any f ∈S, there exists a Loewner chain ft such that f0 = f. 3.2. Evolution families. In Loewner theory, a two-parameter family of holomorphic self-maps of the unit disk (ϕs,t), 0 ≤s ≤t < ∞, called an evolution family, plays a key role. To be precise, (ϕs,t) satisfies the followings; 1. ϕs,s(z) = z; 2. ϕs,t(0) = 0 and ϕ′ s,t(0) = es−t; 3. ϕs,t = ϕu,t ◦ϕs,u for all 0 ≤s ≤u ≤t < ∞.

Theorem 3.6. Suppose that a function p(z, t) is holomorphic in z ∈D and measurable in t ∈[0, ∞) satisfying Re p(z, t) > 0 for all z ∈D and t ∈[0, ∞). Then, for each fixed z ∈D and s ∈[0, ∞), the initial value problem dw dt = −wp(w, t) for almost all t ∈[s, ∞) has a unique absolutely continuous solution w(t) with the initial condi- tion w(s) = z. If we write ϕs,t(z) := w(t), then ϕs,t is an evolution family and univalent on D. Further, the function fs(z) defined by fs(z) := lim t→∞etϕs,t(z) (3.4) e

Theorem 3.7 ([Bec72], [Bec80]). Suppose that ft is a Loewner chain for which p(z, t) in (3.2) satisfying the condition p(z, t) ∈U(k) :=  w ∈C :

Lemma 3.8. Let q(z, t) be a Herglotz function. Suppose that q(0, t) be locally integrable in [0, ∞) with R ∞ 0 Re q(0, t)dt = ∞. Then there exists an inverse Loewner chain wt with (3.7). By applying the notion of an inverse Loewner chain, we obtain a generalization of Becker’s result.

Theorem 3.9 ([Bet92]). Let k ∈[0, 1). Let ft be a Loewner chain for which p(z, t) in (3.2) satisfying the condition

Corollary 3.10 ([Bet92]). Let α ∈[0, 1). Suppose that ft is a Loewner chain for which p(z, t) in (3.2) satisfies p(z, t) ∈∆(−α, α) = n z : −απ 2 ≤arg z ≤απ 2 o for all z ∈D and almost all t ∈[0, ∞). Then ft admits a continuous extension to D for each t ≥0 and f0 has a sin απ/2-quasiconformal extension to C.

Corollary 3.10 does not include Theorem 3.7 in view of the dilatation of the extended quasi- conformal map. In fact, the following relation holds; U(k) ⊂∆(−k0, k0) where k0 := 2 π arcsin  2k 1 + k2  ≥k.

Proposition 3.11. For a function f ∈R, if the boundary of f(D) is locally connected, then eiθ 7→f(eiθ) ∈C is one-to-one. Further, we can make use of (3.12) to observe the shape of f(D) for an f ∈R. We assume that the boundary of f(D) is locally connected. Then the half-line γeiθ := {f(eiθ) + teiθ : t ∈[0, ∞)} is well-defined. Since the inclination of γeiθ is exactly θ, we obtain the following property for R;

Proposition 3.12. Let f ∈S. If f(D) contains some sector domain in C, then f does not belong to R. For example, f(z) = ((1 + z)/(1 −z) −1)/2 maps D onto the half-plane. Hence we immediately conclude that f /∈R (of course in this case it is easy to see that f does not satisfy Re f′ > 0 by calculation).

Theorem 4.1. Let (φt)t≥0 be a one-parameter semigroup of holomorphic self-mappings of D. Then for each z ∈D there exists the limit lim t→0+ φt(z) −z t =: G(z) (4.1) such that G ∈Hol(D, C). The convergence in (4.1) is uniform on each compact subset of D. Moreover, the semigroup (φt)t≥0 can be defined as a unique solution of the Cauchy problem dφt(z) dt = G(φt(z)) (t ≥0) with the initial condition φ0(z) = z.

Theorem 4.2 ([BP78]). A holomorphic function G ∈Hol(D, C) is an infinitesimal generator if and only if there exists a τ ∈D and a function p ∈Hol(D, C) with Re p(z) ≥0 for all z ∈D such that G(z) = (τ −z)(1 −¯τz)p(z) (4.2) for all z ∈D. The equation (4.2) is called the Berkson-Porta representation. In fact, the point τ in (4.2) is the Denjoy-Wolffpoint of the one-parameter semigroup generated with G.

Theorem 4.6 ([BCDM12, Proposition 3.7, Corollary 6.3]). Let (ϕs,t) ∈EF. (i) ϕs,t is univalent in D for all 0 ≤s ≤t < ∞. (ii) For each z0 ∈D and s0 ∈[0, ∞), ϕs0,t(z0) is locally absolutely continuous on t ∈[s0, ∞). (iii) For each z0 ∈D and t0 ∈(0, ∞), ϕs,t0(z0) is absolutely continuous on s ∈[0, t0]. Next, we extend the notion of infinitesimal generators to the same structure as evolution families. Definition 4.7 ([BCDM12, Definition 4.1, Definition 4.3]). A Herglotz vector field on the unit disk D is

Theorem 4.8 ([BCDM12, Theorem 5.2, Theorem 6.2]). For any (ϕs,t) ∈EF, there exists an essentially unique G ∈HV such that dϕs,t(z) dt = G(ϕs,t(z), t) (4.3) for all z ∈D, all s ∈[0, ∞) and almost all t ∈[s, ∞). Conversely, for any G ∈HV, a family of unique solutions of (4.3) with the initial condition ϕs,s(z) = z generates an evolution family.

Theorem 4.10 ([BCDM12, Theorem 4.8]). Let G ∈HV. Then there exists an essentially unique measurable function τ : [0, ∞) →D and p ∈HF such that G(z, t) = (τ(t) −z)(1 −τ(t)z)p(z, t) (4.4) for all z ∈D and almost all t ∈[0, ∞). Conversely, for a given measurable function τ : [0, ∞) → D and p ∈HF, the equation (4.4) forms a Herglotz vector field. For convenience, we call the above measurable function τ : [0, ∞) →D the Denjoy-Wolff function and denote by τ ∈DW. A pair (p, τ) of p ∈HV and τ ∈DW is calle

Theorem 4.12 ([CDMG10b, Theorem 1.3]). For any (ft) ∈LC, if we define ϕs,t(z) := (f−1 t ◦fs)(z) (z ∈D, 0 ≤s ≤t < ∞) then (ϕs,t) ∈EF. Conversely, for any (ϕs,t) ∈EF, there exists an (ft) ∈LC such that the following equality holds (ft ◦ϕs,t)(z) = fs(z) (z ∈D, 0 ≤s ≤t < ∞). (4.6) Differentiate both sides of (4.6) with respect to t then f′ t(ϕs,t) · ˙ϕs,t + ˙ft(ϕs,t) = 0 and therefore combining to (4.5) we have the following generalized Loewner-Kufarev PDE ˙ft(z) = (z −τ(t))(1 −τ(t)z)f′ t(z)p(z, t).

Theorem 4.13 ([CDMG10b, Theorem 1.6 and Theorem 1.7]). Let (ϕs,t) ∈EF. Then there exists a unique normalized (ft) ∈LC such that Ω[(ft)] is either C or an Euclidean disk in C whose center is the origin. Furthermore; • The following 4 statements are equivalent; (i) Ω[(ft)] = C; (ii) L[(ϕs,t)] consists of only one function; (iii) β(z) = 0 for all z ∈D, where β(z) := lim t→+∞ |ϕ′ 0,t(z)| 1 −|ϕ0,t(z)|2 ; (iv) there exists at least one point z0 ∈D such that β(z0) = 0. • On the other hand, if Ω[(ft)] ̸

Theorem 4.16. Let k ∈[0, 1) be a constant. Suppose that (ft) is a Loewner chain of radial type for which p ∈HF associated with (ft) by (4.7), satisfies p(z, t) ∈U(k) for all z ∈D and almost all t ≥0 and τ ∈DW is equal to 0. Then the following assertions hold; (i) ft admits a continuous extension to D for each t ≥0; (ii) F defined in (3.6) gives a k-quasiconformal extension of f0 to C; (iii) Ω[(ft)] = C.

Theorem 4.17 ([GH17]). Suppose that a family of holomorphic functions (ft)t≥0 on the right half-plane H is a Loewner chain of chordal type. If there exists a uniform constant k ∈[0, 1) such that pH, a Herglotz function associated with (ft), satisfies pH(ζ, t) ∈U(k) (4.10) for all ζ ∈H and almost all t ≥0, then (i) ft admits a continuous extension to H ∪iR; (ii) ft has a k-quasiconformal extension to C for each t ≥0. In this case the extension F is explicitly given by F(ζ) :=  f0(ζ), ζ ∈H, f−Re ζ