Loewner Theory for Quasiconformal Extensions: Old and New2010 Mathematics Subject Classification: Primary 30C65, 37F30, Secondary 30C70, 30C45, 30D05. The author was supported by JSPS KAKENHI Grant Numbers 17K14205.

Lemma 2.1 (Weyl’s lemma (see e.g., [IT92, p. 84]). Let f be a continuous function on G whose distributional derivative fz is locally integrable on G. If fz ¼ 0 in the sense of distributions on G, then f is holomorphic on G. Let BðGÞ be the open unit ball f 2 L1ðGÞ : kk1 < 1g of L1ðGÞ, where L1ðGÞ is the complex Banach space of all bounded measurable functions on G, and kk1 :¼ ess supz2G jðzÞj for a  2 L1ðGÞ. An element  2 BðGÞ is called the Beltrami coefficient. If f is a k-quasiconformal

Theorem 2.2 (The measurable Riemann mapping theorem). For a given measurable function  2 BðCÞ, there exists a unique solution f of the equation fz ¼ fz ð2:2Þ for which f : C ! C is a quasiconformal mapping fixing the points 0 and 1. The Eq. ð2:2Þ 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 [IT

Theorem 2.3 (see e.g., [Leh87, Theorem III-1.2]). The complex dilatations  and  are equivalent if and only if f jH  f jH. By the above theorem, the universal Teichmu¨ller space T can be understood as the set of the normalized conformal mappings f jH which can be extended quasiconformally to the upper half-plane Hþ. Recall that for a Mo¨bius transformation f we have Sfg ¼ Sg. Therefore, it is natural to consider the mapping T 3 ½ f 7! Sf jH 2 Q; ð2:4Þ between T and Q, where Q is the s

Theorem 2.4. The mapping ð2:4Þ is a homeomorphism of the universal Teichmu¨ller space T onto its image in Q. The mapping ð2:4Þ is called the Bers embedding of Teichmu¨ller space. We denote the image of T under ð2:4Þ by T 1. It is known that T 1 is a bounded, connected and open subset of Q ([Ahl63]). From the viewpoint of the theory of univalent functions, T 1 is characterized as follows. Let A be the family of functions f holomorphic in D with fð0Þ ¼ 0 and f 0ð0Þ ¼ 1 and S be the subfamily of A

Theorem 3.1 (Gronwall’s area theorem). For a g 2 , we have mðC  gðDÞÞ ¼ 1  X 1 n¼1 njbnj2

Theorem 3.2 ([Bie16]). If f 2 S, then ja2j  2. Equality holds if and only if fðzÞ is a rotation of the Koebe function defined by KðzÞ :¼ z ð1  zÞ2 ¼ 1 4 1 þ z 1  z  2  1

Theorem 3.3 (The Koebe 1/4-theorem). If f 2 S, then fðDÞ contains the disk centered at the origin with radius 1/4. Since the class S is closed with respect to the Koebe transform fKðzÞ :¼ fð zþ 1þ  zÞ  fð Þ ð1  j j2Þf 0ð Þ ¼ z þ 1 2 ð1  j j2Þ f 00ð Þ f 0ð Þ     z2 þ    ; ð3:2Þ

Theorem 3.4. If f 2 S, then zf 00ðzÞ f 0ðzÞ  2jzj2 1  jzj2    4jzj 1  jzj2 ; 1  jzj ð1 þ jzjÞ3  j f 0ðzÞj  1 þ jzj ð1  jzjÞ3 ; jzj

Theorem 3.10. SðkÞ; SðkÞ and ðkÞ are compact families. Ku¨hnau gave a fundamental contribution to the coefficient problem with the variational method.

Theorem 3.11 ([Ku¨h69]). Let fðzÞ ¼ z þ P1 n¼2 anzn 2 SðkÞ and gð Þ ¼ þ P1 n¼0 bnzn 2 ðkÞ. Then the followings hold; jb0j  2k, jb1j  k and ja3  a2 2j  k, in particular ja2j  2k. We note that in the case when k ¼ 1 we obtain estimates for the classes S and . As more general approach to this problem, the distortion theorem for bounded functional was studied. We basically follow the description of the survey paper by Krushkal [Kru05b, Chapter 3]. The reader is also referred to [KK83]. Let

Theorem 3.12. Let F : QðEÞ ! C be bounded. Then we have kFkk  kkFk1. Some applications of the theorem are demonstrated in [Kru05b, Chapter 3.4]. One of them is the distortion theorem for the class SðkÞ (see also [Gut73, Corollary 7]); 1  jzj 1 þ jzj  k  zf 0ðzÞ fðzÞ    1 þ jzj 1  jzj 

Theorem 3.13 ([SS76]). For all f 2 SðkÞ, we have the sharp estimate ja2j  2  4 arccos k

Theorem 3.14 ([Kru88, Kru95]). For a function fðzÞ ¼ z þ a2z2 þ    2 SðkÞ, we have the sharp estimate janj  2k n  1 ð3:3Þ for k  1=ðn2 þ 1Þ. The extremal function of the estimate ð3:3Þ is given by f2ðzÞ :¼ z ð1  kzÞ2 ðk 2 ½0; 1ÞÞ; fnðzÞ :¼ ð f2ðzn1ÞÞ1=ðn1Þ ¼ z þ 2k n  1 zn þ   

Theorem 3.15 ([AW62]). Let f be a non-constant meromorphic function defined on D and k 2 ½0; 1Þ be a constant. If f satisfies kSf k  2k, then f can be extended to a quasiconformal mapping F to bC. In this case the dilatation F is given by FðzÞ :¼ 1 2ðjzj2  1Þ2SF 1 z   1 z4 ; jzj > 1 0; jzj < 1. 8

Theorem 3.16 ([Ahl74]). Let f 2 A. If there exists a k 2 ½0; 1Þ such that for a constant c 2 C the inequality cjzj2 þ ð1  jzj2Þ f 00ðzÞ f 0ðzÞ    k ð3:4Þ holds for all z 2 D, then f 2 SðkÞ. The case when c ¼ 0 is due to Becker [Bec72]. Remark that the condition jcj  k which was stated in the original form is embedded in the inequality ð3:4Þ (see [Hot10]). It is known that many univalence criteria are refined to quasiconformal extension criteria. For instance, Fait, Krzyz˙ and Zygmunt p

Theorem 3.17 ([FKZ76]). Every strongly starlike functions of order  has a sinð =2Þ-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 3.18 ([Sug99]). Let k 2 ½0; 1Þ be a constant. For a given f 2 A, let p denote one of the quantities zf 0ðzÞ=fðzÞ; 1 þ zf 00ðzÞ=f 0ðzÞ and f 0ðzÞ. If 1  pðzÞ 1 þ pðzÞ    k for all z 2 D, then f 2 SðkÞ. We note that in most of the sufficient conditions of quasiconformal extensions including the above theorems the case k ¼ 1 reflects univalence criteria. 4. Classical Loewner Theory The idea of the parametric representation method of conformal maps was introduced by Lo¨wner [Lo¨w23],

Lemma 4.1. For each fixed z 2 D, a Loewner chain ft satisfies j ftðzÞ  fsðzÞj  8jzj ð1  jzjÞ4 jet  esj for all 0  s  t < 1. Hence ft is absolutely continuous on t 2 ½0; 1Þ for all fixed z 2 D. A necessary and sufficient condition for a Loewner chain is shown by Pommerenke.

Theorem 4.2 ([Pom65, Pom75]). Let 0 < r0  1. Let ftðzÞ ¼ etz þ P1 n¼2 anðtÞzn be a function defined on D  ½0; 1Þ. Then ft is a Loewner chain if and only if the following two conditions are satisfied; (i) ft is holomorphic in z 2 Dr0 for each t 2 ½0; 1Þ, absolutely continuous in t 2 ½0; 1Þ for each z 2 Dr0 and satisfies j ftj  K0et ðz 2 Dr0; t 2 ½0; 1ÞÞ

Theorem 4.5. For any f 2 S, there exists a Loewner chain ft such that f0 ¼ f. 4.2 Evolution families In Loewner theory, a two-parameter family of holomorphic self-maps of the unit disk (’s;t), 0  s  t < 1, called an evolution family, plays a key role. To be precise, (’s;t) satisfies the followings; 1. ’s;sðzÞ ¼ z; 2. ’s;tð0Þ ¼ 0 and ’0 s;tð0Þ ¼ est; 3. ’s;t ¼ ’u;t  ’s;u for all 0  s  u  t < 1. We note that ’s;t is not assumed to be univalent on D. By means of the same idea as Lemma 4.1, we

Theorem 4.6. Suppose that a function pðz; tÞ is holomorphic in z 2 D and measurable in t 2 ½0; 1Þ satisfying Re pðz; tÞ > 0 for all z 2 D and t 2 ½0; 1Þ. Then, for each fixed z 2 D and s 2 ½0; 1Þ, the initial value problem dw dt ¼ wpðw; tÞ for almost all t 2 ½s; 1Þ has a unique absolutely continuous solution wðtÞ with the initial condition wðsÞ ¼ z. If we write ’s;tðzÞ :¼ wðtÞ, then ’s;t is an evolution family and univalent on D. Further, the function fsðzÞ defined by fsðzÞ :¼ lim t!1 et’s;tðzÞ ð

Theorem 4.7 ([Bec72], [Bec80]). Suppose that ft is a Loewner chain for which pðz; tÞ in ð4:2Þ satisfying the condition pðz; tÞ 2 UðkÞ :¼ w 2 C : 1  w 1 þ w    k   ð4:5Þ i.e., pðz; tÞ lies in the closed hyperbolic disk UðkÞ in the right half-plane centered at 1 with radius arctanh k, for all z 2 D and almost all t  0. Then ft admits a continuous extension to D for each t  0 and the map F defined by FðreiÞ ¼ f0ðreiÞ;

Lemma 4.8. Let qðz; tÞ be a Herglotz function. Suppose that qð0; tÞ be locally integrable in ½0; 1Þ with R 1 0 Re qð0; tÞdt ¼ 1. Then there exists an inverse Loewner chain wt with ð4:7Þ. By applying the notion of an inverse Loewner chain, we obtain a generalization of Becker’s result.

Theorem 4.9 ([Bet92]). Let k 2 ½0; 1Þ. Let ft be a Loewner chain for which pðz; tÞ in ð4:2Þ satisfying the condition 10 HOTTA

Corollary 4.10 ([Bet92]). Let  2 ½0; 1Þ. Suppose that ft is a Loewner chain for which pðz; tÞ in ð4:2Þ satisfies pðz; tÞ 2 ð; Þ ¼ z :   2  arg z   2   for all z 2 D and almost all t 2 ½0; 1Þ. Then ft admits a continuous extension to D for each t  0 and f0 has a sin  =2-quasiconformal extension to C.

Corollary 4.10 does not include Theorem 4.7 in view of the dilatation of the extended quasiconformal map. In fact, the following relation holds; UðkÞ  ðk0; k0Þ where k0 :¼ 2

Proposition 4.11. For a function f 2 R, if the boundary of fðDÞ is locally connected, then ei 7! fðeiÞ 2 C is one- to-one. Further, we can make use of ð4:12Þ to observe the shape of fðDÞ for an f 2 R. We assume that the boundary of fðDÞ is locally connected. Then the half-line ei :¼ ffðeiÞ þ tei : t 2 ½0; 1Þg is well-defined. Since the inclination of ei is 12 HOTTA

Proposition 4.12. Let f 2 S. If fðDÞ contains some sector domain in C, then f does not belong to R. For example, fðzÞ ¼ ðð1 þ zÞ=ð1  zÞ  1Þ=2 maps D onto the half-plane. Hence we immediately conclude that f =2 R (of course in this case it is easy to see that f does not satisfy Re f 0 > 0 by calculation). IV Bazilevicˇ functions For real constants  > 0 and  2 R, set ¼  þ i. In 1955, Bazilevicˇ [Baz55] showed that the function defined by fðzÞ ¼ ð þ iÞ Z z 0 hðuÞgðuÞui1du  1=ðþiÞ whe

Theorem 4.7 with the chain ftðzÞ :¼ fðetzÞ þ 1 1 þ c ðet  etÞzf 0ðetzÞ; for then 1  pðz; tÞ 1 þ pðz; tÞ ¼ zf 0 t ðzÞ  _f tðzÞ zf 0 t ðzÞ þ _f tðzÞ ¼ c 1 e2t þ 1  1 e2t 

Theorem 5.1. Let ðtÞt0 be a one-parameter semigroup of holomorphic self-mappings of D. Then for each z 2 D there exists the limit lim t!0þ tðzÞ  z t ¼: GðzÞ ð5:1Þ such that G 2 HolðD; CÞ. The convergence in ð5:1Þ is uniform on each compact subset of D. Moreover, the semigroup ðtÞt0 can be defined as a unique solution of the Cauchy problem dtðzÞ dt ¼ GðtðzÞÞ ðt  0Þ

Theorem 5.2 ([BP78]). A holomorphic function G 2 HolðD; CÞ is an infinitesimal generator if and only if there exists a  2 D and a function p 2 HolðD; CÞ with Re pðzÞ  0 for all z 2 D such that GðzÞ ¼ ð  zÞð1  zÞpðzÞ ð5:2Þ for all z 2 D. The Eq. ð5:2Þ is called the Berkson–Porta representation. In fact, the point  in ð5:2Þ is the Denjoy–Wolffpoint of the one-parameter semigroup generated with G. 5.2 Generalized evolution families in the unit disk We have discussed in Sect. 3.1 that a Loewne

Theorem 5.6 ([BCDM12, Proposition 3.7, Corollary 6.3]). Let ð’s;tÞ 2 EF. (i) ’s;t is univalent in D for all 0  s  t < 1. (ii) For each z0 2 D and s0 2 ½0; 1Þ, ’s0;tðz0Þ is locally absolutely continuous on t 2 ½s0; 1Þ. (iii) For each z0 2 D and t0 2 ð0; 1Þ, ’s;t0ðz0Þ is absolutely continuous on s 2 ½0; t0 . Next, we extend the notion of infinitesimal generators to the same structure as evolution families. Definition 5.7 ([BCDM12, Definition 4.1, Definition 4.3]). A Herglotz vector field on the unit

Theorem 5.8 ([BCDM12, Theorem 5.2, Theorem 6.2]). For any ð’s;tÞ 2 EF, there exists an essentially unique G 2 HV such that d’s;tðzÞ dt ¼ Gð’s;tðzÞ; tÞ ð5:3Þ for all z 2 D, all s 2 ½0; 1Þ and almost all t 2 ½s; 1Þ. Conversely, for any G 2 HV, a family of unique solutions of ð5:3Þ with the initial condition ’s;sðzÞ ¼ z generates an evolution family. Here, essentially unique means that if Gðz; tÞ is another Herglotz vector field which satisfies ð5:3Þ, then Gð; tÞ ¼ Gð; tÞ for almost every t  0.

Theorem 5.10 ([BCDM12, Theorem 4.8]). Let G 2 HV. Then there exists an essentially unique measurable function  : ½0; 1Þ ! D and p 2 HF such that Gðz; tÞ ¼ ððtÞ  zÞð1  ðtÞzÞpðz; tÞ ð5:4Þ for all z 2 D and almost all t 2 ½0; 1Þ. Conversely, for a given measurable function  : ½0; 1Þ ! D and p 2 HF, the Eq. ð5:4Þ forms a Herglotz vector field. For convenience, we call the above measurable function  : ½0; 1Þ ! D the Denjoy–Wolfffunction and denote by  2 DW. A pair ðp; Þ of p 2 HV and  2 DW is

Theorem 5.12 ([CDMG10b, Theorem 1.3]). For any ð ftÞ 2 LC, if we define ’s;tðzÞ :¼ ð f 1 t  fsÞðzÞ ðz 2 D; 0  s  t < 1Þ then ð’s;tÞ 2 EF. Conversely, for any ð’s;tÞ 2 EF, there exists an ð ftÞ 2 LC such that the following equality holds ð ft  ’s;tÞðzÞ ¼ fsðzÞ ðz 2 D; 0  s  t < 1Þ: ð5:6Þ Differentiate both sides of ð5:6Þ with respect to t then f 0 t ð’s;tÞ  _’s;t þ _f tð’s;tÞ ¼ 0 and therefore combining to ð5:5Þ we have the following generalized Loewner–Kufarev PDE _f tðzÞ ¼ ðz  ðtÞÞð1 

Theorem 5.13 ([CDMG10b, Theorem 1.6 and Theorem 1.7]). Let ð’s;tÞ 2 EF. Then there exists a unique normalized ð ftÞ 2 LC such that ½ð ftÞ is either C or an Euclidean disk in C whose center is the origin. Furthermore; . The following 4 statements are equivalent; (i) ½ð ftÞ ¼ C; (ii) L½ð’s;tÞ consists of only one function; (iii) ðzÞ ¼ 0 for all z 2 D, where ðzÞ :¼ lim t!þ1 j’0 0;tðzÞj 1  j’0;tðzÞj2 ;

Theorem 5.16. Let k 2 ½0; 1Þ be a constant. Suppose that ( ft) is a Loewner chain of radial type for which p 2 HF associated with ( ft) by ð5:7Þ, satisfies pðz; tÞ 2 UðkÞ for all z 2 D and almost all t  0 and  2 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 ð4:6Þ gives a k-quasiconformal extension of f0 to C; (iii) ½ð ftÞ ¼ C.

Theorem 5.16, g0, and hence f0, has a k-quasiconformal extension to C. 5.5 Quasiconformal extensions for Loewner chains of chordal type A Loewner chain of chordal type (see Definition 5.15) with a quasiconformal extension is discussed by Gumenyuk and the author [GH17]. In the chordal case,  2 DW is a boundary fixed point of D. By some rotation we may assume that  ¼ 1. It is sometimes convenient to discuss the chordal case on the not D but rather the half-plane. In fact, by means of the conjugati

Theorem 5.17 ([GH17]). Suppose that a family of holomorphic functions ð ftÞt0 on the right half-plane H is a Loewner chain of chordal type. If there exists a uniform constant k 2 ½0; 1Þ such that pH, a Herglotz function associated with ( ft), satisfies pHð ; tÞ 2 UðkÞ ð5:10Þ for all 2 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 Fð Þ :¼ f0ð Þ; 2 H, f

Theorem 5.17. In fact, by setting gtðzÞ :¼ ftðzÞ we have gtðzÞ ¼ ðz  Þð1  zÞg0ðzÞpðz; tÞ. After transferring gt to the right half-plane, Theorem 5.17 with the same k as ft is applied. Acknowledgments The author is deeply grateful to Mr. Kazuhiro Morita for his genuine support and continuous encouragement throughout the research work. He would like to thank the anonymous referees for thorough reading of the manuscript and helpful comments. This article is included in the proceedings of t