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Loewner chains and nonlinear resolvents of the Carathéodory family on the unit ball in C n Ian Graham a , Hidetaka Hamada b,∗ , Gabriela Kohr c a

Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4, Canada Faculty of Science and Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku, Fukuoka 813-8503, Japan c Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084 Cluj-Napoca, Romania b

a r t i c l e

i n f o

Article history: Received 26 December 2019 Available online 3 June 2020 Submitted by Z. Cuckovic Keywords: Carathéodory family Herglotz vector ﬁeld Inverse Loewner chain Loewner PDE Nonlinear resolvent family Starlike map

a b s t r a c t In this paper we study various properties of nonlinear resolvents of holomorphic mappings in the Carathéodory family M(Bn ), where Bn is the Euclidean unit ball n in C . First, we prove certain characterizations of inverse Loewner chains f (z, t) = t e− 0 a(τ )dτ z + · · · on Bn × [0, ∞), where a : [0, ∞) → C is a locally Lebesgue integrable function such that a(t) > 0 for a.e. t ≥ 0, and which satisﬁes a natural assumption. Next, we prove that if f ∈ M(Bn ), then the nonlinear resolvent family {Jr }r≥0 is an inverse Loewner chain on Bn and the associated Herglotz vector ﬁeld is of divergence type, where Jr = Jr [f ] = (In + rf )−1 , r ≥ 0. This result is a generalization to higher dimensions of a recent result due to Elin, Shoikhet and Sugawa. We prove that (1 + r)Jr can be embedded as the ﬁrst element of a normal Loewner chain and the family {(1 + r)Jr [f ] : 0 ≤ r < ∞, f ∈ M(Bn )} is a compact subset of S 0 (Bn ). Also, we deduce that the shearing of (1 + r)Jr (r ≥ 0) associated with the family M(B2 ) is quasi-convex of type A and also starlike of order 4/5. We give a suﬃcient condition for (1 + r)Jr to be quasiconformal on Bn and to be extended to a quasiconformal homeomorphism of C n onto itself. Finally, sharp coeﬃcient bounds for the nonlinear resolvent families of certain subsets of M(Bn ), and also examples of support points of the compact families generated by nonlinear resolvent mappings on B2 , will be obtained. © 2020 Elsevier Inc. All rights reserved.

1. Introduction Let U be the unit disc in the complex plane C and let H(U ) be the family of holomorphic functions on U . Also, let * Corresponding author. E-mail addresses: [email protected] (I. Graham), [email protected] (H. Hamada), [email protected] (G. Kohr). https://doi.org/10.1016/j.jmaa.2020.124289 0022-247X/© 2020 Elsevier Inc. All rights reserved.

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I. Graham et al. / J. Math. Anal. Appl. 491 (2020) 124289

f (ζ) N (U ) = f ∈ H(U ) : f (0) = 0, > 0, ζ ∈ U . ζ Here it is understood that f (ζ) = f (0). ζ ζ=0 The above family plays a key role in geometric function theory on U , especially in the theory of univalent functions (see e.g. [25] and [35], and the references therein). On the other hand, the family N (U ) is also important in the study of semigroups and semi-complete vector ﬁelds of holomorphic functions on the unit disc U . It is well known that every function f ∈ N (U ) is the inﬁnitesimal generator of a one parameter continuous semigroup on U (see e.g. [1], [12], [44]; see [38], and the references therein). Moreover, if f : U → C is a holomorphic function such that f (0) = 0 and f (0) > 0, then f is a holomorphic generator of a continuous semigroup on U ﬁxing the origin if and only if f ∈ N (U ) (see e.g. [1], [12] and [37]; see also [38]). The following well known characterization of holomorphic generators was proved in [37] in a more general context of convex domains in complex Banach spaces (see also [12, Proposition 3.1.3] and [38]): Proposition 1.1. If f : U → C is a holomorphic function such that f (0) = 0, then f is a holomorphic generator of a continuous semigroup on U ﬁxing the origin if and only if f satisﬁes the range condition: for every r ≥ 0 and w ∈ U , the equation z + rf (z) = w has a unique solution z := Jr (w) on U . This solution (called the nonlinear resolvent of f ) is a holomorphic self mapping of U , for all r ≥ 0. Generalizations of the above results and various applications in higher dimensions and in the context of complex Banach spaces may be found in [1], [37], [38] (see also [12] and [13] and the references therein). Very recently, Elin, Shoikhet and Sugawa [14] have investigated geometric properties of nonlinear resolvents Jr (r ≥ 0) of the family N (U ). They proved the following interesting and unexpected properties: Proposition 1.2. Let f ∈ N (U ) and let {Jr }r≥0 be the nonlinear resolvent family of f . Then the following statements hold: (i) {Jr }r≥0 is an inverse Loewner chain on U with the associated Herglotz vector ﬁeld of divergence type. (ii) Jr (U ) is a hyperbolically convex domain and the nonlinear resolvent mapping Jr is starlike of order 12 on U , for all r ≥ 0. (iii) If, in addition, there is some α ∈ (0, 1) such that f satisﬁes the condition f (ζ) απ , arg < ζ 2

ζ ∈ U,

then the mapping Jr extends to a k-quasiconformal homeomorphism of C onto itself, for all r ≥ 0, where k = sin( απ 2 ). In this paper, we obtain generalizations of the above statements in Proposition 1.2 to the case of the Euclidean unit ball Bn in C n . We prove that every nonlinear resolvent family {Jr }r≥0 of the family M(Bn ) has the property that {Jr }r≥0 is an inverse Loewner chain on Bn and the associated Herglotz vector ﬁeld is of divergence type, and also (1 + r)Jr may be embedded as the ﬁrst element of a normal Loewner chain on Bn , for all r ≥ 0. Therefore, (1 + r)Jr ∈ S 0 (Bn ), where S 0 (Bn ) is the family of normalized univalent mappings with parametric representation on Bn , but (1 +r)Jr cannot be a support/extreme point of S 0 (Bn ), for all n ∈ N. We prove that the family {(1 + r)Jr [f ] : 0 ≤ r < ∞, f ∈ M(Bn )} is a compact subset of [c] S 0 (Bn ). We also prove that the shearing (1 + r)Jr of (1 + r)Jr is a quasi-convex mapping of type A on B2 and also is a starlike mapping of order 4/5 on B2 . On the other hand, under certain additional assumptions on f ∈ M(Bn ), we deduce that (1 + r)Jr is quasiconformal on Bn and extends quasiconformally to the whole space C n . Finally, we obtain sharp coeﬃcient bounds for the nonlinear resolvent mappings of the

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family M(Bn ) and also examples of support points of the compact families generated by nonlinear resolvent mappings on B2 . We also deduce certain consequences and particular cases of the above results. The main results in this paper can be summarized as follows. The notations will be explained in the next sections. Theorem 1.3. Let f ∈ M(Bn ) and let {Jr }r≥0 be the nonlinear resolvent family of f . Then the following statements hold: (i) {Jr }r≥0 is an inverse Loewner chain on Bn with the associated Herglotz vector ﬁeld h : Bn × [0, ∞) → C n of divergence type given by

h(w, r) =

w−Jr (w) , r

f (w),

r>0 r = 0.

log(1+T ) idBn , M(Bn ) , and thus (1 + T )JT ∈ S 0 (Bn ), for all T ≥ 0. Also, (1 + T )JT ∈ R (ii) The family {(1 + r)Jr [f ] : 0 ≤ r < ∞, f ∈ M(Bn )} is a compact subset of S 0 (Bn ). (iii) If n = 2, then the shearing of (1 + r)Jr is quasi-convex of type A and also starlike of order 4/5 on B2 , for all r ≥ 0. (iv) Let r > 0. If Df (z) − In ≤

1+r c, r

z ∈ Bn ,

for some c ∈ (0, 12 ), then the nonlinear resolvent mapping Jr is quasiconformal on Bn and extends to a quasiconformal homeomorphism of C n onto itself. We also obtain the following sharp coeﬃcient bounds for nonlinear resolvent mappings of the family M(B2 ): Theorem 1.4. Let f ∈ M(B2 ) and let Jr = (Jr1 , Jr2 ) be the nonlinear resolvent mapping of f , for all r ≥ 0. Then the following sharp estimates hold: √ 1 ∂2J 1 3 3r r (0) ≤ , 2 ∂w22 2(1 + r)3

r ≥ 0,

and 1

2r , D2 Jr (0)(u, u), u ≤ 2 (1 + r)3

u ∈ C 2 , u = 1, r ≥ 0.

In addition, if F : B2 → C 2 is given by F (z) =

z1 −

√ 3 3 2 z2 , z2 , 8

z = (z1 , z2 ) ∈ B2 ,

then F is a support point of the compact family F(B2 ), where F(B2 ) = (1 + r)Jr [f ] : 0 ≤ r < +∞, f ∈ M(B2 ) .

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2. Preliminaries Let C n be the space of n-complex variables z = (z1 , . . . , zn ) with the Euclidean inner product z, w = n

z, z be the Euclidean norm in C n and let Bn be the Euclidean unit ball in C n . k=1 zk w k . Let z = n Let L(C ) be the space of linear operators from C n into C n with the standard operator norm. Also, let H(Bn , Ω) be the family of holomorphic mappings from Bn into a domain Ω ⊆ C n . The family H(Bn , Bn ) will be denoted by Hol(Bn ), while the family H(Bn , C n ) will be denoted by H(Bn ). A mapping f ∈ H(Bn ) will be said to be normalized if f (0) = 0 and Df (0) = In , where Df (z) is the Fréchet derivative of f at z ∈ Bn , and In is the identity operator on the space C n . If f : Bn → C n is a holomorphic mapping, we denote by Dk f (z) the k-th Fréchet derivative of f at z ∈ B, for k ∈ N. It is understood that Dk f (0)(wk ) = Dk f (0)(w, . . . , w),

w ∈ Cn.

k−times

Also, let S(Bn ) be the family of normalized univalent mappings on Bn , and let S ∗ (Bn ) (respectively, K(Bn )) be the subset of S(Bn ) consisting of starlike (respectively, convex) mappings on Bn . Note that a mapping f ∈ S(Bn ) is said to be starlike (respectively, convex) if f (Bn ) is a starlike domain in C n with respect to the origin (respectively, a convex domain in C n ). The family S(U ) is denoted by S, where U is the unit disc in C. The family S ∗ (U ) (respectively, K(U )) is denoted by S ∗ (respectively, K). We also consider the following subsets of S ∗ (Bn ). Deﬁnition 2.1. (See e.g. [25].) Let α ∈ [0, 1) and let f : Bn → C n be a normalized locally biholomorphic mapping. We say that f is starlike of order α if

z2 [Df (z)]−1 f (z), z

> α,

z ∈ Bn \ {0}.

Let Sα∗ (Bn ) be the family of starlike mappings of order α on Bn . Then it is clear that Sα∗ (Bn ) ⊆ S ∗ (Bn ), for all α ∈ [0, 1). A special interest in this work is played by the family of quasi-convex mappings of type A, due to Roper and Suﬀridge (see [39, Deﬁnition 3]). Applications of this family in the theory of univalent mappings on Bn may be found in [16], [33], and [39] (see also [25] and the references therein). Deﬁnition 2.2. (See [39, Deﬁnition 3].) A mapping f ∈ S(Bn ) is said to be quasi-convex of type A if the following relation holds:

2α α+β − −1 α−β [Df (αu)] (f (αu) − f (βu)), u

> 0,

(2.1)

for all α, β ∈ U and u ∈ C n , u = 1. Remark 2.3. (i) Let f ∈ S(Bn ) be a quasi-convex mapping of type A. Taking β = 0 in (2.1), we deduce that ∗ f ∈ S1/2 (Bn ) (cf. [39, Theorem 3.2]) (see also, [16]). (ii) Every mapping f ∈ K(Bn ) is also quasi-convex of type A, by [39, Theorem 3.1]. Also, if n = 1, then f ∈ K(U ) iﬀ f is quasi-convex of type A (see e.g. [39]). (iii) If f : Bn → C n is a quasi-convex mapping of type A, then f (z), z >

1 z2 , 2

z ∈ Bn \ {0},

(2.2)

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by [32, Theorem 3.1] (see also [33]; cf. [39]). Deﬁnition 2.4. (See e.g. [2], [37].) A mapping f ∈ H(Bn ) is said to satisfy the range condition (RC) if for each r ≥ 0, (In + rf )(Bn ) ⊇ Bn ,

(2.3)

and the mapping (In + rf )−1 is a well deﬁned holomorphic self-mapping of Bn . In this case, the mapping Jr (= (In + rf )−1 ), r ≥ 0, is called the nonlinear resolvent of f . Remark 2.5. (See e.g. [2], [14], [37].) Note that a mapping f ∈ H(Bn ) satisﬁes the (RC)-condition if for each w ∈ Bn and r ≥ 0, the equation z + rf (z) = w

(2.4)

has a unique solution z = Jr (w)(= (In + rf )−1 (w)) ∈ Bn , and the mapping Jr ∈ Hol(Bn ). In other words, a mapping f ∈ H(Bn ) satisﬁes the (RC)-condition if for each r ≥ 0, there exists a unique mapping Jr ∈ Hol(Bn ) such that the following equality Jr (w) + rf (Jr (w)) = w holds, for all w ∈ Bn . The following known characterization for f ∈ H(Bn ) to satisfy the range condition (RC) is very useful (see [38, Theorems 6.16 and 7.5] and the references therein). Proposition 2.6. Let f ∈ H(Bn ). Then the mapping f satisﬁes the range condition (RC) if and only if f (z), z ≥ (1 − z2 ) f (0), z,

z ∈ Bn .

A particular interest in our discussion consists of the Carathéodory family N (Bn ) given by (see e.g. [25], [36] and [46]): N (Bn ) = h ∈ H(Bn ) : h(0) = 0, h(z), z > 0, z ∈ Bn \ {0} . Also, let M(Bn ) = h ∈ N (Bn ) : Dh(0) = In . Remark 2.7. In view of Proposition 2.6, every mapping f ∈ N (Bn ) satisﬁes the range condition (RC). Moreover, in view of Proposition 2.6 (see also [12, Proposition 3.5.2] and [34, Lemma 2.1]), we deduce that if f ∈ H(Bn ) is such that f (0) = 0 and Df (0) = In , then f satisﬁes the range condition (RC) if and only if f ∈ M(Bn ). We shall also consider the following subset of H(Bn ): R(Bn ) = f ∈ H(Bn ) : f (0) = 0, Df (0) = In , Df (z)(z), z > 0, z ∈ Bn \ {0} . Remark 2.8. In the case of one complex variable, R(U ) S, where S = S(U ) is the usual family of normalized univalent functions on U (see e.g. [35, Theorem 2.11] and [25, Lemma 2.4.1]). In higher dimensions, the family R(Bn ) was recently studied in [15]. The authors in [15] proved that R(Bn ) ⊆ M(Bn ). However, it is not known whether R(Bn ) ⊆ S(Bn ) for n ≥ 2.

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Throughout this paper, if f is a mapping which depends holomorphically on z ∈ Bn , and also depends on other real variables, we denote by Df (z, ·) the Fréchet derivative of f with respect to z ∈ Bn . Deﬁnition 2.9. ([20]; cf. [34]) (i) Let f, g ∈ H(Bn ). We say that f is subordinate to g (f ≺ g) if there exists a Schwarz mapping v (i.e. v ∈ H(Bn ) and v(z) ≤ z, z ∈ Bn ) such that f = g ◦ v. (ii) A mapping f : Bn × [0, ∞) → C n is called a subordination chain if f (0, t) = 0 for t ≥ 0, and f (·, s) ≺ f (·, t), 0 ≤ s ≤ t < ∞. If, in addition, f (·, t) is univalent on Bn for t ≥ 0, then we say that f (z, t) is a univalent subordination chain. A univalent subordination chain f (z, t) is said to be normalized (or a Loewner chain) if Df (0, t) = et In for t ≥ 0. (iii) A Loewner chain f = f (z, t) : Bn × [0, ∞) → C n is said to be normal if {e−t f (·, t)}t≥0 is a normal family on Bn . Remark 2.10. Let f (z, t) be a Loewner chain. Then there exists a unique Schwarz mapping v = v(·, s, t), called the transition mapping associated with f (z, t), such that Dv(0, s, t) = es−t In and f (·, s) = f (v(·, s, t), t), for t ≥ s ≥ 0. Deﬁnition 2.11. ([4], [8], [10]) Let h : Bn × [0, ∞) → C n . We say that h is a Herglotz vector ﬁeld (or a generating vector ﬁeld) if the following conditions hold: (i) h(·, t) ∈ N (Bn ), for a.e. t ≥ 0; (ii) h(z, ·) is measurable on [0, ∞), for all z ∈ Bn . Deﬁnition 2.12. Let h : Bn × [0, ∞) → C n be a Herglotz vector ﬁeld. If the mapping Dh(0, ·) : [0, ∞) → ∞ L(C n ) is locally integrable on [0, ∞) and 0 m(Dh(0, t))dt = +∞, where m(Dh(0, t)) = min Dh(0, t)(z), z : z = 1 ,

t ≥ 0,

then we say that h is a Herglotz vector ﬁeld of divergence type (cf. [14, Deﬁnition 4.1] for n = 1; compare [6] for n = 1). Remark 2.13. (i) Let h : Bn × [0, ∞) → C n be a Herglotz vector ﬁeld such that Dh(0, t) = In for t ≥ 0. Then for each z ∈ Bn , the initial value problem ∂v (z, t) = −h(v(z, t), t), ∂t

a.e.

t ∈ [0, ∞),

v(z, 0) = z,

(2.5)

has a unique Lipschitz continuous solution v = v(z, t) on [0, ∞), such that v(·, t) is a univalent Schwarz mapping and Dv(0, t) = e−t In , t ≥ 0 (see [25, Chapter 8]). (ii) Let f : Bn ×[0, ∞) → C n be a Loewner chain. Then there exists a Herglotz vector ﬁeld h(z, t) = z+· · · , z ∈ Bn , t ≥ 0, such that f satisﬁes the generalized Loewner diﬀerential equation (see [20]; cf. [16, Theorem 1.10]) ∂f (z, t) = Df (z, t)h(z, t), ∂t

a.e.

t ≥ 0,

∀ z ∈ Bn .

(2.6)

(iii) Conversely, if h(z, t) = z + · · · is a Herglotz vector ﬁeld, then every univalent solution f (z, t) = et z + · · · of the generalized Loewner diﬀerential equation (2.6) is a Loewner chain (see [10, Theorem 3.1] and [5]; cf. [16]).

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Deﬁnition 2.14. (See [16].) Let f ∈ H(Bn ) be a normalized mapping. We say that f has parametric representation if there exists a Herglotz vector ﬁeld h(z, t) = z + · · · such that f (z) = lim et v(z, t) t→∞

locally uniformly on Bn , where v = v(z, t) is the unique Lipschitz continuous solution on [0, ∞) of the initial value problem ∂v = −h(v, t), ∂t

a.e.

t ∈ [0, ∞),

v(z, 0) = z,

∀ z ∈ Bn .

We denote by S 0 (Bn ) the family of normalized univalent mappings on Bn which have parametric representation. The following result gives a useful characterization of the family S 0 (Bn ) in terms of normal Loewner chains on Bn (see [20, Corollary 2.9] and [10]; cf. [16] and [21]). Proposition 2.15. Let f ∈ H(Bn ) be a normalized mapping. Then f ∈ S 0 (Bn ) if and only if there exists a normal Loewner chain f (z, t) such that f = f (·, 0). Remark 2.16. (i) It is well known that if n = 1, then S 0 (U ) = S (see [35, Chapter 6]), while if n ≥ 2 then S 0 (Bn ) S(Bn ). (ii) S 0 (Bn ) is a compact subset of H(Bn ) (see [16] and [25, Chapter 8]), while if n ≥ 2 then S(Bn ) is not compact. (iii) For the mappings f ∈ S, the following growth result holds: |z| |z| ≤ |f (z)| ≤ , 2 (1 + |z|) (1 − |z|)2

z ∈ U.

If n ≥ 2, a similar growth result for S(Bn ) does not hold. However, for the mappings f ∈ S 0 (Bn ), the following growth result holds (see [16, Corollary 2.4]; see also [25, Chapter 8]): z z ≤ f (z) ≤ , (1 + z)2 (1 − z)2

z ∈ Bn .

In view of the above remark, it is natural and important to study the family S 0 (Bn ) instead of S(Bn ) if n ≥ 2. Let S 0 (M, Bn ) = f ∈ S 0 (Bn ) : f (z) < M, z ∈ Bn ,

M ∈ [1, ∞).

In the case n = 1, S 0 (M, U ) = S(M ), where S(M ) is the subset of S consisting of bounded functions on U , that is S(M ) = f ∈ S : |f (z)| < M, z ∈ U . Deﬁnition 2.17. (Cf. [22]; see [23].) Let T > 0. A mapping h : Bn × [0, T ] → C n is called a Carathéodory mapping on [0, T ] with values in N (Bn ) if the following conditions hold:

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I. Graham et al. / J. Math. Anal. Appl. 491 (2020) 124289

(i) h(·, t) ∈ N (Bn ) for t ∈ [0, T ]; (ii) h(z, ·) is measurable on [0, T ] for z ∈ Bn . Let C([0, T ], N (Bn )) be the family of all Carathéodory mappings on [0, T ] with values in N (Bn ). Deﬁnition 2.18. ([22], [23]) Let T > 0. Also, let h ∈ C([0, T ], N (Bn )), and let v = v(z, t; h) be the unique Lipschitz continuous solution on [0, T ] of the initial value problem ∂v (z, t; h) = −h(v(z, t; h), t), ∂t

a.e.

t ∈ [0, T ],

v(z, 0; h) = z,

(2.7)

for z ∈ Bn , so that v(·, t; h) is a univalent Schwarz mapping and Dv(0, t; h) = e−t In for t ∈ [0, T ]. 0 (idBn , N (Bn )) = {idBn }, where idBn is the identity map on Bn , and let Let R T (idBn , N (Bn )) = eT v(·, T ; h) : h ∈ C([0, T ], N (Bn )) , R

T ∈ (0, ∞).

T (idBn , N (Bn )) is called the normalized time-T-reachable family of (2.7). The family R log M (idU , M(U )) = S(M ) (see [35] and [40, Theorem 1.48]). Also, the Remark 2.19. It is known that R following relation holds (see [22] and [23]): T (idBn , M(Bn )) S 0 (Bn ), R

T ∈ [0, ∞).

log M idBn , M(Bn ) in terms of normal Loewner chains The following characterization of the family R was obtained in [22, Theorem 3.7] (see also [23, Theorem 4.5]). log M (idBn , M(Bn )) Proposition 2.20. Let M > 1 and let f ∈ H(Bn ) be a normalized mapping. Then f ∈ R if and only if there exists a normal Loewner chain f (z, t) such that f (·, 0) = f and f (·, log M ) = M idBn . log M (idBn , M(Bn )) Remark 2.21. The authors in [22] (see also [23, Corollary 4.7]) proved that the family R n is a compact subset of H(B ), for all M ∈ (1, ∞). log M (idBn , M(Bn )) ⊆ S 0 (M, Bn ) (see [22]), while in [9] it was From Proposition 2.20, we deduce that R n 0 proved that Rlog M idBn , M(B ) = S (M, Bn ), for n ≥ 2 and M ∈ (1, ∞). Next, we recall the notions of extreme/support points associated with a subset of H(Bn ) (see e.g. [38]). Deﬁnition 2.22. Let E ⊆ H(Bn ) be a nonempty set. (i) A mapping f ∈ E is called a support point of E if there exists a continuous linear functional L : H(Bn ) → C such that L|E = constant and L(f ) = max{L(q) : q ∈ E}. We denote by supp E the subset of E consisting of support points of E. (ii) A mapping g ∈ E is called an extreme point of the set E if the equality g = αp + (1 − α)q, with α ∈ (0, 1) and p, q ∈ E, implies that g = p = q. We denote by ex E the subset of E consisting of extreme points of E. Remark 2.23. (i) Since S 0 (Bn ) is a compact subset of H(Bn ), it follows that ex S 0 (Bn ) = ∅ and log M (idBn , M(Bn )), for M ∈ (1, ∞). supp S 0 (Bn ) = ∅. The same is true in the case of the reachable family R (ii) The authors in [9] (see also [19]) proved that if n ≥ 2 and M ∈ (1, ∞), then log M (idBn , M(Bn )) S 0 (M, Bn ) \ ex S 0 (Bn ) ∪ supp S 0 (Bn ) . R

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log M idBn , M(Bn ) and for S 0 (M, Bn ), it is natural and important to In view of the above results for R log M idBn , M(Bn ) , for n ≥ 2. study the family R Note that extremal problems for the family S 0 (Bn ) were studied in [41]. 3. Inverse Loewner chains on Bn × [0, ∞) In connection with the deﬁnition of a univalent subordination chain, we recall the notion of an inverse Loewner chain on Bn × [0, ∞) (cf. [43, Deﬁnition 2.1]; see [6] and [14, Deﬁnition 4.2], for n = 1). We also obtain some results related to inverse Loewner chains on Bn × [0, ∞), which may be also considered of independent interest. Deﬁnition 3.1. A family ft = f (·, t) (t ≥ 0) of holomorphic mappings ft : Bn → C n is called an inverse Loewner chain if the following conditions hold: (i) f (·, t) is a univalent mapping on Bn , for all t ≥ 0; (ii) f (Bn , s) ⊇ f (Bn , t) and f (0, s) = f (0, t), for 0 ≤ s ≤ t < ∞; (iii) Df (0, ·) is a locally absolutely continuous mapping on [0, ∞), and lim Df (0, t) = 0.

t→∞

If, in addition, f (z, 0) = z for z ∈ Bn , and Df (0, t) = e−t In for t ≥ 0, then f (z, t) is called a normalized inverse Loewner chain. Remark 3.2. (i) The condition (ii) in Deﬁnition 3.1 is equivalent to the fact that ft ≺ fs , 0 ≤ s ≤ t < ∞, in view of the univalence of ft = f (·, t), for t ≥ 0. (ii) If f (z, t) is an inverse Loewner chain such that f (z, 0) = z, for all z ∈ Bn , then f (0, t) = f (0, 0) = 0, for all t ≥ 0. Remark 3.3. Betker (see [6, Lemma 1]) proved that if p : U ×[0, ∞) → C is a Herglotz function of divergence type, then there exists a unique inverse Loewner chain f (z, t) : U ×[0, ∞) → C such that f (z, 0) = z, z ∈ U , and ∂f (z, t) = −zf (z, t)p(z, t), a.e. t ≥ 0, ∀ z ∈ U . ∂t Elin, Shoikhet and Sugawa [14, Lemma 4.3] obtained a suﬃcient condition for a family {gt }t≥0 of holomorphic functions gt : U → C to be an inverse Loewner chain, by using a Herglotz function. Schleissinger [43, Theorem 2.6] obtained a necessary and suﬃcient condition for a family {ft }t≥0 of holomorphic mappings ft : Bn → C n to be a normalized inverse Loewner chain, by using a Herglotz vector ﬁeld. Next, we obtain a suﬃcient condition for a mapping f : Bn × [0, ∞) → C n to be an inverse Loewner chain, by using a Herglotz vector ﬁeld (cf. [43, Theorem 2.6] for a(t) = 1, t ≥ 0; see [14, Lemma 4.3] and [6, Lemma 1], for n = 1). Proposition 3.4. Let a : [0, ∞) → C be a measurable function which is locally Lebesgue integrable on [0, ∞) such that a(t) > 0 for a.e. t ≥ 0. Assume that ∞ a(t)dt = ∞. 0

(3.1)

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Also, let f : Bn × [0, ∞) → C n be a mapping such that f (·, t) ∈ H(Bn ), f (0, t) = 0, and Df (0, t) = − 0t a(τ )dτ e In , for all t ≥ 0. Assume that f (z, ·) is locally absolutely continuous on [0, ∞) locally uniformly with respect to z ∈ Bn , and f (z, 0) = z, for all z ∈ Bn . Further, assume that there exists a Herglotz vector ﬁeld h(z, t) = a(t)z + · · · of divergence type such that ∂f (z, t) = −Df (z, t)h(z, t), ∂t

a.e.

t ≥ 0,

∀ z ∈ Bn .

(3.2)

Then f (z, t) is an inverse Loewner chain. Proof. In the ﬁrst part of the proof, we shall use arguments similar to those in the proofs of [14, Lemma 4.3] and [6, Lemma 1]. Fix T > 0, and let F : Bn × [0, ∞) → C n be given by

F (z, t) =

f (z, T − t), z ∈ Bn , t ∈ [0, T ], z ∈ Bn , t ≥ T. et−T z,

Then F (·, t) ∈ H(Bn ), F (0, t) = 0, DF (0, ·) is locally absolutely continuous on [0, ∞), for all t ≥ 0, by the assumption that the function a is locally Lebesgue integrable on [0, ∞). Also, F (z, ·) is locally absolutely continuous on [0, ∞) locally uniformly with respect to z ∈ Bn , by the assumption that f (z, ·) is locally absolutely continuous on [0, ∞) locally uniformly with respect to z ∈ Bn . Taking into account (3.2), we deduce that there exists a null set E ⊂ [0, ∞) such that ∂F (z, t) = DF (z, t)q(z, t), ∂t

∀ t ∈ [0, ∞) \ E,

∀ z ∈ Bn ,

(3.3)

where q : Bn × [0, ∞) → C n is given by

q(z, t) =

h(z, T − t), z,

a.e. t ∈ [0, T ], ∀ z ∈ Bn , a.e. t ≥ T, ∀ z ∈ Bn .

(3.4)

Then q(·, t) ∈ N (Bn ) for a.e. t ≥ 0, q(z, ·) is measurable on [0, ∞), for all z ∈ Bn , and Dq(0, t) = c(t)In , a.e. t ≥ 0, where

c(t) =

a(T − t), 1,

a.e. t ∈ [0, T ], a.e. t ≥ T.

(3.5)

Also, it is clear that q is a Herglotz vector ﬁeld of divergence type. Moreover, F (z, t) is a standard solution of the Loewner diﬀerential equation (3.3). Since the function a : [0, ∞) → C is measurable, locally integrable, and satisﬁes (3.1), it is not diﬃcult to deduce that the function c : [0, ∞) → C is measurable, locally ∞ integrable on [0, ∞), and 0 c(t)dt = +∞. Taking into account [21, Theorem 2.1], there exists a unique Lipschitz continuous solution v(z, s, t) = − st c(τ )dτ e z + · · · on [s, ∞) of the initial value problem ∂v (z, s, t) = −q(v(z, s, t), t) ∂t

a.e.

t ∈ [s, ∞),

v(z, s, s) = z,

such that v(·, s, t) is a univalent Schwarz mapping for each s, t with 0 ≤ s ≤ t < ∞. Let F (z, s, t) = F (v(z, s, t), t),

z ∈ Bn , 0 ≤ s ≤ t < ∞.

(3.6)

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11

∂ In view of (3.3) and (3.6), we deduce that ∂t F (z, s, t) = 0 a.e. t ≥ s, which implies that F (z, t) is a n subordination chain on B ×[0, ∞). In particular, Fs (Bn ) ⊆ Ft (Bn ) for 0 ≤ s ≤ t < ∞, where Ft (z) = F (z, t), for z ∈ Bn and t ≥ 0, and thus f (Bn , T − s) ⊆ f (Bn , T − t), for 0 ≤ s ≤ t ≤ T , i.e. fs (Bn ) ⊇ ft (Bn ), for 0 ≤ s ≤ t ≤ T . Since T > 0 is arbitrary, we obtain that fs (Bn ) ⊇ ft (Bn ), for 0 ≤ s ≤ t < ∞. For each t ≥ 0, we have F (z, t) = F (v(z, t, t + T ), t + T ) for z ∈ Bn . Since F (·, t + T ) is univalent on Bn , we deduce that F (z, t) is a univalent subordination chain. In particular, f (·, t) is univalent on Bn , for all t ∈ [0, T ]. Since T > 0 is arbitrary, we obtain that f (·, t) is univalent on Bn , for all t ≥ 0. t Since Df (0, t) = e− 0 a(τ ) In for t ≥ 0, it is clear that Df (0, ·) is locally absolutely continuous on [0, ∞). Finally, in view of (3.1), we deduce that

lim Df (0, t) = lim e−

t→∞

t 0

a(τ )dτ

t→∞

In = e−

∞ 0

a(τ )dτ

= 0.

Combining the above arguments, we conclude that f (z, t) is an inverse Loewner chain. This completes the proof. For the next result, which is a converse of Proposition 3.4, we shall use arguments similar to those in the proof of Proposition 3.4 (cf. the proof of [43, Theorem 2.6], in the case a(t) = 1 for t ≥ 0). Proposition 3.5. (Cf. [43, Theorem 2.6] for a(t) = 1, t ≥ 0.) Let a : [0, ∞) → C be a measurable function which satisﬁes the assumptions in Proposition 3.4. Also, let f : Bn × [0, ∞) → C n be an inverse Loewner t chain such that f (z, 0) = z, for all z ∈ Bn , and Df (0, t) = e− 0 a(τ )dτ In , for all t ≥ 0. Then f (z, ·) is locally Lipschitz continuous on [0, ∞) locally uniformly with respect to z ∈ Bn , and there exists a Herglotz vector ﬁeld h : Bn × [0, ∞) → C n of divergence type associated with the inverse Loewner chain f (z, t) such that ∂f (z, t) = −Df (z, t)h(z, t), ∂t

a.e. t ≥ 0, ∀ z ∈ Bn .

Proof. Fix T > 0. Let F : Bn × [0, ∞) → C n be given by

F (z, t) =

f (z, T − t), et−T z,

z ∈ Bn , t ∈ [0, T ], z ∈ Bn , t ≥ T.

Since f (z, t) is an inverse Loewner chain and f (z, 0) = z, for all z ∈ Bn , it is not diﬃcult to deduce that t F (z, t) is a univalent subordination chain on Bn × [0, ∞). Also, DF (0, t) = e 0 c(τ )dτ In , for all t ≥ 0, where c : [0, ∞) → C is given by (3.5). Then DF (0, t) is locally absolutely continuous of t ∈ [0, ∞), and thus F (z, ·) is locally Lipschitz continuous on [0, ∞) locally uniformly with respect to z ∈ Bn (see e.g. the proof of [20, Theorem 2.8]; see also e.g. [25, Chapter 8] and [47, Proposition 1.3.4]). Hence, f (z, ·) is also Lipschitz continuous on [0, T ] locally uniformly with respect to z ∈ Bn . Moreover, ∂F ∂t (·, t) exists and is holomorphic n on B for a.e. t ∈ [0, ∞), and there exists a Herglotz vector ﬁeld q(z, t) : Bn × [0, ∞) → C n such that q(z, t) = z for z ∈ Bn , t ≥ T , and (see e.g. the proof of [16, Theorem 1.10]; see also [47, Proposition 1.3.6]) ∂F (z, t) = DF (z, t)q(z, t), a.e. t ∈ [0, ∞), ∀ z ∈ Bn . ∂t Now, if h(z, t) = q(z, T − t) = a(t)z + · · · , for all z ∈ Bn and t ∈ [0, T ], then h(·, t) ∈ N (Bn ) for t ∈ [0, T ], h(z, ·) is measurable on [0, T ] for z ∈ Bn , and it is not diﬃcult to deduce that ∂f (z, t) = −Df (z, t)h(z, t), a.e. t ∈ [0, T ], ∀ z ∈ Bn . ∂t

(3.7)

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I. Graham et al. / J. Math. Anal. Appl. 491 (2020) 124289

∞ Finally, since T > 0 is arbitrary, h is well-deﬁned on Bn × [0, ∞) by (3.7), and since 0 a(t)dt = +∞, it t follows that limt→∞ 0 m(Dh(0, τ ))dτ = +∞. Thus, h(z, t) is a Herglotz vector ﬁeld of divergence type and the conclusion follows, as desired. This completes the proof. We deduce the following existence and uniqueness result for the solutions of the inverse Loewner diﬀerential equation (3.2) (cf. [43, Theorem 2.6], in the case a(t) = 1 for t ≥ 0). Proposition 3.6. Let a : [0, ∞) → C be a measurable function which satisﬁes the assumptions in Proposition 3.4. Also, let h : Bn × [0, ∞) → C n be a Herglotz vector ﬁeld of divergence type such that Dh(0, t) = a(t)In for a.e. t ≥ 0. Then there is a unique inverse Loewner chain f : Bn × [0, ∞) → C n − 0t a(τ )dτ n such that f (z, 0) = z, for all z ∈ B , and Df (0, t) = e In , for all t ≥ 0, which is the locally absolutely continuous solution on [0, ∞) of the inverse Loewner diﬀerential equation (3.2). Proof. We divide the proof into three steps, as follows. Step I. Fix T > 0. We prove the existence of the inverse Loewner chain fT (z, t) on [0, T ], which satisﬁes the conditions in the statement of Proposition 3.6 on [0, T ]. To this end, let

q(z, t) =

h(z, T − t), z,

a.e. t ∈ [0, T ], ∀ z ∈ Bn , ∀ t > T, ∀ z ∈ Bn .

Since q satisﬁes the assumptions of h in [18, Lemma 2.4], in view of [18, Lemma 2.4], for each s ∈ [0, ∞) and z ∈ Bn , the initial value problem ∂v = −q(v, t), a.e. t ∈ [s, ∞), v(z, s, s) = z, ∂t has a unique solution v = v(·, s, t) which is a univalent Schwarz mapping, v(z, s, ·) is Lipschitz continuous on t [s, ∞) locally uniformly with respect to z ∈ Bn , and such that Dv(0, s, t) = e− s a(T −τ )dτ In , for t ∈ [s, T ]. Moreover, there exists a univalent subordination chain g(z, t) such that g(z, s) = g(v(z, s, t), t) for z ∈ Bn , g(z, ·) is locally Lipschitz continuous on [0, ∞) locally uniformly with respect to z ∈ Bn and g satisﬁes the Loewner diﬀerential equation: ∂g (z, t) = Dg(z, t)q(z, t), a.e. t ∈ [0, ∞), ∀ z ∈ Bn . ∂t As in the proof of [25, Corollary 8.1.10], we deduce for all t ∈ [0, ∞) and z ∈ Bn that ∂v (z, s, t) = Dv(z, s, t)q(z, s), a.e. s ∈ [0, t]. ∂s

(3.8)

Next, let fT (z, t) = v(z, T − t, T ), for z ∈ Bn and t ∈ [0, T ]. Then fT (·, t) is a univalent mapping on − 0t a(τ )dτ n B , fT (0, t) = 0 for all t ∈ [0, T ], DfT (0, t) = e In for all t ∈ [0, T ], fT (Bn , s) ⊇ fT (Bn , t) for n 0 ≤ s ≤ t ≤ T , and fT (z, 0) = z, for all z ∈ B . Taking into account the relation (3.8), it is not diﬃcult to deduce that fT (z, t) satisﬁes the inverse Loewner diﬀerential equation (3.2) on [0, T ]. Hence fT (z, t) is an inverse Loewner chain on [0, T ], which satisﬁes the conditions in the statement of Proposition 3.6 on [0, T ]. Step II. Fix T > 0. We prove the uniqueness of the inverse Loewner chain fT (z, t) on [0, T ], which satisﬁes the conditions in the statement of Proposition 3.6 on [0, T ]. To this end, assume that fT (z, t) is another inverse Loewner chain on [0, T ] which satisﬁes the statement of Proposition 3.6 on [0, T ]. Let

F (z, t) =

fT (z, T − t), z ∈ Bn , t ∈ [0, T ], et−T z, z ∈ Bn , t ≥ T,

I. Graham et al. / J. Math. Anal. Appl. 491 (2020) 124289

13

and

F(z, t) =

fT (z, T − t), et−T z,

z ∈ Bn , t ∈ [0, T ], z ∈ Bn , t ≥ T.

Then F (z, t) and F (z, t) are univalent subordination chains which satisfy the same Loewner diﬀerential equation on Bn × [0, ∞). Since the assumptions of [18, Corollary 4.4] are satisﬁed, in view of [18, Corollary 4.4], we obtain that F (·, t) = F (·, t), for all t ∈ [0, T ]. This implies that fT (z, t) ≡ fT (z, t) for t ∈ [0, T ], as desired. Step III. By Steps I and II, the mapping f (z, t) = fk (z, t)

for t ∈ [0, k], z ∈ Bn , k ∈ N,

is well deﬁned and f (z, t) is an inverse Loewner chain which satisﬁes the conditions in the statement of Proposition 3.6. This completes the proof. Remark 3.7. The results contained in Propositions 3.4, 3.5 and 3.6 may be extended to the case of inverse − 0t A(τ )dτ Loewner chains f (z, t) = e z + · · · such that f (z, 0) = z for z ∈ Bn , where A : [0, ∞) → L(C n ) is measurable and locally Lebesgue integrable mapping on [0, ∞) such that m(A(t)) > 0 for a.e. t ≥ 0, ∞ m(A(t))dt = +∞, and such that the mapping A satisﬁes certain regularity conditions as in [21, Theorem 0 2.3]. 4. Nonlinear resolvents, Loewner chains, and starlikeness on Bn Let {Jr }r≥0 be the nonlinear resolvent family of f ∈ M(Bn ). In this section, we prove that the family {Jr }r≥0 is an inverse Loewner chain on Bn with the Herglotz vector ﬁeld h given by (4.1) which is of divergence type. In addition, we prove that (1 + r)Jr (r ≥ 0) may be embedded in normal Loewner chains on Bn . We also prove that if n = 2, then the shearing of (1 + r)Jr is quasi-convex of type A and also starlike of order 4/5 on B2 . Finally, we obtain suﬃcient conditions for the nonlinear resolvents to be quasiconformal on Bn and to extend quasiconformally to the whole space C n . 4.1. Nonlinear resolvents and inverse Loewner chains We begin this section with the following result, which is a generalization to higher dimensions of [14, Proposition 4.5] (compare with [14, Theorem 2.5] and [45, Theorem 23]). Theorem 4.1. If f ∈ M(Bn ), then the nonlinear resolvent family {Jr }r≥0 of f is an inverse Loewner chain on Bn with the Herglotz vector ﬁeld of divergence type h : Bn × [0, ∞) → C n given by

h(w, r) =

w−Jr (w) , r

f (w),

r>0 r = 0.

(4.1)

log(1+T ) idBn , M(Bn ) , and In particular, Jr (Bn ) ⊆ Js (Bn ), for 0 ≤ s ≤ r < ∞. In addition, (1 + T )JT ∈ R thus (1 + T )JT ∈ S 0 (Bn ), for all T ≥ 0. Proof. First, we prove that {Jr }r≥0 is an inverse Loewner chain, by an argument similar to that in the proof of [14, Proposition 4.5].

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I. Graham et al. / J. Math. Anal. Appl. 491 (2020) 124289

Taking into account the deﬁnition of the nonlinear resolvent mapping Jr , we deduce that Jr is univalent on Bn for r ≥ 0, and J0 (w) = w, for all w ∈ Bn . Also, since f (0) = 0 and Df (0) = In , we deduce that 1 Jr (0) = 0 and DJr (0) = 1+r In , for all r ≥ 0. n Now, we prove that Js (B ) ⊆ Jr (Bn ), for 0 ≤ r ≤ s < ∞. To this end, note that z ∈ Jr (Bn ) if and only if (In + rf )(z) ∈ Bn . Hence, it is enough to prove that (cf. the proofs of [14, Theorem 2.5] and [45, Theorem 23]) z + rf (z) ≤ z + sf (z),

z ∈ Bn ,

0 ≤ r ≤ s < ∞.

Indeed, using an elementary computation and the fact that f (z), z > 0, for all z ∈ Bn \ {0}, we deduce that z + rf (z)2 = z2 + 2r f (z), z + r2 f (z)2 ≤ z2 + 2s f (z), z + s2 f (z)2 = z + sf (z)2 . 1 Thus, Js (Bn ) ⊆ Jr (Bn ), for 0 ≤ r ≤ s < ∞. In addition, since DJr (0) = 1+r In , r ≥ 0, it is clear that DJ(0, ·) is locally absolutely continuous on [0, ∞), where J(w, r) = Jr (w) for w ∈ Bn and r ≥ 0, and DJr (0) → 0 as r → ∞. In view of the above arguments, we deduce that {Jr }r≥0 is an inverse Loewner chain. Next, we prove that the mapping h : Bn × [0, ∞) → C n given by (4.1) is a Herglotz vector ﬁeld of divergence type, associated with {Jr }r≥0 . Indeed, since

Jr (w) + rf (Jr (w)) = w,

w ∈ Bn ,

(4.2)

we deduce that (In + rDf (Jr (w)))DJr (w) = In ,

w ∈ Bn .

(4.3)

On the other hand, diﬀerentiating both sides of (4.2) with respect to r ≥ 0 (note that J(w, ·) is locally Lipschitz continuous in r ≥ 0 in view of Proposition 3.5), we obtain that (In + rDf (Jr (w)))

∂Jr (w) = −f (Jr (w)), ∂r

a.e. r ≥ 0.

Hence, in view of (4.3) and the above relation, we deduce that ∂Jr (w) = −DJr (w)f (Jr (w)), ∂r

a.e. r ≥ 0,

w ∈ Bn .

Further, using (4.2) and the above equality, we deduce that ∂Jr (w) = −DJr (w)h(w, r), ∂r

a.e. r ≥ 0,

w ∈ Bn ,

(4.4)

where h : Bn × [0, ∞) → C n is given by (4.1). Note that limr0 w−Jrr (w) = f (w), for all w ∈ Bn . We have that h(·, r) ∈ H(Bn ), h(0, r) = 0, r ≥ 0, and since Jr (w) ≤ w, w ∈ Bn , it follows that 1 h(w, r), w ≥ 0, w ∈ Bn , r ≥ 0. Also, since Dh(0, r) = 1+r In , r ≥ 0, we deduce in view of the above inequality and the minimum principle for harmonic functions that

I. Graham et al. / J. Math. Anal. Appl. 491 (2020) 124289

h(w, r), w > 0,

w ∈ Bn \ {0},

15

r ≥ 0.

Hence h(·, r) ∈ N (Bn ), r ≥ 0. Also, h(w, ·) is measurable on (0, ∞) for w ∈ Bn , since J(w, ·) is measurable on (0, ∞), w ∈ Bn . Thus, the mapping h given by (4.1) is a Herglotz vector ﬁeld. Further, since Dh(0, r) = 1 1+r In , r ≥ 0, we have that Dh(0, ·) is locally integrable on [0, ∞) and ∞

∞ m(Dh(0, r))dr =

0

dr = +∞. 1+r

0

Consequently, from (4.4) and the above arguments (see also Propositions 3.4 and 3.5), we deduce that {Jr }r≥0 is an inverse Loewner chain with the Herglotz vector ﬁeld h given by (4.1), which is of divergence type. Next, ﬁx T > 0 and let q(z) = (1 + T )JT (z), for all z ∈ Bn . Then it is clear that q ∈ S(Bn ). We prove that the mapping q may be embedded as the ﬁrst element of a normal Loewner chain {f (z, t∗ )}t∗ ≥0 on Bn ∗ such that f (z, T ∗ ) = eT z, for all z ∈ Bn , where T ∗ = log(1 + T ). Let g = g(z, t) : Bn × [0, ∞) → C n be given by

g(z, t) =

(1 + T )JT −t (z), (1 + T )et−T z,

z ∈ Bn , t ∈ [0, T ], z ∈ Bn , t ≥ T.

(4.5)

Then gt = g(·, t) is a univalent mapping on Bn , gt (0) = 0, and after elementary computations we deduce t that Dgt (0) = e 0 a(τ )dτ In , t ≥ 0, where

a(t) =

1 1+T −t ,

1,

t ∈ [0, T ], t ≥ T.

(4.6)

On the other hand, since Js (Bn ) ⊇ Jt (Bn ) for 0 ≤ s ≤ t < ∞, it is not diﬃcult to deduce that gs (Bn ) ⊆ gt (Bn ), for 0 ≤ s ≤ t < ∞, where gt is given by (4.5). Moreover, since Jr (Bn ) ⊆ Bn , r ≥ 0, we easily deduce that

(1 + T )z, z ∈ Bn , t ∈ [0, T ], − 0t a(τ )dτ e gt (z) ≤ 1, z ∈ Bn , t ∈ [T, ∞). Finally, we prove that q may be embedded as the ﬁrst element of a normal Loewner chain. Indeed, let t α(t) = 0 a(τ )dτ , for t ≥ 0, where a : [0, ∞) → [0, ∞) is given by (4.6). Then α(0) = 0, α(t) → ∞ as t → ∞, and α is a continuous and increasing function on [0, ∞). Thus α : [0, ∞) → [0, ∞) is one-to-one of [0, ∞) onto [0, ∞). Therefore, if t∗ = α(t) for t ≥ 0, then t∗ ∈ [0, ∞), and if f (z, t∗ ) : Bn × [0, ∞) → C n is given by f (z, t∗ ) = g(z, t), for z ∈ Bn , then f (·, t∗ ) is a univalent mapping on Bn , f (0, t∗ ) = 0, and ∗

Df (0, t∗ ) = Dg(0, t) = eα(t) In = et In ,

t∗ ∈ [0, ∞).

∗

Hence e−t ft∗ (·) ∈ S(Bn ), for all t∗ ≥ 0, where ft∗ (·) = f (·, t∗ ). On the other hand, since gs (Bn ) ⊆ gt (Bn ), 0 ≤ s ≤ t < ∞, we obtain that fs∗ (Bn ) ⊆ ft∗ (Bn ), ∗

0 ≤ s∗ ≤ t∗ < ∞. ∗

Since {e−t ft∗ (·)}t∗ ≥0 = {e−α(t) gt (·)}t≥0 , we have that {e−t ft∗ (·)}t∗ ≥0 is a locally uniformly bounded family on Bn . Therefore, {ft∗ }t∗ ≥0 is a normal Loewner chain. Since f (·, 0) = g(·, 0) = (1 + T )JT , it follows that q is the ﬁrst element of the normal Loewner chain {ft∗ }t∗ ≥0 .

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16

log(1+T ) (idBn , M(Bn )). To this end, let T ∗ = α(T ). Then T ∗ = log(1 + T ) and Next, we prove that q ∈ R f (·, T ∗ ) = g(·, T ) = (1 + T )idBn = e

T 0

a(τ )dτ

∗

idBn = eT idBn ,

where idBn (z) = z, for z ∈ Bn . Since {ft∗ }t∗ ≥0 is a normal Loewner chain and q = f (·, 0), we deduce in view of Proposition 2.20 that log(1+T ) (idBn , M(Bn )), q∈R as desired. This completes the proof. In view of Theorem 4.1, we obtain the following particular cases (compare with [14], in the case n = 1). Example 4.2. Let f : Bn → C n be a quasi-convex mapping of type A (in particular, let f ∈ K(Bn )), and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Then {Jr }r≥0 is an inverse Loewner chain on log(1+T ) idBn , M(Bn ) , and thus (1 + T )JT ∈ S 0 (Bn ), for all T ≥ 0. Bn . Also, (1 + T )JT ∈ R Proof. Since f is quasi-convex of type A, it follows that f (z), z ≥ 12 z2 , for all z ∈ Bn . Hence, it suﬃces to apply Theorem 4.1. This completes the proof. Example 4.3. Let f ∈ R(Bn ). Then f (z), z ≥ k0 z2 , z ∈ Bn by [13, Theorem 2.7.1] (cf. [15, Proposition 1.6]), where k0 = 2 ln 2 − 1. In particular, f ∈ M(Bn ), and if {Jr }r≥0 is the nonlinear resolvent family of log(1+T ) idBn , M(Bn ) , and thus f , then {Jr }r≥0 is an inverse Loewner chain on Bn . Also, (1 + T )JT ∈ R (1 + T )JT ∈ S 0 (Bn ) for all T ≥ 0. 4.2. Compactness of the family generated by nonlinear resolvent mappings Next, we obtain the following compactness result for the family F(Bn ) ⊂ S 0 (Bn ) given by (4.7). Proposition 4.4. Let F(Bn ) = (1 + r)Jr [f ] : 0 ≤ r < +∞, f ∈ M(Bn ) ,

(4.7)

where Jr [f ] is the nonlinear resolvent mapping of f ∈ M(Bn ). Then the family F(Bn ) is compact in S 0 (Bn ) ⊂ H(Bn ). Proof. We divide the proof into two steps, as follows. Step I. Fix T > 0 and let FT (Bn ) = Jr [f ] : 0 ≤ r ≤ T, f ∈ M(Bn ) . We prove that the family FT (Bn ) is a compact subset of H(Bn ). Since the family {Jr [f ] : 0 ≤ r ≤ T, f ∈ M(Bn )} is a locally uniformly bounded family on Bn , it suﬃces to show that it is closed in H(Bn ). Let {Jrk [fk ]}k≥1 be such that 0 ≤ rk ≤ T , fk ∈ M(Bn ) and Jrk [fk ] → G n locally uniformly on Bn . Then G(Bn ) ⊂ B . Since Jrk [fk ](0) = 0 for k ≥ 1, we have G(0) = 0 and this implies that G(Bn ) ⊂ Bn . Since M(Bn ) is compact, we may assume that fk → f locally uniformly on Bn . Also, we may assume that rk → r∞ ∈ [0, T ] as k → ∞. Letting k → ∞ in the equality Jrk [fk ](w) + rk fk (Jrk [fk ](w)) = w,

k ≥ 1, w ∈ Bn ,

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we have G(w) + r∞ f (G(w)) = w,

w ∈ Bn .

Therefore, we have G = Jr∞ [f ] and this implies that the family FT (Bn ) is closed, as desired. Step II. Next, we prove that F(Bn ) is compact in H(Bn ). Indeed, since the family F(Bn ) ⊆ S 0 (Bn ) is a locally uniformly bounded family on Bn , it suﬃces to show that it is closed in S 0 (Bn ). Let {(1+rk )Jrk [fk ]}k≥1 be such that rk ≥ 0, fk ∈ M(Bn ) and (1 + rk )Jrk [fk ] → G ∈ S 0 (Bn ) locally uniformly on Bn . By the ﬁrst step of the proof, it suﬃces to consider the case rk → ∞ as k → ∞. Since M(Bn ) is compact, we may assume that fk → f locally uniformly on Bn . Letting k → ∞ in the equality In + rk Dfk (Jrk [fk ](w)) (1 + rk )DJrk [fk ](w) = In , 1 + rk

k ≥ 1, w ∈ Bn ,

we have Df (0)DG(w) = DG(w) = In . Therefore, we have G(w) = w = (1 + 0)J0 (w) and this implies that the family F(Bn ) is closed, as desired. This completes the proof. Remark 4.5. The family {Jr [f ] : r ≥ 0, f ∈ M(Bn )} is not compact in H(Bn ), where Jr [f ] is the nonlinear resolvent mapping of f ∈ M(Bn ). Indeed, let f (z) = (z1 + αz22 , z ),

z = (z1 , z ) ∈ Bn ,

√ where |α| ≤ 3 3/2. Then f ∈ M(Bn ) and Jr [f ](w) =

1 1+r

w1 −

r 2 , αw , w 2 (1 + r)2

∀ w = (w1 , w ) ∈ Bn .

(4.8)

Letting r → ∞, we have Jr [f ] → 0 locally uniformly on Bn . This implies that the family {Jr [f ] : r ≥ 0, f ∈ M(Bn )} is not compact. w In particular, if we consider f (z) = z, ∀ z ∈ Bn , then f ∈ M(Bn ) and Jr [f ](w) = 1+r → 0 locally n uniformly on B , as r → ∞, and we obtain the non-compactness of the family {Jr [f ] : r ≥ 0, f ∈ M(Bn )}. Note that the nonlinear resolvent mapping Jr [f ] deﬁned in (4.8) will be used in the proofs of Propositions 4.11, 5.1, 5.6 and 5.7. Next, let Aut(C n ) be the family of automorphisms of C n , and let Aut0 (C n ) be the subset of Aut(C n ) consisting of the normalized automorphisms of C n . Also, let SR (Bn ) = f ∈ S(Bn ) : f (Bn ) is a Runge domain in C n be the family of normalized univalent mappings on Bn with Runge image. Remark 4.6. Let f ∈ M(Bn ) and let Jr [f ], r ≥ 0, be the nonlinear resolvent mapping of f . In view of Theorem 4.1, we have that (1 + r)Jr [f ] ∈ S 0 (Bn ), and thus (1 + r)Jr [f ] ∈ SR (Bn ), by [42, Theorem 2.3] (cf. [5, Proposition 5.1]). Assume that n ≥ 2. Then for every r ≥ 0, there is a sequence {Ψk }k∈N ⊆ Aut(C n ) such that Ψk |Bn → (1 + r)Jr [f ] locally uniformly on Bn as k → ∞, by [3, Theorem 2.1].

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Taking into account Proposition 4.4 and Remark 4.6, it would be interesting to give an answer to the following question (compare with [26]): Question 4.7. Let n ≥ 2. Also, let F(Bn ) be the compact subset of S 0 (Bn ) given by (4.7). Then is it true that F(Bn ) = F(Bn ) ∩ Aut0 (C n )? 4.3. Nonlinear resolvents and starlikeness on Bn Next, we are concerned with the following question, whose answer is positive in the case n = 1 (see [14]). Question 4.8. Let f ∈ M(Bn ) and let {Jr }r≥0 be the nonlinear resolvent family of f . Then is it true that (1 + r)Jr is starlike on Bn , for all r ≥ 0 and n ≥ 2? Next, we give some partial answers to the above question (compare with [14, Corollary 2.6] and [45, Theorem 25]). Proposition 4.9. Let f (z) = F (z)z, where F : Bn → C is a holomorphic function. Assume that f ∈ M(Bn ). Let {Jr }r≥0 be the nonlinear resolvent family of f . Then (1 + r)Jr is a starlike mapping on Bn , for all r ≥ 0. Proof. Since f ∈ M(Bn ), it follows that F (0) = 1 and F (z) > 0, for z ∈ Bn \ {0}. Let h(z) = 1 + rF (z) and let ϕ(z) = 1/h(z). Then, for z ∈ Bn , z ∈ Jr (Bn ) if and only if z + rf (z) = h(z)z ∈ Bn . Therefore, we obtain that Jr (Bn ) = {z ∈ Bn : z < |ϕ(z)|} (cf. the proof of [14, Theorem 2.5]). As in the proof of [31, Theorem 2.1], we obtain that Ω = {z ∈ Bn : z < |ϕ(z)|} is a starlike domain. This completes the proof. Before to prove the next result, we recall the notion of shearing of a mapping f ∈ H(B2 ), due to Bracci (see [7, Deﬁnition 1.3]). This notion was very useful in the study of extremal problems associated to the family S 0 (Bn ) (see [7] and [9]). Deﬁnition 4.10. Let f ∈ H(B2 ) be a normalized mapping, and let f (z) = z1 + αz22 + O(|z1 |2 , |z1 z2 |, z3 ), z2 + O(z2 ) , z = (z1 , z2 ) ∈ B2 , be the power series of f on B2 . The shearing f [c] of f is given by f [c] (z) = z1 + αz22 , z2 ,

z = (z1 , z2 ) ∈ B2 .

[c]

Next, we prove that (1 + r)Jr is quasi-convex of type A, and it is also starlike of order 4/5 on B2 for all [c] r ≥ 0, where Jr is the shearing of the nonlinear resolvent mapping Jr of a mapping f ∈ M(B2 ) (compare with [14, Corollary 2.6] and [45, Theorem 25]). Proposition 4.11. Let f ∈ M(B2 ) and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . [c] [c] Also, let Jr be the shearing of Jr , r ≥ 0. Then (1 + r)Jr is quasi-convex of type A, r ≥ 0. Moreover, [c] ∗ (1 + r)Jr ∈ Sβ∗ (B2 ) ⊆ S4/5 (B2 ) with β = (1 + r)2 /(r2 + 3r + 1), for all r > 0.

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Proof. Let (1) (1) (1) (2) (2) (2) f (z) = z1 + a2,0 z12 + a1,1 z1 z2 + a0,2 z22 + · · · , z2 + a2,0 z12 + a1,1 z1 z2 + a0,2 z22 + · · · , be the power series expansion of f on B2 . In view of Deﬁnition 4.10, the shearing f [c] of f is given by f [c] (z) = z1 + αz22 , z2 ,

z = (z1 , z2 ) ∈ B2 ,

(1)

where α = a0,2 . Since f ∈ M(B2 ), it follows that f [c] ∈ M(B2 ), in view of [7, Proposition 2.1]. Next, ﬁx r ≥ 0 and let w = (w1 , w2 ) ∈ B2 . Then Jr (w) + rf (Jr (w)) = w.

(4.9)

Since f [c] ∈ M(B2 ), there is a unique solution z = Jr (w) ∈ B2 of the equation z + rf [c] (z) = w. By (4.8), we obtain that Jr (w) =

αr w2 w1 − w2 , 1 + r (1 + r)3 2 1 + r

,

w = (w1 , w2 ) ∈ B2 .

√

(4.10) √

|α|r 3 3 Let gr : B2 → C 2 be given by gr = (1 + r)Jr . Since |α| ≤ 3 2 3 , by [7, Corollary 2.2], we have (1+r) 2 ≤ 8 . Therefore, √we obtain in view of [39, Example 9] that gr is a quasi-convex mapping of type A. Also, since |α|r 3 3 r 2 (1+r)2 ≤ 2 (1+r)2 and r/(1 + r) = (1 − β)/β, we obtain in view of [24, Corollary 4.6] that (1 + r)Jr ∈ ∗ 2 Sβ (B ), for all r > 0. 1 On the other hand, from (4.9), we obtain that DJr (0) = 1+r I2 and

D2 Jr (0)(u, u) = −

r D2 f (0)(u, u), (1 + r)3

∀ u ∈ C2.

(4.11)

Thus, Jr has the following power series expansion: Jr (w) =

1 1 r w− · D2 f (0)(w, w) + · · · , 3 1+r (1 + r) 2

w ∈ B2 .

[c]

Finally, we deduce the shearing Jr of Jr is given by (see also [7, Deﬁnition 1.3]) Jr[c] (w)

1 = 1+r

r 1 2 a w , w2 , w1 − (1 + r)2 0,2 2

∀ w = (w1 , w2 ) ∈ B2 ,

[c] and thus, Jr (w) = Jr (w), for all r ≥ 0 and w ∈ B2 , where Jr is given by (4.10). This completes the proof.

Corollary 4.12. Let f ∈ M(B2 ) be given by f (z) = z1 + αz22 , z2 ,

z = (z1 , z2 ) ∈ B , 2

√ 3 3 |α| ≤ , 2

and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Then (1 + r)Jr is quasi-convex of type ∗ A, for all r ≥ 0. Moreover, (1 + r)Jr ∈ Sβ∗ (B2 ) ⊆ S4/5 (B2 ), for all r > 0, where β = (1 + r)2 /(r2 + 3r + 1).

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In particular, let f (z) =

z1 +

√ 3 3 2 z2 , z2 ∈ M(B2 ), 2

z = (z1 , z2 ) ∈ B2 ,

and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Then (1 + r)Jr ∈ supp Sβ∗ (B2 ) \ supp Sγ∗ (B2 ), ∀ γ ∈ [0, β), r > 0. Proof. It suﬃces to show that (1 + r)Jr ∈ / supp Sγ∗ (B2 ) for all γ ∈ [0, β) with r > 0. The other part directly follows from the proof of Proposition 4.11. Let r > 0 and γ ∈ [0, β) be ﬁxed. Assume that (1 + r)Jr ∈ supp Sγ∗ (B2 ). Then, there exists a continuous linear functional L : H(B2 ) → C such that L|E = constant and L((1 + r)Jr ) = max{L(q) : q ∈ E}, where E = Sγ∗ (B2 ). We have √ r 3 3 L((1 + r)Jr ) = L(w) − w22 L(e1 ) , 2 2 (1 + r) where e1 = (1, 0). Since L|E = constant, there exists a homogeneous polynomial Pm of degree m with m ≥ 2 such that L(Pm ) > 0. This implies that idB2 is not a support point of E with respect to L. Then, we have w22 L(e1 ) < 0. Let F (w) =

√ 3 3 2 w1 − a0 (γ)w2 , w2 , 2

w = (w1 , w2 ) ∈ B2 ,

where

a0 (γ) =

γ ∈ [0, 12 ] γ ∈ ( 12 , β).

1, 1−γ γ ,

Then F ∈ E by [24], and since a0 (γ) > a0 (β) = contradiction. This completes the proof.

r (1+r)2 ,

we have L(F ) > L((1 + r)Jr ). This is a

Corollary 4.13 below shows that the nonlinear resolvent Jr of a mapping in M(Bn ) has the property that (1 + r)Jr is not a support/extreme point of the family S 0 (Bn ), for all r ≥ 0 and n ≥ 1. Corollary 4.13. Let f ∈ M(Bn ), and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Also, let M = 1 + T . Then (1 + T )JT ∈ S 0 (M, Bn ) \ (ex S 0 (Bn ) ∪ supp S 0 (Bn )), ∀ T ∈ [0, ∞). Proof. First, assume that n = 1. Then (1 + T )JT ∈ S(M ) is a bounded map, and thus (1 + T )JT cannot be a support/extreme point of S = S 0 (U ), for all T ≥ 0 (see [35]). Next, assume that n ≥ 2. If T = 0, then J0 = idBn , which cannot be an extreme/support point of S 0 (Bn ), in view of [22, Propositions 2.2, 2.4] (see log(1+T ) (idBn , M(Bn )) S 0 (M, Bn ), by Theorem 4.1. In view of also [23]). If T > 0, then (1 + T )JT ∈ R [19, Corollary 7] (see also [9, Corollary 5.6]), the conclusion follows, as desired. 4.4. Nonlinear resolvents and quasiconformal extensions Question 4.14. Under which additional conditions related to f ∈ M(Bn ), the associated nonlinear resolvent n Jr : Bn → Bn extends to a homeomorphism of B onto Jr (Bn ), for all r ≥ 0? A similar question in the case of a quasiconformal homeomorphism of Jr to C n onto itself.

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Before to prove the following result, we recall the notion of a quasiconformal mapping in C n (see e.g. [25], and [27] and the references therein). Deﬁnition 4.15. Let Ω and Ω be domains in the space Rm endowed with the Euclidean norm · , and let K > 0 be a constant. A homeomorphism f : Ω → Ω is said to be K-quasiconformal if it is diﬀerentiable a.e., ACL (absolutely continuous on lines), and D(f ; x)m ≤ K| det D(f ; x)|,

a.e. x ∈ Ω,

where D(f ; x) denotes the (real) Jacobian matrix of f at x, and D(f ; x) = sup{D(f ; x)(w) : w = 1}. A mapping F : Ω → C n is said to be quasiconformal on Ω if there exists a constant K > 0 such that F is K-quasiconformal on Ω. Elin, Shoikhet and Sugawa [14, Theorem 4.6] proved that if f ∈ N (U ) satisﬁes the following condition: f (ζ) απ , arg < ζ 2

ζ ∈ U,

for some α ∈ (0, 1), then the associated nonlinear resolvent mapping Jr , r ≥ 0, extends to a k-quasiconformal homeomorphism of C onto itself, where k = sin απ 2 . Next, we obtain the following suﬃcient condition for the nonlinear resolvent mapping Jr of a mapping f ∈ M(Bn ) that satisﬁes a natural condition on Bn , to extend to a quasiconformal homeomorphism of C n onto itself. Theorem 4.16. Let f ∈ M(Bn ) and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Let r > 0. (i) If there exists a constant c ∈ (0, 12 ) such that Df (z) − In ≤

1+r c, r

z ∈ Bn ,

then (1 + r)Jr is quasiconformal on Bn and extends to a quasiconformal homeomorphism of C n onto itself. (ii) If √ 2( 5 − 2)(1 + r) Df (z) − In ≤ , r

z ∈ Bn ,

then (1 + r)Jr is starlike and quasiconformal on Bn and extends to a quasiconformal homeomorphism of C n onto itself. Proof. (i) Let w ∈ Bn be ﬁxed and let z = (In + rf )−1 (w). By the deﬁnition of Jr , we have (1 + r)DJr (w) = (1 + r)[In + rDf (z)]−1 −1 r = In − In − Df (z) 1+r

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∞ k r = In + . In − Df (z) 1+r k=1

Therefore, we have ∞ r k c (1 + r)DJr (w) − In ≤ 1 + r In − Df (z) ≤ 1 − c < 1. k=1

Then the above relation implies that (1 + r)Jr is quasiconformal on Bn and extends to a quasiconformal homeomorphism of C n onto itself, by [27, Corollary 4.2] (see also [17, Lemma 2.2]). (ii) By using arguments similar to the above, we have that (1 + r)DJr (w) − In ≤

√ 2( 5 − 2) 2 √ =√ , 1 − 2( 5 − 2) 5

w ∈ Bn .

√ By [30, Corollary 2], we obtain that (1 + r)Jr ∈ S ∗ (Bn ). Finally, let c = 2( 5 − 2). Then c < 1/2, and from the ﬁrst part of the proof, we deduce that Jr extends to a quasiconformal homeomorphism of C n onto itself. This completes the proof. Example 4.17. Let f ∈ H(Bn ) be a normalized mapping. Assume that f satisﬁes the condition Df (z) − In ≤ k,

z ∈ Bn ,

for some k ∈ (0, 1), then f ∈ R(Bn ) ⊆ M(Bn ). Let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Then, we obtain the following results from Theorem 4.16 (i). (i) If k ∈ ( 12 , 1), then (1 + r)Jr is quasiconformal on Bn and extends to a quasiconformal homeomorphism of C n onto itself for 0 ≤ r < rk , where rk = 1/(2k − 1). (ii) If k ∈ (0, 12 ], then (1 + r)Jr is quasiconformal on Bn and extends to a quasiconformal homeomorphism of C n onto itself for all r ≥ 0. In particular, from Theorem 4.16, we obtain the following suﬃcient condition for quasiconformal extension to C n of the nonlinear resolvents of the family M(Bn ). Corollary 4.18. Let f ∈ M(Bn ) and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Also, let r > 0. If there exists a constant c ∈ (0, 14 ) such that 1+r c, Df (z)(u), u − 1 ≤ r

z ∈ Bn ,

u = 1,

then (1 + r)Jr is quasiconformal on Bn and extends to a quasiconformal homeomorphism of C n onto itself. Proof. Fix z ∈ Bn and let A = Df (z) − In . Then A ∈ L(C n ) and |V (A)| ≤ numerical radius of the operator A given by (see e.g. [28])

1+r r c,

where |V (A)| is the

|V (A)| = max{| A(u), u)| : u = 1}. Since A ≤ 2|V (A)| (see e.g. [13], [28]), we deduce that A ≤ 2c 1+r r . Hence, the result follows from Theorem 4.16 (i), as desired. Using arguments similar to those in the proof of Corollary 4.18, we obtain the following consequence of Theorem 4.16 (ii).

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Corollary 4.19. Let f ∈ M(Bn ) and let {Jr }r≥0 be the nonlinear resolvent family of the mapping f . Also, let r > 0. If (√5 − 2)(1 + r) , Df (z)(u), u − 1 ≤ r

z ∈ Bn ,

then (1 + r)Jr is starlike and quasiconformal on Bn , and extends to a quasiconformal homeomorphism of C n onto itself. 5. Coeﬃcient bounds for nonlinear resolvents of M(Bn ) In this section, we obtain various coeﬃcient bounds for nonlinear resolvents of certain subsets of M(B2). In the case n = 1, related results may be found in a recent work of Elin and Jacobzon [11]. Taking into account the proof of Proposition 4.11, we obtain the following sharp coeﬃcient bounds for the resolvent mapping Jr (r ≥ 0) of M(B2 ). Proposition 5.1. Let f ∈ M(B2 ) and let Jr = (Jr1 , Jr2 ) be the nonlinear resolvent mapping of f , for all r ≥ 0. Then the following estimates hold: √ 1 ∂2J 1 3 3r r (0) ≤ , 2 ∂w22 2(1 + r)3

r ≥ 0,

(5.1)

and 1

2r , D2 Jr (0)(u, u), u ≤ 2 (1 + r)3

u ∈ C 2 , u = 1, r ≥ 0.

(5.2)

These estimates are sharp. In addition, 1 8r , D2 Jr (0) ≤ 2 (1 + r)3

r ≥ 0.

(5.3)

√ 2 Proof. Since f ∈ M(B2 ), it follows that 12 ∂∂zf21 (0) ≤ 3 2 3 by [7]. Then the estimate (5.1) follows from 2 (4.11) and the above relation. To prove sharpness of (5.1), it suﬃces to consider the mapping f : B2 → C 2 be given by

√ 3 3 2 f(z) = z1 + z , z2 , 2 2

z = (z1 , z2 ) ∈ B2 .

(5.4) √

then 1 ∂ J2r (0) = − 3 3r 3 , in view Then f ∈ M(B2 ) and if {Jr }r≥0 is the nonlinear resolvent family of f, 2 ∂w2 2(1+r) of (4.10). On the other hand, since f ∈ M(B2 ), it follows that (see [16, Theorem 1.2]) 2

1

D2 f (0)(u, u), u ≤ 2, 2

∀ u ∈ C2,

1

u = 1.

Hence, the estimate (5.2) follows from the relation (4.11) and the above inequality. Sharpness of (5.2) follows by considering the mapping F : B2 → C 2 given by F (z) =

1 + z1 1 + z2 z1 , z2 1 − z1 1 − z2

,

z = (z1 , z2 ) ∈ B2 .

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Then F ∈ M(B2 ) and 12 D2 F (0)(e1 , e1 ), e1 = 2, where e1 = (1, 0). Next, let {Jr }r≥0 be the nonlinear resolvent family of F . Taking into account (4.11) and the above relation, we obtain that 1

2r . D2 Jr (0)(e1 , e1 ), e1 = 2 (1 + r)3 Finally, we prove the estimate (5.3). To this end, ﬁx r ≥ 0, and let Pr (u) = 12 D2 Jr (0)(u, u), for all u ∈ C 2 . Then Pr is a homogeneous polynomial of degree 2, and the following estimate holds (see e.g. [28] and [13]): Pr ≤ 4|V (Pr )|,

(5.5)

where Pr = max{Pr (w) : w = 1} and |V (Pr )| is the numerical radius of the polynomial Pr given by |V (Pr )| = max{| Pr (w), w| : w = 1}. Hence, in view of (5.2) and (5.5), we obtain that Pr (u) ≤

8r , (1 + r)3

u ∈ C 2 , u = 1, r ≥ 0.

Finally, let Ar : C 2 × C 2 → C 2 be given by Ar = 12 D2 Jr (0). Then Ar is a bilinear symmetric operator, and Ar = Pr , by [29, Theorem 4]. Therefore, the estimate (5.3) holds, as desired. This completes the proof. Next, we obtain examples of support points of the compact families generated by nonlinear resolvent mappings on B2 . Example 5.2. Fix T > 0, and let FT (B2 ) = (1 + T )JT [f ] : f ∈ M(B2 ) ,

(5.6)

where JT [f ] is the nonlinear resolvent mapping of f ∈ M(B2 ). Since M(B2 ) is a compact subset of H(B2 ), we obtain as in the proof of Proposition 4.4 that FT (B2 ) is also a compact subset of H(B2 ). 2 Let Λ : H(B2 ) → C be given by Λ(g) = − 12 ∂∂zg21 (0), g = (g1 , g2 ) ∈ H(B2 ). Then 2

√ 3 3T Λ(gT ) ≤ = Λ(GT ), 2(1 + T )2

∀ gT ∈ FT (B2 ),

where f˜ ∈ M(B2 ) is given by (5.4), GT = (1 + T )JT [f˜], and (1 + T )JT [f˜](z) = If f = idB2 , z ∈ B2 , then JT [f ](z) = Λ F (B2 ) = constant.

√ 3 3T 2 z1 − z , z2 , 2(1 + T )2 2

1 1+T

z = (z1 , z2 ) ∈ B2 .

(5.7)

z, z ∈ B2 , and Λ((1 + T )JT [f ]) = 0 < Λ(GT ). Hence,

T

In view of the above arguments, we deduce that GT ∈ supp FT (B2 ).

Question 5.3. Let T > 0 and let GT = (1 + T )JT [f˜] be given by (5.7). Also, let FT (B2 ) be the family given by (5.6). Then, is it true that GT ∈ ex FT (B2 )?

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Example 5.4. Let F : B2 → C 2 be given by F (z) =

√ 3 3 2 z , z2 , z1 − 8 2

z = (z1 , z2 ) ∈ B2 .

(5.8)

Then F ∈ supp F(B2 ), where F(B2 ) is the compact family given by (4.7), that is F(B2 ) = (1 + r)Jr [f ] : 0 ≤ r < +∞, f ∈ M(B2 ) . Proof. It is easily seen that F = G1 , where GT = (1 + T )JT [f˜] (T ≥ 0) is given by (5.7). Hence F ∈ F(B2 ). In view of (5.1), we have that √ √ 2 1 ∂ q1 ≤ 3 3r ≤ 3 3 , ∀ q = (q1 , q2 ) = (1 + r)Jr [f ] ∈ F(B2 ), r ≥ 0. (0) 2(1 + r)2 2 ∂w2 8 2 2

Next, let Λ : H(B2 ) → C be given by Λ(g) = − 12 ∂∂zg21 (0), g = (g1 , g2 ) ∈ H(B2 ). Then 2

√ 3 3 Λ(q) ≤ = Λ(F ), 8

∀ q ∈ F(B2 ),

where F is given by (5.8). Also, since Λ((1 +r)Jr [idB2 ]) = 0 < Λ(F ), it follows that Λ F(B2 ) = constant. In view of the above arguments, we deduce that F ∈ supp F(B2 ), as desired. This completes the proof. Question 5.5. Let F : B2 → C 2 be given by (5.8). Is it true that F ∈ ex F(B2 )? Next, we obtain the following sharp coeﬃcient bounds for the nonlinear resolvent mapping of a quasiconvex mapping f of type A on B2 (in particular, for f ∈ K(B2 )). We have Proposition 5.6. Let f : B2 → C 2 be a quasi-convex mapping of type A, and let Jr = (Jr1 , Jr2 ) be the nonlinear resolvent mapping of f , for all r ≥ 0. Then the following estimates hold: √ 1 ∂2J 1 3 3r r (0) ≤ , 2 ∂w22 4(1 + r)3

r ≥ 0,

(5.9)

and 1

r , D2 Jr (0)(u, u), u ≤ 2 (1 + r)3

u ∈ C 2 , u = 1, r ≥ 0.

(5.10)

These estimates are sharp. In addition, 1 4r , D2 Jr (0) ≤ 2 (1 + r)3

r ≥ 0.

(5.11)

2 Proof. Since f is quasi-convex of type A, it follows from (2.2) that g ∈ M(B g(z) = 2f (z) − z, √ ), 3where 2 g ∂ 1 2 [c] 2 1 z ∈ B . Then the shearing g of g also belongs to M(B ), and hence 2 ∂z2 (0) ≤ 2 3 , by [7]. Consequently, 2 √ 1 ∂2f 3 3 1 and this estimate is sharp (see also [24, Proposition 4.2]). Therefore, the relation (5.9) 2 (0) ≤ 2 ∂z2

4

follows from (4.11) and the previous inequality. On the other hand, since f is quasi-convex of type A, it follows that f is starlike of order 1/2 on B2 . Hence, in view of [16, Corollary 2.19], we obtain that

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26

1

D2 f (0)(u, u), u ≤ 1, 2

∀ u ∈ C2,

u = 1.

Thus, the relation (5.10) follows from (4.11) and the above inequality. Finally, we prove that the estimates (5.9) and (5.10) are sharp. To this end, let f : B2 → C 2 be given by f (z) =

√ 3 3 2 z1 + z , z2 , 4 2

z = (z1 , z2 ) ∈ B2 .

Then f is quasi-convex of type A, in view of [39, Example 9]. Hence, if {Jr }r≥0 is the nonlinear resolvent √

∂2J 1

3 3r family of f , then 12 ∂w2r (0) = − 4(1+r) 3 , by (4.10). To prove sharpness of (5.10), let f : B2 → C 2 be given by

f (z) =

z1 z2 , 1 − z1 1 − z1

,

z = (z1 , z2 ) ∈ B2 .

Then f ∈ K(B2 ) (see [25] and [39]), and thus f is also quasi-convex of type A on B2 . In this case, 1

D2 f (0)(e1 , e1 ), e1 = 1. 2 Let {Jr }r≥0 be the nonlinear resolvent family of f . Then from (4.11) and the above relation, we obtain that 1

r . D2 Jr (0)(e1 , e1 ), e1 = 2 (1 + r)3 Finally, the estimate (5.11) follows by using arguments similar to those in the proof of (5.3). This completes the proof. Using arguments similar to those in the above proofs, we obtain the following sharp coeﬃcient bounds for the nonlinear resolvent mapping Jr , r ≥ 0, of R(B2 ). Proposition 5.7. Let f ∈ R(B2 ), and let Jr = (Jr1 , Jr2 ) be the nonlinear resolvent mapping of f , for all r ≥ 0. Then the estimates (5.9) and (5.10) hold, and these estimates are sharp also for the nonlinear resolvent mappings of the family R(B2 ). Also, the estimate (5.11) holds for the nonlinear resolvent mappings of the family R(B2 ). Proof. Since f ∈ R(B2 ), we have that g ∈ M(B2 ), where g(z) = Df (z)(z), z ∈ B2 . Then the shearing g [c] √ 2 2 2 g ∂ 3 3 1 2 1 of g also belongs to M(B ), and hence 2 ∂z2 (0) ≤ 2 , by [7]. Since 12 ∂∂zg21 (0) = ∂∂zf21 (0), we have that 2 2 2 3√3 1 ∂2f 1 . Thus, the relation (5.9) follows from the above inequality and (4.11), as desired. 2 (0) ≤ 2 ∂z2

4

On the other hand, since g(z) = Df (z)(z), z ∈ B2 , we obtain by an elementary computation that 1 2 D g(0)(u, u) = D2 f (0)(u, u), 2

u ∈ C2.

Since g ∈ M(B2 ), it follows that 1

D2 g(0)(u, u), u ≤ 2, 2

u = 1,

1

D2 f (0)(u, u), u ≤ 1, 2

u = 1.

and thus

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27

Next, in view of (4.11) and the above inequality, we deduce (5.10), as desired. To prove sharpness of (5.9) for the family R(B2 ), let f : B2 → C 2 be given by √ 3 3 2 f (z) = z1 + z , z2 , 4 2

z = (z1 , z2 ) ∈ B2 .

Then f is a normalized holomorphic mapping on B2 , and it is clear that √ 3 3 2 Df (z)(z) = z1 + z , z2 ∈ M(B2 ), 2 2

z = (z1 , z2 ) ∈ B2 .

Hence f ∈ R(B2 ), and if {Jr }r≥0 is the nonlinear resolvent family of f , then 12 (4.10). To prove sharpness of (5.10) for the family R(B2 ), let f : B2 → C 2 be given by f (z) = − z1 − 2 log(1 − z1 ), −z2 − 2 log(1 − z2 ) ,

∂ 2 Jr1 ∂w2 (0)

√

3 3r = − 4(1+r) 3 , by

z = (z1 , z2 ) ∈ B2 ,

where we choose the branch of the logarithmic function log(1 − ζ) on U such that log(1 − ζ) ζ=0 = 0. Then f is a normalized holomorphic mapping on B2 , and it is easy to see that Df (z)(z), z > 0, z ∈ B2 \ {0}. Thus f ∈ R(B2 ). In this case, 1

D2 f (0)(e1 , e1 ), e1 = 1, 2 and if {Jr }r≥0 is the nonlinear resolvent family of f , then in view of (4.11) and the above relation, we obtain that 1

r . D2 Jr (0)(e1 , e1 ), e1 = 2 (1 + r)3 Finally, the estimate (5.11) follows by using arguments similar to those in the proof of (5.3). This completes the proof. Remark 5.8. From the proofs of Propositions 5.1, 5.6, and 5.7, we deduce that the sharp coeﬃcient bounds (5.2) and (5.10) hold in C n , for all n ≥ 2. Also, the coeﬃcient bounds (5.3) and (5.11) hold in C n , for all n ≥ 2. For example, if f ∈ M(Bn ), then we have that (see [16, Theorem 1.2]) 1

D2 f (0)(u, u), u ≤ 2, 2

u ∈ Cn,

u = 1.

Then using an argument similar to that in the proof of (5.2), based on the above inequality and the relation (4.11), we deduce that (5.2) remains true in higher dimension n ≥ 2. Sharpness follows by considering the mapping F ∈ M(Bn ) given by F (z) =

z1

1 + z1 1 + zn , . . . , zn 1 − z1 1 − zn

,

z = (z1 , . . . , zn ) ∈ Bn .

The proof of the extension of (5.10) to C n , n ≥ 2, is the same as in the case n = 2. Similar ideas apply in the case of the coeﬃcient bounds (5.3) and (5.11).

28

I. Graham et al. / J. Math. Anal. Appl. 491 (2020) 124289

Example 5.9. Let G : B2 → C 2 be given by √ 3 3 2 z , z2 , G(z) = z1 − 16 2

z = (z1 , z2 ) ∈ B2 .

Then G ∈ supp G(B2 ) \ supp F(B2 ), where G(B2 ) = (1 + r)Jr [f ] : f ∈ R(B2 ), r ≥ 0 and F(B2 ) is given by (4.7). Proof. In view of the relation (5.9) and Proposition 5.7, we deduce that G ∈ supp G(B2 ). By using an argument similar to that in the proof of Corollary 4.12, we obtain that G ∈ / supp F(B2 ), as desired. Remark 5.10. In the case n = 1, the authors in [11, Corollary 3.1] obtained the sharp coeﬃcient bound |a2 | ≤ 2a21 (1 − a1 ) for the nonlinear resolvent mapping Jr (w) = a1 w + a2 w2 + · · · of f ∈ M(U ), where 1 a1 = 1+r . Acknowledgments Some of the research for this paper was carried out in May, 2019, when Gabriela Kohr visited the Department of Mathematics of the University of Toronto. She expresses her gratitude to the members of this department, in particular to Professor John Bland, for useful advices and his hospitality during that visit. I. Graham was partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. H. Hamada was partially supported by JSPS KAKENHI Grant Number JP19K03553. The authors are very grateful to Toshiyuki Sugawa for helpful discussions during the preparation of this paper. The authors would like to thank the referee for a careful reading and useful suggestions which improved the paper. References [1] M. Abate, Iteration Theory of Holomorphic Maps on Taut Manifolds, Mediterranean Press, Rende, 1989. [2] D. Aharonov, M. Elin, S. Reich, D. Shoikhet, Parametric representation of semi-complete vector ﬁelds on the unit balls in C n and in Hilbert space, Rend. Mat. Acc. Lincei 10 (1999) 229–253. [3] E. Andersén, L. Lempert, On the group of holomorphic automorphisms of C n , Invent. Math. 110 (1992) 371–388. [4] L. Arosio, F. Bracci, H. Hamada, G. Kohr, An abstract approach to Loewner chains, J. Anal. Math. 119 (2013) 89–114. [5] L. Arosio, F. Bracci, F.E. Wold, Solving the Loewner PDE in complete hyperbolic starlike domains of C n , Adv. Math. 242 (2013) 209–216. [6] T. Betker, Löwner chains and quasiconformal extensions, Complex Var. 20 (1992) 107–111. [7] F. Bracci, Shearing process and an example of a bounded support function in S 0 (B2 ), Comput. Methods Funct. Theory 15 (2015) 151–157. [8] F. Bracci, M.D. Contreras, S. Diaz-Madrigal, Evolution families and the Loewner equation II: complex hyperbolic manifolds, Math. Ann. 344 (2009) 947–962. [9] F. Bracci, I. Graham, H. Hamada, G. Kohr, Variation of Loewner chains, extreme and support points in the class S 0 in higher dimensions, Constr. Approx. 43 (2016) 231–251. [10] P. Duren, I. Graham, H. Hamada, G. Kohr, Solutions for the generalized Loewner diﬀerential equation in several complex variables, Math. Ann. 347 (2010) 411–435. [11] M. Elin, F. Jacobzon, Coeﬃcient body for nonlinear resolvents, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 73 (2019) 46–57. [12] M. Elin, S. Reich, D. Shoikhet, Complex dynamical systems and the geometry of domains in Banach spaces, Diss. Math. 427 (2004) 1–62. [13] M. Elin, S. Reich, D. Shoikhet, Numerical Range of Holomorphic Mappings and Applications, Birkhäuser/Springer, Cham, 2019.

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