Support points for families of univalent mappings on bounded symmetric domains

Theorem 1.2. Let g : U →C satisfy the conditions of Assumption 1.1, g(ζ) = g(ζ) for all ζ ∈U, g′(0) < 0 and a0(g) = dist(1, ∂g(U)). Also, let f = (f1, f2) ∈ S0 g(B2). Then

Theorem 1.3. Let g : U →C satisfy the conditions of Assumption 1.1, g(ζ) = g(ζ) for all ζ ∈U, g′(0) < 0 and a0(g) = dist(1, ∂g(U)). Also, let f = (f1, f2) ∈ S0 g(U2). Then

Theorem 1.6. Let BX be the unit ball of an n-dimensional JB∗-triple X = (Cn, ∥· ∥X) of rank r ≥2. Let g : U →C satisfy the conditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). Also, let f = (f1, . . . , fn) ∈S0 g(BX). Then

Theorem 1.7. Let g : U →C satisfy the conditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). Also, let n ≥2 and let f = (f1, . . . , fn) ∈S0 g(Bn). Then

Theorem 1.8. Let B be the unit ball of Cn with respect to an arbitrary norm on Cn. Let g : U →C be a univalent holomorphic function such that g(0) = 1, g(ζ) = g(ζ), and ℜg(ζ) > 0, ζ ∈U. Assume that g(ρ) = O(1 −ρ) as ρ →1 −0. Then there exists an unbounded support point for S0 g(B). 2 Preliminaries First, we give the relation of a0(g) and dist(1, ∂g(U)) for functions g which satisfy the conditions of Assumption 1.1.

Proposition 2.1. Let g : U →C be a convex (univalent) function, which satisfies the conditions of Assumption 1.1. Also, let a0(g) be given by (1.1). Then we have a0(g) ≥dist(1, ∂g(U)).

Lemma 2.14. Let BX be a bounded symmetric domain realized as the open unit ball of a JB∗-triple X = (Cn, ∥· ∥), and let e be an arbitrary tripotent in X. Then we have |h0(x, e)| ≤∥x∥h0(e, e), x ∈X.

Lemma 2.16. For z = (z1, z2, 0′′) ∈X \ {0}, let l(1) z (w) = |z1| z1 w1, w ∈X, for z1 ̸= 0, and l(2) z (w) = |z2| z2 w2, w ∈X, for z2 ̸= 0. Then l(1) z ∈T (z) for |z1| = ∥z∥, and l(2)

Proposition 3.2. Let BX be as in Definition 3.1. Let g : U →C sat- isfy the conditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). If h = (h1, h2, . . . , hn) ∈Mg(BX), then

Lemma 2.16, we obtain that hk(z) zk = 1 + X α∈N2,|α|≥2 qk α zα zk ∈g(U), for |zk| = ∥z∥= ρ ∈(0, 1), k = 1, 2. Next, let η ∈[0, 2π) be such that q1 0,2 = |q1

Proposition 3.3. Let BX be as in Definition 3.1. Let g : U →C satisfy the conditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). Let hi,j[g] : BX → X be given by hi,j[g](z) = z ± d1(g)z2 j ei, z ∈BX, 1 ≤i ̸= j ≤r. Then hi,j[g] ∈Mg(BX) for 1 ≤i ̸= j ≤r. Also, if fi,j[g] = fi,j[g](z, t) : BX × [0, ∞) →X are given by fi,j[g](z, t) = etz ∓d1(g)z2 j ei
, z ∈BX,

Proposition 2.1 and Corollary 2.2] and [18], in the case of the Euclidean unit ball B2; see [14, Corollary 4.8] and [32, Lemma 5], in the case of the unit bidisc U2).

Corollary 3.4. Let BX be as in Definition 3.1. Let h = (h1, h2, . . . , hn) ∈ M(BX). Then h[c] i,j ∈M(BX) for 1 ≤i ̸= j ≤r, and the following estimates hold:

Corollary 3.5. Let BX be as in Definition 3.1. Let α ∈[0, 1) and let h = (h1, h2, . . . , hn) ∈Mα(BX). Then h[c] i,j ∈Mα(BX) for 1 ≤i ̸= j ≤r, and the following estimates hold:

Corollary 3.6. Let BX be as in Definition 3.1. Let α ∈[0, 1) and h ∈Mg(BX), where g(ζ) = 1−(1−2α)ζ 1+ζ , ζ ∈U. Then h[c] i,j ∈Mg(BX) for 1 ≤i ̸= j ≤r, and the following sharp estimates hold:

Corollary 3.7. Let BX be as in Definition 3.1. Let α ∈(0, 1] and let h ∈ Mg(BX), where g(ζ) =  1−ζ 1+ζ α , ζ ∈U. Then h[c] i,j ∈Mg(BX) for 1 ≤i ̸= j ≤r, and the following sharp estimates hold:

Corollary 3.8. Let BX be as in Definition 3.1. The following statements hold: (i) Let α ∈[0, 1) and Φα i,j : BX →X be given by Φα i,j(z) = z ± d1(α)z2 j ei, z = (z1, z2, . . . , zn) ∈BX, (3.4) for 1 ≤i ̸= j ≤r, where d1(α) is given by (3.3). Then Φα i,j ∈S∗ α(BX), and thus Φα i,j ∈S0 α(BX) for 1 ≤i ̸= j ≤r. (ii) Let α ∈[0, 1) and Ψα

Proposition 3.10. Let BX be as in Definition 3.1. Let g : U →C be a univalent holomorphic function with g(0) = 1 and ℜg(ζ) > 0 for ζ ∈U, and let h = (h1, . . . , hn) ∈Mg(BX). Then

Theorem 4.1. Let BX be as in Definition 3.1. Let g : U →C satisfy the conditions of Assumption 1.1. Also, let f(z, t) : BX × [0, ∞) →X be a g- Loewner chain. Then f [c] i,j(z, t) are also g-Loewner chains, for 1 ≤i ̸= j ≤r. In particular, if f ∈S0 g(BX), then f [c] i,j ∈S0 g(BX), for 1 ≤i ̸= j ≤r. Next, we prove the following sharp coefficient bounds for the family S0 g(BX), where g : U →C satisfies the conditions of Assumption 1.1 (compare [2, Theo- rem 3.1] and [18], in the case of the unit ball B

Theorem 4.2. Let BX be as in Definition 3.1. Let g : U →C satisfy the con- ditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). Let f = (f1, . . . , fn) ∈ S0 g(BX). Then

Corollary 4.3. Let BX be as in Definition 3.1. The following statements hold: (i) Let α ∈[0, 1) and let f = (f1, f2, . . . , fn) ∈S0 α(BX). Also, let d1(α) be given by (3.3). Then

Theorem 4.4. Let BX be as in Definition 3.1. Let g : U →C satisfy the conditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). Let Fi,j[g](z) = z + d1(g)z2 j ei, z ∈BX, (4.3) for 1 ≤i ̸= j ≤r. Then Fi,j[g] ∈S∗ g(BX) are bounded support points for S0 g(BX). In particular, Fi,j[g] are also bounded support points for S∗ g(BX), for 1 ≤i ̸= j ≤r.

Corollary 4.6. Let BX be as in Definition 3.1. The following statements hold: (i) Let α ∈[0, 1) and let Φα i,j be the mapping given by (3.4). Then Φα i,j are bounded support points for the family S0 α(BX), for 1 ≤i ̸= j ≤r. Moreover, Φα i,j are also bounded support points for the family S∗ α(BX), for 1 ≤i ̸= j ≤r. (ii) Let α ∈[0, 1) and let Ψα i,j be the mapping given by (3.5). Then Ψα i,j are bounded support points for the family AS0 α(BX) for 1 ≤i ̸= j ≤r. Moreover, Ψα

Theorem 5.7], [12, Theorem 2.14], [13, Theorem 10]; cf. [5], [32, Theorem 3], for g(ζ) = 1−ζ 1+ζ , ζ ∈U).

Theorem 4.9. Let BX be as in Definition 3.1. Let g : U →C be a univalent holomorphic function such that g(0) = 1 and ℜg(ζ) > 0, ζ ∈U. Also, let f ∈S0 g(BX). Then

Theorem 4.11. Let B be the unit ball of Cn with respect to an arbitrary norm on Cn. Let g : U →C be a univalent holomorphic function such that g(0) = 1, g(ζ) = g(ζ), and ℜg(ζ) > 0, ζ ∈U. Assume that g(ρ) = O(1 −ρ) as ρ →1 −0. Then there exists an unbounded support point for S0 g(B).

Proposition 5.1. Let g : U →C satisfy the conditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). Let n ≥2. (i) If h = (h1, h2, . . . , hn) ∈Mg(Bn), then

Corollary 5.2. Let n ≥2, α ∈[0, 1), and let h ∈Mg(Bn), where g(ζ) = 1−ζ 1+(1−2α)ζ , ζ ∈U. Then h[c] i,j ∈Mg(Bn) for 1 ≤i ̸= j ≤n, and the following sharp estimates hold:

Corollary 5.3. Let n ≥2, α ∈(0, 1], and let h ∈Mg(Bn), where g(ζ) =  1−ζ 1+ζ α , ζ ∈U, and the branch of the power function is chosen such that g(0) = 1. Then h[c] i,j ∈Mg(Bn) for 1 ≤i ̸= j ≤n, and the following sharp estimates hold:

Theorem 5.4. , Let g : U →C satisfy the conditions of Assumption 1.1 and let d1(g) = dist(1, ∂g(U)). Let n ≥2. (i) Let f = (f1, . . . , fn) ∈S0 g(Bn). Then

Corollary 5.5. Let n ≥2, α ∈[0, 1). (i) Let f ∈S0 α(Bn). Then

Corollary 5.6. Let n ≥2, α ∈(0, 1]. (i) Let f ∈SS0 α(Bn). Then