Boundary subordination

Theorem 2.1 ([P2, p. 79]). A function f ∈A has a finite angular derivative f′ ∠(ζ) at ζ ∈T if and only if f′ has a finite angular limit at ζ. Moreover f′ ∠(ζ) = ∠f′(ζ). 3. Julia functions 3.1. Basic facts. Denote by B the class of functions ω ∈A with |ω(z)| < 1 for z ∈D and by P the class of functions p ∈A with Re p(z) > 0 for z ∈D.

Proposition 3.3. Fix a ∈H. 1. If ω ∈B1, then p = ha ◦ω ∈P0. 2. If p ∈P0, then ω = h−1 a ◦p ∈B1. In particular, for a = 1: 3. If ω ∈B1, then (3.1) p = h1 ◦ω = 1 −ω 1 + ω ∈P0. 4. If p ∈P0, then (3.2) ω = h−1 1 ◦p = 1 −p

Theorem 3.4. Let ω ∈B1. 1. The angular derivative ω′ ∠(1) exists and 0 < ω′ ∠(1) = ∠lim z→1 1 −ω(z) 1 −z = lim r→1− 1 −ω(r) 1 −r = lim r→1− 1 −|ω(r)|

Theorem 3.5. Let p ∈P0. 1. The angular derivative p′ ∠(1) exists and −∞≤p′ ∠(1) = ∠lim z→1 p(z) z −1 < 0.

Proposition 3.7. Let a ∈H and λ ∈(0, ∞]. 1. If ω ∈B1(λ), then p = ha ◦ω ∈P0(λ|a|2/Re a). 2. If p ∈P0(λ), then ω = h−1 1 ◦p ∈B1(λ Re a/|a|2). In particular, for a = 1: 3. If ω ∈B1(λ), then p = h1 ◦ω ∈P0(λ). 4. If p ∈P0(λ), then ω = h−1 1 ◦p ∈B1(λ). Observe that from Theorems 3.4 and 3.5 we have

Theorem 3.8. Let λ ∈(0, ∞). 1. If ω ∈B1(λ), then, for every k > 0, (3.4) ω(Ok) ⊂Oλk. 2. If ω ∈P0(λ), then, for every k > 0, (3.5) p(Ok) ⊂D(λk/2, λk/2). 3.2. Julia functions and Schwarz functions. Now we describe some relations for functions which satisfy the Schwarz and the Julia lemmas. First we distinguish some sets of functions in B.

Lemma 3.10. Let n ∈N. If ω ∈B(n) 0 , then (3.6) |ω(z)| ≤|z|n, z ∈D, and (3.7) 1 n!|ω(n)(0)| ≤1. Equality in (3.6) for some z ̸= 0 or in (3.7) can occur only for ω(z) = κzn, z ∈D, where κ ∈T. The following lemma can be found in [P2, p. 84].

Lemma 3.11. If ω ∈B, then ω has at most one attractive fixed point in D, i.e. a point ξ ∈D such that • ω(ξ) = ξ and |ω′(ξ)| < 1 when ξ ∈D, • ω∠(ξ) = ξ and |ω′ ∠(ξ)| < 1 when ξ ∈T. The above lemma implies at once

Lemma 3.12. If ω ∈B1(λ), λ ∈(0, 1), then ω has no fixed point in D.

Lemma 3.11 also yields the following result for Schwarz functions in the class B1.

Lemma 3.13. If ω ∈B0,1, then ω ∈B1(λ) for some λ ≥1. The last result can be improved:

Lemma 3.14. Let n ∈N. If ω ∈B(n) 0,1 , then (3.8) n ≤ω′ ∠(1) ≤∞, and ω ∈B1(λ) for some λ ≥n.

Lemma 3.16. Let n ∈N. Let f be analytic in the disk Dr, r > 0, with (3.9) f(0) = f′(0) = · · · = f(n−1)(0) = 0, f(n)(0) ̸= 0. Assume that at z0 ∈∂Dr the limit f∠(z0) exists and is finite with (3.10) |f∠(z0)| = sup{|f(z)| : z ∈Dr}. Then f′ ∠(z0) exists and (3.11) z0f′ ∠(z0) f∠(z0) = λ, where

Theorem 3.18. Let q ∈S and assume that q′(ζ0) ̸= 0 at ζ0 ∈∂D exists. Let n ∈N and p be a function analytic in the disk Dr, r > 0, with p ̸≡p(0), p′(0) = · · · = p(n−1)(0) = 0, p(n)(0) ̸= 0, such that p(0) = q(0) and p(Dr) ⊂q(D). Assume that at z0 ∈∂Dr the limit (3.17) p(z0) = lim z→z0 p(z) exists and is finite, and (3.18) p(z0) = q(ζ0). If p′ ∠(z0) exists and is finite, then

Theorem 4.2. Let f, F ∈A satisfy (4.1). If f ⋞F, then f(D) ⊂F(D).

Theorem 4.3. Let λ ∈(0, ∞) and f, F ∈A satisfy (4.1). If f ⋞λ F, then f(Ok) ⊂F(Oλk) for every k > 0.

Proposition 4.4. p ∈P0 if and only if p(z) ⋞1 −z 1 + z , z ∈D. 4.2. Boundary subordination of univalent functions. If F is uni- valent we can obtain additional properties of boundary subordination.

Theorem 4.5. Let λ ∈(0, 1), f ∈A and F ∈S. If f ⋞λ F, then f(z) ̸= F(z) for every z ∈D.

Theorem 4.7. Let f ∈A and F ∈S satisfy (4.1). If F(D) is a Jordan domain and f(D) ⊂F(D), then f ⋞λ F for some λ ∈(0, ∞].

Theorem 4.8. Let f, F ∈A. Let F ′ be normal in D and let f ⋞λ F for some λ ∈(0, ∞). • If f′ ∠(1) exists and is finite, then F ′ ∠(1) exists and is finite. • If f′ ∠(1) = 0, then F ′ ∠(1) = 0.

Theorem 4.10. Let f ∈A. Then the following conditions are equivalent: (1) f ∈S∗ 0 and f∠(1) = 0. (2) There exists λ ∈(0, 1] such that (4.2) −(1 −z)2 f′(z) f(z) ⋞λ 41 −z 1 + z , z ∈D. Multivalent starlike functions with respect to a boundary point were con- sidered in [ESZ1], where the main result can be formulated now as follows.

Theorem 4.11. Let λ ∈(0, ∞). If f ∈A satisfies (4.2), then (1) f(D) ∈Z∗; (2) f is p-valent function if and only if p −1 < λ ≤p. 4.3.2. Functions convex in the positive direction of the real axis. The class of functions convex in the direction of the imaginary axis was in- troduced by Robertson [R1]. The classes of functions convex in the posi- tive (negative) direction of the imaginary (real) axis as the subclasses of Robertson’s class were distinguished by Hengartner and Schober [HS] who propose

Theorem 4.13. Let f ∈A. Then the following conditions are equivalent: (1) f ∈CR+. (2) There exists λ ∈(0, ∞] such that (4.4) (1 −z)2f′(z) ⋞λ 41 −z 1 + z , z ∈D. Elin and Shoikhet [ES] studied the problem of finding the horizontal strip of minimal width containing f(D) and of maximal width lying in f(D) when f ∈CR+. They introduced the following subclasses in CR+.

Theorem 2 of [BL] using boundary subordination.

Theorem 4.15. Let 0 < ν ≤λ < ∞and let f ∈A. Then f ∈CR+(λ, ν) if and only if (4.4) holds and there exists ζ = eiφ ̸= 1 such that ∠lim z→ζ 1 −ζz 1 + ω(z) = ν sin2 φ 2 .

Definition 3.1. Let B1 = {ω ∈B : ω∠(1) = 1}. Let us call functions in the class B1 Julia functions. Similarly, let P0 = {p ∈P : p∠(1) = 0}.

Definition 3.6. For λ ∈(0, ∞] let B1(λ) = {ω ∈B1 : ω′ ∠(1) = λ}, P0(λ) = {p ∈P0 : p′ ∠(1) = −λ/2}. Clearly, B1 = [ λ∈(0,∞] B1(λ),

Definition 3.9. Let B0 = {ω ∈B : ω(0) = 0}. Functions in the class B0 are called Schwarz functions. For n ∈N let B(n) 0 = {ω ∈B0 : ω′(0) = · · · = ω(n−1)(0) = 0, ω(n)(0) ̸= 0}. Let B0,1 = B0 ∩B1. For each n ∈N let B(n)

Definition 4.1. Let f, F ∈A and suppose that f∠(1) and F∠(1) exist with (4.1) f∠(1) = F∠(1). • We say that f is boundary subordinated to F if there exists ω ∈B1 such that f = F ◦ω in D. We then write f ⋞F. • Let λ ∈(0, ∞]. We say that f is λ-boundary subordinated to F if there exists ω ∈B1(λ) such that f = F ◦ω in D. We then write f ⋞λ F. As an immediate consequence of Definition 4.1 we have

Definition 4.9. A simply connected domain Ω⊂C, Ω̸= C, with 0 ∈∂Ωis called starlike with respect to the boundary point (at the origin) if for every w ∈Ω, (0, w] = {tw : t ∈(0, 1]} ⊂Ω. The class of all such domains will be denoted by Z∗. Let S∗ 0 ⊂S be the class of all functions f such that 0 ∈∂f(D) and f(D) ∈Z∗ 0. Functions belonging to S∗ 0 will be called starlike with respect to

Definition 4.12. A simply connected domain Ω⊂C, Ω̸= C, is called convex in the positive direction of the real axis if {w + t : t ≥0} ⊂Ω for every w ∈Ω.

Definition 4.14. Let 0 < ν ≤λ ≤∞, with ν < λ when λ = ∞. A function f ∈CR+ belongs to CR+(λ, ν) if • f(D) lies in a horizontal strip of minimal width 2λπ; • f(D) contains a horizontal strip of maximal width 2νπ. Some properties of functions in CR+(λ, ν) were demonstrated in [ES]. An analytical characterization of CR+(λ, ν) was given in [BL]. We rewrite