Uniformly locally univalent harmonic mappings associated with the pre-Schwarzian norm

Theorem 2.1. Let f = h+ g be a sense-preserving harmonic mapping in D. Then either ||Ph+εg|| = ||Pf|| = ∞or both ||Ph+εg|| and ||Pf|| are finite for each ε ∈D. If ||Pf|| < ∞, then the inequality (2.1) ||Ph+εg|| −||Pf|| ≤1 holds for each ε ∈D. In particular, ||Ph|| −||Pf|| ≤1. The constant 1 is sharp in the two estimates.

Corollary 2.1. Let f = h + g be a sense-preserving harmonic mapping in D. If ||Ph|| < ∞, then for any ε1, ε2 ∈D, we have the following inequalities. (1) The sharp inequality ||Ph+ε1g|| −||Ph+ε2g|| ≤2 holds. (2) If |ε1| = |ε2|, then ||Ph+ε1g|| = ||Ph+ε2g||. If |ε1| ̸= |ε2|, then ||Ph+ε1g|| −||Ph+ε2g|| ≤|ε1| + |ε2| < 2. (3) If |ε1| ≤|ε2|, then we have the sharp inequality ||Ph+ε1g|| −||Ph+ε2g|| ≤1. If |ε1| > |ε2|, then ||Ph+ε1g|| −||Ph+ε2g|| ≤1 + |ε1| + |ε2| < 3.

Corollary 2.2. Let f = h+g be a sense-preserving harmonic mapping in D. If h+ε1g is univalent (resp. convex) in D for some ε1 ∈D, then ||Ph+εg|| < 9 (resp. 7) and ||Ph+εg|| ≤ 8 (resp. 6) for each ε ∈D. Furthermore, the constants 8 and 6 are sharp. Conversely, if either ||Ph+ε1g|| ≥9 (resp. 7) or ||Ph+ε2g|| > 8 (resp. 6) for some ε1, ε2 ∈D, then h + εg is not univalent (resp. convex) in D for any ε ∈D. The harmonic Koebe function K = hK +gK and the harmonic half-plane mapping L = hL +gL still sho

Theorem 3.1. Let f = h + g be a sense-preserving and ULU harmonic mapping in D. Then we have (3.1) (1 −|z|2)|Ph(z)| ≤2(α/t + |z|) and (1 −|z|2)|Pf(z)| ≤2(α0/t + |z|) for every z ∈D, where t =    eρ(f) −1 eρ(f) + 1 if ρ(f) < ∞, 1

Theorem 3.2. Let f = h+ g be a sense-preserving harmonic mapping in D. If f is SHU (resp. SHC), then we have ||Ph+εg|| ≤6 (resp. 4) and ||Ph+εg|| ≤6 (resp. 4) for each ε ∈D. All estimates are sharp.

Theorem 3.3. Let f = h + g be a sense-preserving harmonic mapping in D. If ||Pf|| ≤ M, then f is univalent in the hyperbolic disk Dh(z, t) for each z ∈D. Consequently, f is ULU in D and its uniformly hyperbolic radius ρ(f) is no less than t. Here t = 2 tanh−1 (1/(8(M + 1))).

Lemma 4.1. ([17, p. 44] and [40, Theorem 2]) Let f be a locally univalent analytic function in D. Then the following are equivalent.

Theorem 4.1. (Equivalent conditions) Let f = h + g be a sense-preserving harmonic mapping in D. Then the following conditions are equivalent. (1) h + g is SAULU; (2) h + g is SAULC; (3) h + g has SBAPSN; (4) h + g has SBASN; (5) For any two points ε1, ε2 ∈D with ε1 ̸= ε2, there exists a constant m1 > 0, and a univalent analytic function F1 such that (h + ε1g)′ = (F ′ 1)m1 if and only if there exists a constant m2 > 0, and a univalent analytic function F2 such that (h + ε2g)′ = (F ′ 2)m2; (6) f i

Theorem 3.3, we see that for each z ∈D, h + λg is convex in Dh(z, (2 − √ 3)t) for every |λ| = 1 by the classical result on the radius of convexity (see [17, p. 44]), where t = 2 tanh−1(1/(8(M + 2))). It follows from [20, Theorem 3.1] that f is convex in the hyperbolic disk Dh(z, (2 − √ 3)t) for each z ∈D, which means that f is ULC in D. Again, by the bridge (AB), we prove that (6) ⇔(7) ⇔(8) ⇔(9). This completes the proof. □ Remarks. In the remarks below, let f = h + g be sense-preserving in D. (

Proposition 5.1. For the functions Ha,b and Ha defined by (5.2), we have the following properties: (1) ∥PHa,b∥= 2 max{|a|, |b|}. Thus, if max{|a|, |b|} ≤1/2, then the functions Ha,b are univalent in D. If max{|a|, |b|} > 3, then the functions Ha,b are not univalent in D. (2) If min{|a|, |b|} + |a −b| ≤1, then the functions Ha,b are close-to-convex and univalent in D. (3) If a ≤0 ≤b ≤a + 3, then the functions Ha,b are convex in one direction and univalent in D. Furthermore, if a ≤0 ≤b ≤a + 2, then

Proposition 5.2. For all θ ∈R, the family of harmonic mappings Fa,b,θ defined by (5.1) has the following properties:

Theorem 5.1. The following conditions are equivalent. (1) λ ≥1; (2) There exists a ω ∈A0(λ) with |ω′(0)| = 1; (3) The set {µ · I : |µ| = 1} is contained in A0(λ); (4) Every automorphism σ of the unit disk is an admissible dilatation in BH(λ).

Proposition 6.1. A harmonic mapping f ∈H belongs to BH(λ) if and only if for each pair of points z, z0 in D, the inequality |A(z) −A(z0)| ≤λdh(z, z0) holds, where A(z) = log Jf(z).

Theorem 6.1. (Distortion theorem) Let f = h + g ∈BH(λ) for some λ ≥0 with b1 = g′(0), and let Ha,b and Ha be defined by (5.2). Then for each z ∈D, we have (1) (1 −|b1|2)H′ λ(−|z|) ≤Jf(z) ≤(1 −|b1|2)H′ λ(|z|); (2) p 1 −|b1|2H′ λ 2 (−|z|) ≤|h′(z)| ≤(1 + |b1z|)H′ λ−1 2 , λ+1 2 (|z|); (3) |g′(z)| ≤(|z| + |b1|)H′ λ−1

Proposition 5.2. Similarly, for each λ ≥0, the function H λ 2 provides the sharpness for the left sides of (2) and (4) at z = −r ∈(−1, 0] when f ∈B0 H(λ). It follows from Proposition 5.2 that F λ−1 2 , λ+1 2 ,0 is univalent in D for λ = 1. The equality in the left side of (6) occurs for f = F0,1,0 −|b1|F0,1,0 ∈BH(1) and z = −r ∈(−1, 0]. We complete the proof. □

Corollary 6.1. (Growth and covering theorem) Let f = h + g ∈BH(λ) with b1 = g′(0), and let Ha,b and Ha be defined by (5.2). If λ > 1, then f, h and g satisfy the same growth condition f(z) (h(z), g(z)) = O(1 −|z|) 1−λ 2 as |z| →1. If λ < 1, then f (resp. h, g) is bounded by (1+|b1|)H λ+1 2 (1) (resp. (1−|b1|)H λ−1 2 , λ+1 2 (1)+|b1|H λ+1 2 (1), H λ+1 2 (1)−(1−|b1|)H λ−1 2 , λ+1

Proposition 6.2. Let f = h + g be a sense-preserving harmonic mapping in D. If f satisfies the condition β(f) := lim |z|→1−((1 −|z|2)|Pf(z)| −1) log 1 1 −|z|2 < −2, then f, h and g are bounded in D.

Theorem 6.2. Let f = h + g ∈BH(λ) for some λ ∈[0, 1). Then h + εg is H¨older continuous of exponent 1−λ 2 in D for each ε ∈D. Moreover, f is H¨older continuous of exponent 1−λ 2 in D. 7. Coefficient estimates for the class BH(λ) Throughout the section we consider f = h + g ∈BH, where h(z) = ∞ X n=1 anzn and g(z) = ∞

Theorem 7.1. If f ∈BH(λ), then we have (7.1) |a2| ≤1 2 min  (1 −|b1|2)λ + 2|b1b2|, min ε∈D {|1 + εb1|(λ + 1) + 2|εb2|}  . If f ∈B0 H(λ), then |a2| ≤λ/2 and the estimate is sharp for all λ > 0.

Theorem 7.2. Let f = h + g ∈BH(λ). Then, for any a > 0 and a real number p, we have (7.2) Ip(r, h′ + εg′) = O (1 −r)−α(|p|(λ+1)/2)−a
, for each ε ∈D and thus, in particular, |an| + |bn| = O nα((λ+1)/2)−1+a
. For p > 0, we get that (7.3) Ip(r, f) = O (1 −r)p−α(|p|(λ+1)/2)−a
.

Theorem 7.3. Let f = h + g ∈BH(λ) for some λ with 1.982 < λ ≤5. If there exists a constant ε ∈D such that h+εg is univalent in D, then |an+εbn| = O n(λ−3)/2
as n →∞. In particular, if h is univalent in D, then |an| = O n(λ−3)/2
as n →∞. Moreover, if h + εg is univalent in D for every |ε| = 1, then |an| + |bn| = O n(λ−3)/2
as n →∞. The three estimates are sharp.

Theorem 7.4. For each λ ∈(0, ∞) and ε ∈D, we have max{(λ −1)/2, 0} ≤γ(BH(λ)) (γ(AH(λ, ε))) ≤α((λ + 1)/2), where α(λ) = √ 1+4λ2−1 2 . In particular, γ(BH(λ)) = O((λ + 1)2) and γ(AH(λ, ε)) = O((λ + 1)2) as λ →0. We continue the discussion by mentioning a connection with integral means for univalent analytic functions. For a univalent harmonic mapping f = h+g ∈SH, a complex number ε ∈D and a real number p, we let βfε(p) = lim r→1−

Theorem 7.5. For f ∈BH(λ) and a real number p, βfε(p) ≤α(|p|(λ + 1)/2) = p 1 + p2(1 + λ)2 −1 2 holds for each ε ∈D. In particular, the Brennan conjecture is true for every univalent harmonic mapping f with ∥Pf∥≤ √ 2 −1. 8. The space BH(λ) and the Hardy space For a harmonic mapping f = h + g in D, the Bloch seminorm is given by (see [14]) ∥f∥BH = sup z∈D (1 −|z|2) |h′(z)| + |g′(z)|

Theorem 8.1. (1) If λ < 1, then BH(λ) ∩SH ⊂h∞. (2) If λ = 1, then BH(λ) ∩SH ⊂BMOH. (3) If λ > 1, then BH(λ) ∩SHK ⊂hp for every 0 < p < 2/(λ −1), where K ≥1 and SHK = {f = h + g ∈SH : f is K-quasiconformal}.

Theorem 8.2. Let λ ≥1. Then BH(λ) ⊂hp with 0 < p < p0(λ) = 4 (λ+3)(λ−1), where p0(λ) = ∞if λ = 1.

Theorem 7.2.

Corollary 8.1. A uniformly locally univalent harmonic mapping f in D is contained in the Hardy space hp for some p = p(f) > 0. 9. Subordination principles for the estimate of PSN In this section, AD denotes the class of analytic functions φ from D into itself and A0 D denotes the subclass of AD with the normalization φ(0) = 0. If f and F are restricted to be analytic, then we say that f is said to be subordinate (resp. weakly subordinate) to F (written f ≺F (resp. f ⪯F)) if there exists a functi

Theorem 9.1. (Subordination principle I) Let f = h + g be a harmonic mapping in D and F = H + G ∈BH. If h′ + g′ ⪯H′ + G′, then we have ||Pf|| ≤||PF||. In this case, f is ULU in D.

Theorem 9.2. (Subordination principle II) Let f = h+g be a sense-preserving harmonic mapping in D and F = H + G ∈BH such that h′ ⪯H′. Then we have ||Pf|| ≤||PF|| + 2. Thus, f is ULU in D.