Abstract
The primary aim of this paper is to characterize the uniformly locally univalent harmonic mappings in the unit disk. Then, we obtain sharp distortion, growth and covering theorems for one parameter family ${\mathcal B}_{H}(λ)$ of uniformly locally univalent harmonic mappings. Finally, we show that the subclass of $k$-quasiconformal harmonic mappings in ${\mathcal B}_{H}(λ)$ and the class ${\mathcal B}_{H}(λ)$ are contained in the Hardy space of a specific exponent depending on the $λ$, respectiv
Results & Lemmas (15)
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Corollary 1.1.
Corollary 1.1. A locally univalent harmonic mapping f = h + g ∈H belongs to BH(λ) if and only if, for each pair of points z1, z2 in D and θ…
Corollary 1.1. A locally univalent harmonic mapping f = h + g ∈H belongs to BH(λ) if and only if, for each pair of points z1, z2 in D and θ ∈[0, 2π], |uθ(z1) −uθ(z2)| ≤2λdh(z1, z2), where uθ(z) = log h′(z) + eiθg′(z) . 2. Characterizations of Uniformly locally univalent harmonic mappings We now state our first result which is indeed a generalization of [24, Theorem 1] to the case of harmonic mappings.
Theorem 2.1.
Theorem 2.1. A harmonic mapping f = h + g is uniformly locally univalent in D if and only if ∥Tf∥< ∞. For the proof of the sufficiency of…
Theorem 2.1. A harmonic mapping f = h + g is uniformly locally univalent in D if and only if ∥Tf∥< ∞. For the proof of the sufficiency of Theorem 2.1, we need the following classical result due to Noshiro [19]. Lemma A. Let f(z) = z + P∞ k=2 akzk be analytic for |z| < R and |f ′(z)| < M for |z| < R. Then the disk |z| < R/M is mapped on a starlike domain with respect to the origin by f and also by all its polynomial sections fn(z) = z + Pn k=2 akzk (n = 2, 3, . . .).
Theorem 3.1
Theorem 3.1 (Distortion theorem). Let λ be a nonnegative real number and f(z) = h(z) + g(z) = P∞ n=1 anzn + P∞ n=1 bnzn ∈BH(λ). Then for z…
Theorem 3.1 (Distortion theorem). Let λ be a nonnegative real number and f(z) = h(z) + g(z) = P∞ n=1 anzn + P∞ n=1 bnzn ∈BH(λ). Then for z ∈D, we have λf(z) = |h′(z)| −|g′(z)| ≥|1 −|b1| | 1 −|z| 1 + |z| λ = |1 −|b1| |H′ λ(−|z|), Λf(z) = |h′(z)| + |g′(z)| ≤(1 + |b1|)
Corollary 3.1.
Corollary 3.1. For λ > 1, each f(z) = h(z) + g(z) = P∞ n=1 anzn + P∞ n=1 bnzn ∈ BH(λ) satisfies the growth condition f(z) = O (1 −|z|)1−λ…
Corollary 3.1. For λ > 1, each f(z) = h(z) + g(z) = P∞ n=1 anzn + P∞ n=1 bnzn ∈ BH(λ) satisfies the growth condition f(z) = O (1 −|z|)1−λ as |z| →1. On the other hand, for λ < 1, each mapping f ∈BH(λ) is bounded with the bound (1 + |b1|)Hλ(1). Moreover, if λ > 0 and f ∈S0 H ∩BH(λ) in D, then the image f(D) contains the disk {w : |w| < −Hλ(−1)}. By [4, 5], for λ ≤1/2, B(λ) ⊂S and so, by Theorem 2.1, for λ ≤1/2, f ∈BH(λ) must be univalent in D. We also note that, for 0 ≤λ ≤1, we have −Hλ(−1) ≥−H1
Theorem 3.2.
Theorem 3.2. If a harmonic mapping f = h + g in D satisfies the condition lim |z|→1− (1 −|z|2)
Theorem 3.2. If a harmonic mapping f = h + g in D satisfies the condition lim |z|→1− (1 −|z|2)
Theorem 3.3.
Theorem 3.3. Let 0 ≤λ < 1. Then each mapping f ∈BH(λ) is H¨older continuous of exponent 1 −λ in D. The proof follows from Theorem 2.1 and…
Theorem 3.3. Let 0 ≤λ < 1. Then each mapping f ∈BH(λ) is H¨older continuous of exponent 1 −λ in D. The proof follows from Theorem 2.1 and [15, Theorem 2.6] and so, we omit its detail. 4. The space BH(λ) and the Hardy space We begin this section with the following concepts. Definition 4.1. For 0 < p < ∞, the Hardy space Hp is the set of all functions f analytic in D for which Mp(r, f) = 1 2π Z 2π 0 |f(reiθ)|p dθ 1/p
Theorem 4.1.
Theorem 4.1. (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 < 1/(λ…
Theorem 4.1. (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 < 1/(λ −1). For the proof of Theorem 4.1, we need some preparation.
Lemma 4.1.
Lemma 4.1. If f = h + g ∈BH(1), then ∥f∥BH ≤4(1 + |b1|).
Lemma 4.1. If f = h + g ∈BH(1), then ∥f∥BH ≤4(1 + |b1|).
Lemma 4.2.
Lemma 4.2. Let f ∈SHk and p > 0. Then Ip(r) ≤2(1 + k2)(|p −2| + 1) 1 −k2 Z r 0 M(ρ)pρ−1 dρ (0 ≤r < 1), where M(r):= M(r, f) = max 0≤θ≤2π…
Lemma 4.2. Let f ∈SHk and p > 0. Then Ip(r) ≤2(1 + k2)(|p −2| + 1) 1 −k2 Z r 0 M(ρ)pρ−1 dρ (0 ≤r < 1), where M(r) := M(r, f) = max 0≤θ≤2π |f(reiθ)|.
Corollary 3.1
Corollary 3.1, f is bounded. Next we assume that f = h + g ∈BH(1) ∩SH. Then, by Lemma 4.1, it follows that f is Bloch and thus, h is Bloch,…
Corollary 3.1, f is bounded. Next we assume that f = h + g ∈BH(1) ∩SH. Then, by Lemma 4.1, it follows that f is Bloch and thus, h is Bloch, since, for f = h + g ∈SH, h is Bloch if and only if h is BMOA if and only if f is BMOH (cf. [1]). Consequently, f ∈BMOH. Finally, we assume that f ∈BH(λ) ∩SHk for some λ > 1. Then, by Theorem 3.1, we deduce that f(z) = O((1 −|z|)1−λ) and thus, M(r) = O((1 −|z|)1−λ). Furthermore, using Lemma 4.2, we find that Ip(r) ≤ 2(1 + k2)(|p −2| + 1) 1 −k2 Z r 0 M(ρ)pρ−1
Theorem 4.2.
Theorem 4.2. Let λ ≥1. Then BH(λ) ⊂hp with 0 < p < 1 λ2−1. In the above, the expression 1 λ2−1 is interpreted as ∞when λ = 1.
Theorem 4.2. Let λ ≥1. Then BH(λ) ⊂hp with 0 < p < 1 λ2−1. In the above, the expression 1 λ2−1 is interpreted as ∞when λ = 1.
Corollary 4.1. · coeff
Corollary 4.1. A uniformly locally univalent harmonic mapping f in D is contained in the Hardy space hp for some p = p(f) > 0. In [14], Kim…
Corollary 4.1. A uniformly locally univalent harmonic mapping f in D is contained in the Hardy space hp for some p = p(f) > 0. In [14], Kim also conjectured that the assertion (3) in Theorem B holds for B(λ). 5. Coefficient estimates for the class BH(λ) Let f(z) = h(z) + g(z) = P∞ n=1 anzn + P∞ n=1 bnzn with a1 = 1 and b1 = 0. If f ∈BH(λ), then by Theorem 2.1, for each θ ∈[0, 2π],
Theorem 5.1.
Theorem 5.1. Let f(z) = h(z) + g(z) = P∞ n=1 anzn + P∞ n=1 bnzn ∈BH(λ) with a1 = 1 and b1 = 0. Then, for each ε > 0, a real number p and…
Theorem 5.1. Let f(z) = h(z) + g(z) = P∞ n=1 anzn + P∞ n=1 bnzn ∈BH(λ) with a1 = 1 and b1 = 0. Then, for each ε > 0, a real number p and uniformly for θ ∈[0, 2π], we have Ip(r, h′ + eiθg′) = O (1 −r)α(|p|λ)−ε , and thus, Ip(r, f) = O (1 −r)−α(|p|λ)−ε , |an| + |bn| = O nα(λ)−1+ε , where α(λ) =
Theorem 5.2.
Theorem 5.2. For each λ ∈(0, ∞), we have max λ −1, 0 ≤γ(BH(λ)) ≤α(λ), where α(λ) = √ 1+4λ2−1 2. In particular, γ(BH(λ)) = O(λ2) as λ →0.…
Theorem 5.2. For each λ ∈(0, ∞), we have max{λ −1, 0} ≤γ(BH(λ)) ≤α(λ), where α(λ) = √ 1+4λ2−1 2 . In particular, γ(BH(λ)) = O(λ2) as λ →0. Now we mention a connection with integral means for univalent analytic functions. For a univalent harmonic mapping f ∈SH and a real number p, we let βf,θ(p) = lim r→1− log Ip(r, h′ + eiθg′) log 1 1−r
Theorem 5.3.
Theorem 5.3. For f ∈BH(λ) and a real number p, βf, θ(p) ≤α(|p|λ) = p 1 + 4p2λ2 −1 2 holds for each θ ∈[0, 2π]. In particular, the Brennan…
Theorem 5.3. For f ∈BH(λ) and a real number p, βf, θ(p) ≤α(|p|λ) = p 1 + 4p2λ2 −1 2 holds for each θ ∈[0, 2π]. In particular, the Brennan conjecture is true for univalent functions f with ∥Tf∥≤ √ 2.
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