Abstract
We consider the class univalent log-harmonic mappings on the unit disk. Firstly, we obtain necessary and sufficient conditions for a complex-valued continuous function to be starlike or convex in the unit disk. Then we present a general idea, for example, to construct log-harmonic Koebe mapping, log-harmonic right half-plane mapping and log-harmonic two-slits mapping and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike mappings ar
Results & Lemmas (8)
Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.
Proposition 1.1.
Proposition 1.1. Let f(z) = ϕ(z)|g(z)|2 be a complex-valued function on D, where ϕ, g ∈A and ϕ(z), g(z) ̸= 0 in D 0. Then f ∈FS∗(α) if and…
Proposition 1.1. Let f(z) = ϕ(z)|g(z)|2 be a complex-valued function on D, where ϕ, g ∈A and ϕ(z), g(z) ̸= 0 in D\{0}. Then f ∈FS∗(α) if and only if ϕ ∈FS∗(α).
Theorem 1.2.
Theorem 1.2. Let f(z) = ϕ(z)|z|2(p−1) (p ≥1). Then f ∈FC(α) if and only if ϕ ∈FC(α).
Theorem 1.2. Let f(z) = ϕ(z)|z|2(p−1) (p ≥1). Then f ∈FC(α) if and only if ϕ ∈FC(α).
Lemma 3.1.
Lemma 3.1. ([11, Theorem 6.4]) Let s1(z) ≺s2(z), where s1(0) = s′ 1(0) −1 = 0 and s2(0) = s′ 2(0) −1 = 0. Then we have the following: (1)…
Lemma 3.1. ([11, Theorem 6.4]) Let s1(z) ≺s2(z), where s1(0) = s′ 1(0) −1 = 0 and s2(0) = s′ 2(0) −1 = 0. Then we have the following: (1) if s2 ∈C, then |an| ≤1 for n = 2, 3, · · · ; (2) if s2 ∈S∗, then |an| ≤n for n = 2, 3, · · · .
Theorem 3.2.
Theorem 3.2. Let f(z) = zh(z)g(z) belong to S∗ Lh(α) (0 ≤α < 1), where h(z) and g(z) are given by (1.2). Then |an −bn| ≤2(1 −α) n for n ≥1.…
Theorem 3.2. Let f(z) = zh(z)g(z) belong to S∗ Lh(α) (0 ≤α < 1), where h(z) and g(z) are given by (1.2). Then |an −bn| ≤2(1 −α) n for n ≥1. Equality holds if f(z) = fα(z) or one of its rotation, where where fα(z) is given by (2.5).
Corollary 3.3.
Corollary 3.3. Let f(z) = zh(z)g(z) be an element of S∗ Lh, where h(z) and g(z) are given by (1.2). Then |an −bn| ≤2 n for n ≥1. Equality…
Corollary 3.3. Let f(z) = zh(z)g(z) be an element of S∗ Lh, where h(z) and g(z) are given by (1.2). Then |an −bn| ≤2 n for n ≥1. Equality holds if f(z) = f0(z) or one of its rotation, where f0(z) is given by (2.7).
Theorem 3.4.
Theorem 3.4. Let f(z) = zh(z)g(z) be a log-harmonic mapping in D, where h(z) and g(z) are given by (1.2), and satisfy the condition ∞ X n=1…
Theorem 3.4. Let f(z) = zh(z)g(z) be a log-harmonic mapping in D, where h(z) and g(z) are given by (1.2), and satisfy the condition ∞ X n=1 n |an −bn| ≤1 −α (3.1) for some α ∈[0, 1). Then f ∈S∗ Lh(α).
Theorem 4.1.
Theorem 4.1. Let f(z) = zh(z)g(z) ∈CLh. Then for z ∈D we have (1) |h(z)| ≤ 1 1 −|z| exp |z| 1 −|z| ; (2) |g(z)| ≤exp |z| 1 −|z| ;
Theorem 4.1. Let f(z) = zh(z)g(z) ∈CLh. Then for z ∈D we have (1) |h(z)| ≤ 1 1 −|z| exp |z| 1 −|z| ; (2) |g(z)| ≤exp |z| 1 −|z| ;
Theorem 4.2.
Theorem 4.2. Let f(z) = zh(z)g(z) ∈CLh. Then for z ∈D we have (1) |fz(z)| ≤ 1 (1 −|z|)3 exp 2|z| 1 −|z| ; (2) |fz(z)| ≤ |z| (1 −|z|)3…
Theorem 4.2. Let f(z) = zh(z)g(z) ∈CLh. Then for z ∈D we have (1) |fz(z)| ≤ 1 (1 −|z|)3 exp 2|z| 1 −|z| ; (2) |fz(z)| ≤ |z| (1 −|z|)3 exp 2|z| 1 −|z| ;
Function classes studied:
Related Papers