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Abstract
We determine the representation theorem, distortion theorem, coefficients estimate and Bohr's radius for log-harmonic starlike mappings of order $α$, which are generalization of some earlier results. In addition, the inner mapping radius of log-harmonic mappings is also established by constructing a family of $1$-slit log-harmonic mappings. Finally, we introduce pre-Schwarzian, Schwarzian derivatives and Bloch's norm for non-vanishing log-harmonic mappings, several properties related to these ar
Results & Lemmas (9)
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Lemma 2.1.
Lemma 2.1. ([22, Corollary 3.6]) Let p(z) be analytic in D with p(0) = 1. Then Re p(z) > 0 in D if and only if there is a probability…
Lemma 2.1. ([22, Corollary 3.6]) Let p(z) be analytic in D with p(0) = 1. Then Re p(z) > 0 in D if and only if there is a probability measure δ on ∂D such that p(z) = Z ∂D 1 + ηz 1 −ηz dδ(η), z ∈D. Since each p has the form p(z) = 1 + µ(z) 1 −µ(z) = 1 + 2µ(z) 1 −µ(z) for some µ ∈B, we have the following equivalent version of Lemma 2.1.
Lemma 2.2.
Lemma 2.2. If µ ∈B with µ(0) = 0, then µ(z) 1 −µ(z) = Z ∂D ξz 1 −ξz dκ(ξ), z ∈D, for some probability measure κ on ∂D.
Lemma 2.2. If µ ∈B with µ(0) = 0, then µ(z) 1 −µ(z) = Z ∂D ξz 1 −ξz dκ(ξ), z ∈D, for some probability measure κ on ∂D.
Theorem 2.3.
Theorem 2.3. A log-harmonic mapping f(z) = zh(z)g(z) ∈S∗ Lh(α) if and only if there are two probability measures δ and κ on ∂D such that…
Theorem 2.3. A log-harmonic mapping f(z) = zh(z)g(z) ∈S∗ Lh(α) if and only if there are two probability measures δ and κ on ∂D such that h(z) = exp Z ∂D Z ∂D K1(z, η, ξ) dδ(η) dκ(ξ) , (2.1) where K1(z, η, ξ) =
Theorem 2.4.
Theorem 2.4. Let f(z) = zh(z)g(z) ∈S∗ Lh(α) with µ(0) = 0. Then for z ∈D, (1) 1 1+|z| exp (1 −α) −2|z| 1+|z| ≤|h(z)| ≤ 1 1−|z| exp (1…
Theorem 2.4. Let f(z) = zh(z)g(z) ∈S∗ Lh(α) with µ(0) = 0. Then for z ∈D, (1) 1 1+|z| exp (1 −α) −2|z| 1+|z| ≤|h(z)| ≤ 1 1−|z| exp (1 −α) 2|z| 1−|z|
Corollary 2.5.
Corollary 2.5. Let f(z) = zh(z)g(z) ∈S∗ Lh(α). Also, let H(z) = zh(z) and G(z) = zg(z). Then (1) 1 2 e1−α ≤d(0, ∂H(D)) ≤1; (2) 1 22α−1 e1−α…
Corollary 2.5. Let f(z) = zh(z)g(z) ∈S∗ Lh(α). Also, let H(z) = zh(z) and G(z) = zg(z). Then (1) 1 2 e1−α ≤d(0, ∂H(D)) ≤1; (2) 1 22α−1 e1−α ≤d(0, ∂G(D)) ≤1; (3) 1 22α e2(1−α) ≤d(0, ∂f(D)) ≤1. The equalities occur if and only if f(z) is one of the functions of the form ηfα(ηz), |η| = 1, where fα(z) is given by (1.5).
Theorem 2.6.
Theorem 2.6. Let f(z) = zh(z)g(z) ∈S∗ Lh(α). Then |an| ≤2(1 −α) + 1 n and |bn| ≤2(1 −α) + 2α −1 n (2.8) for all n ≥1. The equalities occur…
Theorem 2.6. Let f(z) = zh(z)g(z) ∈S∗ Lh(α). Then |an| ≤2(1 −α) + 1 n and |bn| ≤2(1 −α) + 2α −1 n (2.8) for all n ≥1. The equalities occur if and only if f(z) is one of the functions of the form ηfα(ηz), |η| = 1, where fα(z) is given by (1.5).
Theorem 3.1.
Theorem 3.1. Let f(z) = zh(z)g(z) ∈S∗ Lh(α), H(z) = zh(z) and G(z) = zg(z). Then (1) |z| exp ∞ X n=1 |an||z|n ! ≤d(0, ∂H(D)) for |z| ≤rH,…
Theorem 3.1. Let f(z) = zh(z)g(z) ∈S∗ Lh(α), H(z) = zh(z) and G(z) = zg(z). Then (1) |z| exp ∞ X n=1 |an||z|n ! ≤d(0, ∂H(D)) for |z| ≤rH, where rH is the unique root in (0, 1) of the equation r 1 −r exp (1 −α) 2r
Theorem 5.1.
Theorem 5.1. Suppose that f(z) = h(z)g(z) is a sense-preserving log-harmonic mapping in D. Then pre-Schwarzian derivative Pf of f(z) is…
Theorem 5.1. Suppose that f(z) = h(z)g(z) is a sense-preserving log-harmonic mapping in D. Then pre-Schwarzian derivative Pf of f(z) is harmonic if and only if the dilatation µ(z) of f(z) is constant.
Proposition 6.1.
Proposition 6.1. If f(z) = h(z)g(z) ∈BLh, then (i) f af b ∈BLh for any a, b ∈C (affine invariance) (ii) f ◦φα ∈BLh for any α ∈D (linear…
Proposition 6.1. If f(z) = h(z)g(z) ∈BLh, then (i) f af b ∈BLh for any a, b ∈C (affine invariance) (ii) f ◦φα ∈BLh for any α ∈D (linear invariance).
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