Abstract
Let $h$ and $g$ be two analytic functions in the unit disc $Δ$ that $g(0)=1$. Also let $β$ be a complex number with ${\rm Re}\{β\}>-1/2$. A function $f$ is said to be log--harmonic mapping if it has the following representation \begin{equation*}
f(z)=z |z|^{2β} h(z)\overline{g(z)}\quad (z\in Δ). \end{equation*} A log--harmonic mapping $f$ is said to be starlike log--harmonic mapping of order $α$, where $0\leq α<1$, if \begin{equation*}
{\rm Re}\left\{\frac{zf_z -\overline{z}f_{\overline{z}}}
Results & Lemmas (7)
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Lemma 1.1.
Lemma 1.1. [8, p. 35] Let Ξ be a simply connected domain in the complex plane C, Ξ ̸= C and let b be a complex number with Re b > 0.…
Lemma 1.1. [8, p. 35] Let Ξ be a simply connected domain in the complex plane C, Ξ ̸= C and let b be a complex number with Re{b} > 0. Suppose that a function ψ : C2 × ∆→C satisfies the condition ψ(iρ, σ; z) ̸∈Ξ for all real ρ, σ ≤−| b−iρ |2 /(2Re{b}) and all z ∈∆. If the function p(z) defined by p(z) = b + b1z + b2z2 + · · · is analytic in ∆and if ψ(p(z), zp′(z); z) ∈Ξ, then Re{p(z)} > 0 in ∆.
Theorem 2.1.
Theorem 2.1. Let α ∈[0, 1) and Re β > −1/2. Then the function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α) if, and only if, (2.1) …
Theorem 2.1. Let α ∈[0, 1) and Re{β} > −1/2. Then the function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α) if, and only if, (2.1) z h′(z) h(z) −z g′(z) g(z) ≺2(1 −α)z 1 −z (z ∈∆).
Lemma 2.1.
Lemma 2.1. The function f(z) = z|z|2βh(z)g(z) (Re β > −1/2) belongs to the class S∗ LH(α) if, and only if, (2.5) 1 + Re z h′(z) h(z) −z…
Lemma 2.1. The function f(z) = z|z|2βh(z)g(z) (Re{β} > −1/2) belongs to the class S∗ LH(α) if, and only if, (2.5) 1 + Re z h′(z) h(z) −z g′(z) g(z) > α (0 ≤α < 1, z ∈∆). Following, we give the representation theorem for the function h of the mappings f of the form (1.2) in the set S∗ LH(α).
Theorem 2.2.
Theorem 2.2. A function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α) if, and only if, (2.6) h(z) = g(z) exp Z z 0 2(1 −α)ψ(t) t(1…
Theorem 2.2. A function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α) if, and only if, (2.6) h(z) = g(z) exp Z z 0 2(1 −α)ψ(t) t(1 −ψ(t)) dt (z ∈∆), where ψ is a Schwarz function, 0 ≤α < 1 and Re{β} > −1/2.
Theorem 2.3.
Theorem 2.3. Let α ∈[0, 1) and Re β > −1/2. Let a function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α) where h(z) = 1 + ∞ X n=1…
Theorem 2.3. Let α ∈[0, 1) and Re{β} > −1/2. Let a function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α) where h(z) = 1 + ∞ X n=1 anzn and g(z) = 1 + ∞ X n=1 bnzn,
Theorem 2.4.
Theorem 2.4. Let α ∈(1/2, 1). If the function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α), then (2.12) Re h(z) g(z) > µ(α):= 1 3…
Theorem 2.4. Let α ∈(1/2, 1). If the function f(z) = z|z|2βh(z)g(z) belongs to the class S∗ LH(α), then (2.12) Re h(z) g(z) > µ(α) := 1 3 −2α (z ∈∆).
Theorem 2.5.
Theorem 2.5. If f(z) = z|z|2βh(z)g(z) is log–harmonic mapping, then (1 −|γ|2)(1 −|z|2) (1 + |γ||z|)2 |fz|2 ≤Jf(z) ≤ …
Theorem 2.5. If f(z) = z|z|2βh(z)g(z) is log–harmonic mapping, then (1 −|γ|2)(1 −|z|2) (1 + |γ||z|)2 |fz|2 ≤Jf(z) ≤ (1−|γ|2)(1−|z|2) (1−|γ||z|)2 |fz|2 |z| < |γ|, |fz|2 |z| ≥|γ|,
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