On the Pre-Schwarzian Norm and Starlikeness of Certain Logharmonic Mappings

Theorem 2.1. Let $f \in L_{\mathcal{R}}$ be a sense-preserving logharmonic mapping in the unit disk $\mathbb{D}$ . Then the pre-Schwarzian norm $||P_f|| \leq 11$ . The estimate is sharp.

Theorem 2.2. Let $f \in L_{\mathcal{R}}$ be a sense-preserving logharmonic mapping with dilatation $\omega$ . Then for |z| = r the sharp inequalities $$(2.8) |f(z)| \le \begin{cases} \frac{e^{-r/\alpha(1+\alpha)}}{(1-r)^4} e^{(1/\alpha-\alpha)^2 \log(1+\alpha r)}, & when \ |\omega(0)| = \alpha \ne 0\\ \frac{e^{-3r-r^2/2}}{(1-r)^4}, & when \ \omega(0) = 0 \end{cases}$$ hold.

Lemma 2.1. Let $f = h + \overline{g}$ be a sense-preserving harmonic mapping in the unit disk $\mathbb{D}$ . Then $||P_f|| < \infty$ if and only if f is uniformly locally univalent. We use this result to generate a family of uniformly locally univalent harmonic mappings.

Theorem 2.3. Let $f \in L_{\mathcal{R}}$ be sense-preserving logharmonic mapping in $\mathbb{D}$ . Then $\log f$ is locally uniformly univalent in $\mathbb{D}$ .

Theorem 2.4. Let $f = ze^{h(z)}\overline{e^{g(z)}} \in L^0_{\mathcal{R}}$ be a sense-preserving logharmonic mapping in $\mathbb{D}$ of the form (2.14) such that $$(2.16) |1 - b_1| + \sum_{n=2}^{\infty} n|a_n - b_n| \le 1.$$ Then f is fully starlike in $\mathbb{D}$ .

Definition 2.1. Suppose that a function $f \in C^1(\mathbb{D})$ satisfies the condition that f(z) = 0 if and only if z = 0, and that $J_f = |f_z|^2 - |f_{\overline{z}}|^2 > 0$ on $\mathbb{D}$ . Then f is one-one in $\mathbb{D}$ and $f(\mathbb{D}_r)$ is starlike for each 0 < r < 1 if and only if Re $$\left(\frac{Df(z)}{f(z)}\right) > 0, \quad z \in \mathbb{D} \setminus \{0\}.$$ Example 2.1. There are logharmonic mappings $f \in L^0_{\mathcal{R}}$ which are not hereditarily starlike in the unit disk $\mathbb{D}$ . Let us consider a logharmonic mapping $f(z)=ze^{h(z)}\overline{e^{g(z)}}$ where $h(z)=\log{(1+z)}$ and the dilatation $\omega(z)=-z$ . Here, h'(z)=1/(1+z) and so h(0)=h'(0)-1=0, and $\operatorname{Re} h'(z)$ is positive in the unit disk $\mathbb D$ . This implies that $f\in L^0_{\mathcal R}$ . Next, from (2.15) it is follows that $$\operatorname{Re} \frac{Df(z)}{f(z)} = \operatorname{Re} [(1 - \omega(z))(1 + zh'(z))] = \operatorname{Re} (1 + 2z)$$ which is non positive in $\mathbb{D} \cap \{z \in \mathbb{C} : -1 < \operatorname{Re} z < -1/2\}.$ Our next result provides sufficient condition under which a logharmonic mapping $f \in L^0_{\mathcal{P}}$ is hereditarily starlike.