Geometric Subfamily of Functions Convex in Some Direction and Blaschke Products

Theorem 2.1. Suppose that $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1]$ . Then $$\operatorname{Re}\left(\frac{zh''(z)}{h'(z)}\right) \le \frac{\alpha}{2} - \frac{1}{2\alpha}(1 - |z|^2) \left|\frac{h''(z)}{h'(z)}\right|^2 \quad \text{for all } z \in \mathbb{D}. \tag{2.1}$$ Strict inequality holds for all $z \in \mathbb{D}$ unless $h'(z) = (1 - \zeta z)^{\alpha}$ for some $\zeta \in \mathbb{T}$ .

Theorem 2.2. Let $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1]$ . Then $$h'(z) = \exp\left(\alpha \int_{\mathbb{T}} \log(1 - \zeta z) \, d\mu(\zeta)\right) \quad \text{for } z \in \mathbb{D},$$ (2.3) where $\mu$ is a probability measure on $\mathbb{T}$ so that $\int_{\mathbb{T}} d\mu(\zeta) = 1$ . Also, we have $h'(z) \prec H_{\alpha}(z)$ for $z \in \mathbb{D}$ , where $H_{\alpha}(z) = (1-z)^{\alpha}$ and $\prec$ is the usual subordination [10, 23].

Theorem 2.3. Suppose that $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1]$ , and satisfies (2.2) for some $\phi \in \mathbb{B}$ . Then $\phi$ is a finite Blaschke product with degree $m \geq 1$ if and only if $$h'(z) = \prod_{k=1}^{m+1} (1 - \zeta_k z)^{\alpha t_k},$$ where $\zeta_k \in \mathbb{T}$ are distinct points, $0 < t_k < 1$ and $\sum_{k=1}^{m+1} t_k = 1$ .

Corollary 3.1. Suppose that $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1]$ . If $0 < \alpha < \frac{1}{2}$ , then $h(\mathbb{T})$ is a quasicircle and h has a $(1+2\alpha)/(1-2\alpha)$ -quasiconformal extension to the whole complex plane $\mathbb{C}$ . For $\alpha \in [1/2, 1)$ , we have the following result.

Theorem 3.2. Suppose that $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1]$ . Then $h(\mathbb{D})$ is a quasidisk.

Lemma 3.3. Let $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1)$ such that $h'(z) = (1-\zeta z)^{\alpha}$ for all $z \in \mathbb{D}$ and $\zeta \in \mathbb{T}$ . Then $||S_h|| = 2\alpha(2+\alpha)$ .

Theorem 3.4. Suppose that $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1]$ . Then $||S_h|| \le 2\alpha(2+\alpha)$ , where the equality is attained by h(z) which is obtained from $h'(z) = (1-\zeta z)^{\alpha}$ for some $\zeta \in \mathbb{T}$ .

Theorem 4.1. Suppose that $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0,1]$ . Then there exists c > 0 such that every harmonic mapping $f = h + \overline{g}$ with dilatation $|\omega(z)| < c$ , is univalent in $\mathbb{D}$ , where the constant c depends only on the domain $h(\mathbb{D})$ .

Theorem 4.2. Let $h \in \mathcal{G}(\alpha)$ for some $\alpha \in (0, 1/2)$ , and $f = h + \overline{g}$ be a sense-preserving harmonic mapping with dilatation $\omega = g'/h'$ . If $$|\omega(z)| \le 1 - \alpha |z|(1+|z|) \quad \text{for all } z \in \mathbb{D}, \tag{4.1}$$ then f is univalent in $\mathbb{D}$ .