On quasiconformal close-to-convex harmonic mappings involving starlike functions

Theorem 1. $\mathcal{G}(\alpha) \not\subset \mathcal{S}$ for $\alpha \in (3/2, +\infty)$ . <span id="page-3-0"></span> Figure 1. The image of D under the mapping f(z) = z − 1 2 z ² + 1 4 z ² − 1 6 z 3. Proof. We consider the analytic function h^β ∈ A given by $$h_{\beta}(z) = \frac{1}{\beta} \left[ 1 - (1 - z)^{\beta} \right] \quad (2 < \beta < 3; \ z \in \mathbb{D}).$$ It follows that $$1 + \frac{zh_{\beta}''(z)}{h_{\beta}'(z)} = \frac{1 - \beta z}{1 - z},$$ and therefore, $$\operatorname{Re}\left(1 + \frac{zh_{\beta}''(z)}{h_{\beta}'(z)}\right) < \frac{1+\beta}{2} \quad \left(\frac{3}{2} < \frac{1+\beta}{2} < 2\right),$$ which implies that $$h_{\beta} \in \mathcal{G}((1+\beta)/2) = \mathcal{G}(\alpha) \quad \left(\frac{3}{2} < \alpha < 2\right).$$ In what follows, we shall prove that the function h^β is not univalent in D. It easily to verify that h^β have real coefficients, and thus, hβ(z) = hβ(z) for all z ∈ D. In particular, we see that $$\operatorname{Re}\left(h_{\beta}\left(re^{i\theta}\right)\right) = \operatorname{Re}\left(h_{\beta}\left(re^{-i\theta}\right)\right)$$ for some $r \in (0,1)$ and $\theta \in (-\pi,0) \cup (0,\pi)$ . It is sufficient to show that there exist $r_0 \in (0,1)$ and $\theta_0 \in (-\pi,0) \cup (0,\pi)$ such that $$\operatorname{Im}\left(h_{\beta}\left(r_{0}e^{i\theta_{0}}\right)\right) = \operatorname{Im}\left(h_{\beta}\left(r_{0}e^{-i\theta_{0}}\right)\right) = 0.$$ In view of $$\operatorname{Im}(h_{\beta}(z)) = \operatorname{Im}\left(\frac{1 - (1 - z)^{\beta}}{\beta}\right) = -\operatorname{Im}\left(\frac{e^{\beta \log(1 - z)}}{\beta}\right),$$ we see that $$\operatorname{Im}\left(h_{\beta}\left(re^{i\theta}\right)\right) = -\operatorname{Im}\left(\frac{e^{\beta\log\left(1-re^{i\theta}\right)}}{\beta}\right)$$ $$= -\frac{e^{\beta\log\left|1-re^{i\theta}\right|}}{\beta}\sin\left[\beta\arg\left(1-re^{i\theta}\right)\right],$$ and $$-\operatorname{Im}\left(h_{\beta}\left(re^{-i\theta}\right)\right) = \frac{e^{\beta\log|1-re^{-i\theta}|}}{\beta}\sin\left[\beta\arg\left(1-re^{-i\theta}\right)\right] = \operatorname{Im}\left(h_{\beta}\left(re^{i\theta}\right)\right).$$ By noting that $$\arg\left(1-re^{i\theta}\right)\in\left(-\frac{\pi}{2},0\right)\cup\left(0,\frac{\pi}{2}\right),$$ we deduce that for each $\beta \in (2,3)$ , there exist $r_0 \in (0,1)$ and $\theta_0 \in (-\pi,0) \cup (0,\pi)$ such that $$\sin\left[\beta\arg\left(1-r_0e^{i\theta_0}\right)\right]=0.$$ It follows that $$\operatorname{Im}\left(h_{\beta}\left(r_{0}e^{i\theta_{0}}\right)\right) = \operatorname{Im}\left(h_{\beta}\left(r_{0}e^{-i\theta_{0}}\right)\right) = 0.$$ Therefore, we see that there exist two distinct points $z_1 = r_0 e^{i\theta_0}$ and $z_2 = r_0 e^{-i\theta_0}$ in $\mathbb{D}$ such that $h_{\beta}(z_1) = h_{\beta}(z_2)$ , which shows that the function $h_{\beta}(z)$ is not univalent in $\mathbb{D}$ . Thus, we deduce that the class $\mathcal{G}(\alpha)$ always contains a non-univalent function for each $\alpha \in (3/2, 2)$ . Moreover, by noting that the class $\mathcal{G}(\alpha)$ is not univalent in $\mathbb{D}$ for $\alpha \in [2, +\infty)$ (see [16, Example 2.1]), we deduce that the assertion of Theorem 1 holds. To illustrate our counterexample, we present the image domain of $\mathbb{D}$ under the function $h_{5/2}(z) = 2/5 \left[1 - (1-z)^{5/2}\right]$ (see Figure 2).

Lemma 2. Let $f \in \mathcal{G}(\alpha)$ with $\alpha \in (1, 3/2]$ . Then $$\left|a_{3}-\delta a_{2}^{2}\right| \leq \begin{cases} \frac{\alpha-1}{3}\left|3+\delta-(2+\delta)\alpha\right| & \left(\left|\delta-\frac{3-2\alpha}{3(\alpha-1)}\right| \geq \frac{1}{3(\alpha-1)}\right), \\ \frac{\alpha-1}{3} & \left(\left|\delta-\frac{3-2\alpha}{3(\alpha-1)}\right| < \frac{1}{3(\alpha-1)}\right). \end{cases}$$ (3.4) Equality in the Fekete-Szegö functional is attained in each case.

Theorem 3. Let $f \in \mathcal{F}(\alpha, \lambda)$ be of the form (1.1). Then $$|b_3 - \delta b_2^2| \le \frac{2(\alpha - 1)|\lambda|}{3} + \frac{|\delta||\lambda|^2}{4}.$$ (3.5) The inequality is sharp.

Theorem 5. Let $f \in \mathcal{F}(\alpha, \lambda, n)$ . Then <span id="page-8-3"></span> $$r\left[|\lambda|\left(\frac{r}{n+2} - \frac{1}{n+1}\right)r^n - \frac{r}{2} + 1\right] \le |f(z)|$$ $$\le r\left[|\lambda|\left(\frac{r}{n+2} + \frac{1}{n+1}\right)r^n + \frac{r}{2} + 1\right].$$ (3.14) The inequalities are sharp.

Theorem 6. Let $f \in \mathcal{F}(\alpha, \lambda, n)$ . Then for 0 < r < 1, we have $$2\pi \int_0^r \left(1 - |\lambda|^2 \xi^{2n}\right) (1 - \xi)^2 \xi d\xi \le \mathcal{A}\left(f(\mathbb{D}_r)\right) \le 2\pi \int_0^r \left(1 - |\lambda|^2 \xi^{2n}\right) (1 + \xi)^2 \xi d\xi. \tag{3.17}$$

Theorem 7. Let $f \in \mathcal{F}(\alpha, \lambda)$ be of the form (1.1). Then for each $m \geq 1, l \geq 2$ , <span id="page-9-1"></span> $$S_{m,l}(f)(z) = \sum_{k=1}^{m} a_k z^k + \sum_{k=2}^{l} b_k z^k \quad (a_1 = 1)$$ is close-to-convex in $|z| < r_c \approx 0.503$ , where $r_c$ is the least positive real root in the interval (0,1) of the equation: $$2 + 2\ln(1-r) + r\ln(1-r) - r + r^2 = 0. (3.19)$$ The bound $r_c$ is sharp.

Definition 1. A harmonic mapping $f = h + \overline{g} \in \mathcal{H}$ is said to be in the class $\mathcal{F}(\alpha, \lambda, n)$ if h and g satisfy the conditions <span id="page-2-3"></span> $$\operatorname{Re}\left(1 + \frac{zh''(z)}{h'(z)}\right) < \alpha \quad \left(1 < \alpha \le \frac{3}{2}\right),$$ (1.3) <span id="page-2-2"></span>and $$g'(z) = \lambda z^n h'(z) \quad \left(\lambda \in \mathbb{C} \text{ with } |\lambda| \le \frac{1}{n+1}; n \in \mathbb{N}\right).$$ (1.4) For simplicity, we denote the class $\mathcal{F}(\alpha, \lambda, 1)$ by $\mathcal{F}(\alpha, \lambda)$ . The image of $\mathbb{D}$ under the mapping $$f(z) = z - \frac{1}{2}z^2 + \frac{1}{4}z^2 - \frac{1}{6}z^3 \in \mathcal{F}(3/2, 1/2)$$ is presented as Figure 1. This paper is organized as follows. In Section 2, we provide a counterexample to illustrate the non-univalency of the class $\mathcal{G}(\alpha)$ for $\alpha \in (3/2,2)$ . In Section 3, we prove several basic properties of the class $\mathcal{F}(\alpha,\lambda,n)$ of quasiconformal close-to-convex harmonic mappings with starlike analytic part, such results as coefficient inequalities, an integral representation, a growth theorem, an area theorem, and radii of close-to-convexity of partial sums of the class, are derived. 2. Non-univalency of the class $$\mathcal{G}(\alpha)$$ for $\alpha \in (3/2, +\infty)$ <span id="page-2-0"></span>Obradović et al. [30] stated that the class $\mathcal{G}(\alpha)$ is not univalent in $\mathbb{D}$ for $\alpha \in (3/2, +\infty)$ , but they did not give detailed proof about the non-univalency. We note that Kargar et al. [16] given a counterexample to prove the class $\mathcal{G}(\alpha)$ is not univalent in $\mathbb{D}$ for $\alpha \in [2, +\infty)$ , in this section, we shall give a counterexample to illuminate the non-univalency of the class $\mathcal{G}(\alpha)$ for $\alpha \in (3/2, 2)$ .