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

Theorem 1. $\mathcal{G}(\alpha) \not\subset \mathcal{S}$ for $\alpha \in (3/2, +\infty)$ .  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 1. If h(z) = z + P^∞ ^k=2 akz k satisfies the condition [\(1.2\)](#page-1-1) with 1 < α ≤ 3/2, then $$|a_k| \le \frac{2(\alpha - 1)}{(k - 1)k} \quad (k \ge 2),$$ (3.1) with the extremal function given by $$h(z) = \int_0^z \left(1 - t^{k-1}\right)^{\frac{2(\alpha - 1)}{k - 1}} dt \quad (k \ge 2).$$

Theorem 2. Let f = h + g ∈ F(α, λ, n) be of the form [\(1.1\)](#page-0-1). Then the coefficients a^k (k ≥ 2) of h satisfy [\(3.1\)](#page-5-2), furthermore, the coefficients b^k (k = n+1, n+2, . . . ; n ∈ N) of g satisfy $$|b_{n+1}| \le \frac{|\lambda|}{n+1} \quad (n \in \mathbb{N}) \quad and \quad |b_{k+n}| \le \frac{2|\lambda|(\alpha-1)}{(k-1)k(k+n)} \quad (k \in \mathbb{N} \setminus \{1\}; n \in \mathbb{N}).$$ $$(3.2)$$ The bounds are sharp for the extremal function given by $$f(z) = \int_0^z \left(1 - t^{k-1}\right)^{\frac{2(\alpha - 1)}{k - 1}} dt + \overline{\int_0^z \lambda t^n \left(1 - t^{k-1}\right)^{\frac{2(\alpha - 1)}{k - 1}} dt} \quad (n \in \mathbb{N}).$$

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.

Corollary 1. Let $f \in \mathcal{F}(\alpha, \lambda)$ be of the form (1.1). Then $$|a_3 - a_2^2| \le \begin{cases} \frac{\alpha - 1}{3} |4 - 3\alpha| & \left( \left| 1 - \frac{3 - 2\alpha}{3(\alpha - 1)} \right| \ge \frac{1}{3(\alpha - 1)} \right), \\ \frac{\alpha - 1}{3} & \left( \left| 1 - \frac{3 - 2\alpha}{3(\alpha - 1)} \right| < \frac{1}{3(\alpha - 1)} \right), \end{cases}$$ and $$|b_3 - b_2^2| \le \frac{2(\alpha - 1)|\lambda|}{3} + \frac{|\lambda|^2}{4} \le \frac{11}{48}.$$ The inequalities are sharp. Now, we give an integral representation of the mapping f ∈ F(α, λ, n).

Theorem 4. Let f ∈ F(α, λ, n). Then $$\begin{split} f(z) = & \int_0^z \exp\left(2(1-\alpha)\int_0^\zeta \frac{\varpi(t)}{t(1-\varpi(t))}dt\right)d\zeta \\ & + \lambda \int_0^z \zeta^n \cdot \exp\left(2(1-\alpha)\int_0^\zeta \frac{\varpi(t)}{t(1-\varpi(t))}dt\right)d\zeta, \end{split}$$ where \$ is the Schwarz function with \$(0) = 0 and |\$(z)| < 1 (z ∈ D). Proof. Suppose that f ∈ F(α, λ, n). It follows from [\(1.3\)](#page-2-3) that $$1 + \frac{zh''(z)}{h'(z)} \prec \frac{1 - (2\alpha - 1)z}{1 - z} \quad (z \in \mathbb{D}), \tag{3.7}$$ where " ≺ " denotes the familiar subordination of analytic functions. By virtue of [\(3.7\)](#page-7-0), we see that $$1 + \frac{zh''(z)}{h'(z)} = \frac{1 - (2\alpha - 1)\varpi(z)}{1 - \varpi(z)} \quad (z \in \mathbb{D}), \tag{3.8}$$ where \$ is the Schwarz function with \$(0) = 0 and |\$(z)| < 1 (z ∈ D). From [\(3.8\)](#page-7-1), we have $$\frac{(zh'(z))'}{zh'(z)} - \frac{1}{z} = \frac{2(1-\alpha)\varpi(z)}{z(1-\varpi(z))},$$ which, upon integration, yields $$\log(h'(z)) = 2(1 - \alpha) \int_0^z \frac{\varpi(t)}{t(1 - \varpi(t))} dt.$$ (3.9) We thus find from [\(3.9\)](#page-7-2) that $$h(z) = \int_0^z \exp\left(2(1-\alpha)\int_0^\zeta \frac{\varpi(t)}{t(1-\varpi(t))}dt\right)d\zeta. \tag{3.10}$$ Combining [\(1.4\)](#page-2-2) with [\(3.10\)](#page-7-3), we obtain $$g(z) = \lambda \int_0^z \zeta^n \cdot \exp\left(2(1-\alpha) \int_0^\zeta \frac{\varpi(t)}{t(1-\varpi(t))} dt\right) d\zeta.$$ (3.11) Thus, the assertion of Theorem [4](#page-7-4) follows from [\(3.10\)](#page-7-3) and [\(3.11\)](#page-7-5). Remark 1. Theorem [4](#page-7-4) provides a direct integration method for constructing quasiconformal close-to-convex harmonic mappings by choosing suitable Schwarz functions \$. The following lemma due to Maharana et al. [\[25\]](#page-13-9) will play a crucial role in the proof of our last three results.$

Lemma 3. If h ∈ G, then for |z| = r < 1, the following statements are true. $$\left|\frac{zh''(z)}{h'(z)}\right| \le \frac{r}{1-r}.$$ The inequality is sharp and equality is attained for the function $$h(z) = z - \frac{z^2}{2}. (3.12)$$ $$1 - r \le |h'(z)| \le 1 + r. \tag{3.13}$$ The inequalities are sharp and equalities are attained for the function given by (3.12). (3) If $$h(z) = S_n(z) + \Sigma_n(z)$$ , with $\Sigma_n(z) = \sum_{k=n+1}^{\infty} a_k z^k$ , then $$\left|\Sigma'_n(z)\right| \le r^n \phi(r, 1, n) \text{ and } \left|z\Sigma''_n(z)\right| \le \frac{r^n}{1 - r},$$ where $\phi(r, 1, n)$ is the unified zeta function which is defined by the series $$\phi(z, s, a) = \sum_{k=0}^{\infty} \frac{z^k}{(k+a)^s} \quad (|z| < 1; \Re(s) > 1; a \neq 0, -1, -2, \ldots).$$ We now give the growth theorem for the class $\mathcal{F}(\alpha, \lambda, n)$ .

Theorem 5. Let $f \in \mathcal{F}(\alpha, \lambda, n)$ . Then $$r\left[|\lambda|\left(\frac{r}{n+2} - \frac{1}{n+1}\right)r^n - \frac{r}{2} + 1\right] \le |f(z)|$$ $$\le r\left[|\lambda|\left(\frac{r}{n+2} + \frac{1}{n+1}\right)r^n + \frac{r}{2} + 1\right].$$ (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 $m \geq 1, l \geq 2$ , $$S_{m,l}(f)(z) = \sum_{k=1}^{m} a_k z^k + \sum_{k=2}^{l} b_k z^k \quad (a_1 = 1)$$ 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 $$\operatorname{Re}\left(1 + \frac{zh''(z)}{h'(z)}\right) < \alpha \quad \left(1 < \alpha \le \frac{3}{2}\right),$$ (1.3) 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 $\alpha \in (3/2, +\infty)$ 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 $\alpha \in (3/2, 2)$ .