Univalence of horizontal shear of Cesàro type transforms

Theorem 3.1. If $\varphi \in \mathcal{S}$ , then $C_{\alpha\beta}[\varphi]$ is contained in $\mathcal{S}$ for $|\alpha| \leq 1/[2(2+|\beta|)]$ . Proof. By the definition of $C_{\alpha\beta}[\varphi]$ , the concept of logarithmic derivative followed by the triangle inequality leads to $$(1-|z|^2)\left|\frac{z(C_{\alpha\beta}[\varphi])''(z)}{(C_{\alpha\beta}[\varphi])'(z)}\right| \le (1-|z|^2)|\alpha|\left(\left|\frac{z\varphi'(z)}{\varphi(z)}-1\right|+\left|\frac{\beta z}{1-z}\right|\right).$$ If $\varphi \in \mathcal{S}$ , then Theorem 9 of [18, p 69] gives that $\left| \frac{z\varphi'(z)}{\varphi(z)} - 1 \right| \leq 2/(1-|z|)$ and so it follows that $$(1 - |z|^2) \left| \frac{z(C_{\alpha\beta}[\varphi])''(z)}{(C_{\alpha\beta}[\varphi])'(z)} \right| \le |\alpha| \left( 2(1 + |z|) + |\beta|(1 + |z|) \right) < 2|\alpha|(2 + |\beta|).$$ Now, by the Becker criterion [5] for the univalence of an analytic function (see also [39, Theorem 6.7, p. 172] and [20, Theorem 3.3.1, p. 130]), $C_{\alpha\beta}[\varphi]$ is univalent in $\mathbb{D}$ provided $2|\alpha|(2+|\beta|) \leq 1$ and hence the result follows. Remark 1. We assume that the bound for $\alpha$ in Theorem 3.1 may be improved further, however, for $\alpha, \beta$ satisfying $|\alpha|(2+|\beta|) \geq 2$ , we ensure the existence of a function $\varphi \in \mathcal{S}$ such that $C_{\alpha\beta}[\varphi] \notin \mathcal{S}$ . This can be seen by considering the Koebe function $\varphi(z) = z/(1-z)^2$ , $z \in \mathbb{D}$ . Indeed, the corresponding integral transform $$C_{\alpha\beta}[\varphi(z)] = \int_0^z (1-\zeta)^{-\alpha(2+\beta)} d\zeta$$ is trivially not univalent for $-\alpha(2+\beta)=2$ . Remark 2. For the choice $\beta = 0$ , Theorem 3.1 is equivalent to [26, Theorem 3]. As a consequence of Theorem 3.1, one may generate a number of integral transforms that are indeed univalent. Our next purpose is to construct harmonic mappings corresponding to the integral transforms $C_{\alpha\beta}$ through shear construction. From the algorithm described in Section 1, we require to show that $C_{\alpha\beta}$ is CHD.

Theorem 3.3. If $\varphi \in \mathcal{S}^*(\delta)$ , then $C_{\alpha\beta}[\varphi]$ is convex in one direction in $\mathbb{D}$ for all $\alpha, \beta \geq 0$ satisfying $\alpha(\beta + 2(1 - \delta)) \leq 3$ . Proof. By the definition of $C_{\alpha\beta}[\varphi]$ , we have $$1 + \operatorname{Re}\left[\frac{z(C_{\alpha\beta}[\varphi])''(z)}{(C_{\alpha\beta}[\varphi])'(z)}\right] = 1 + \alpha \operatorname{Re}\left[\frac{z\varphi'(z)}{\varphi(z)} - 1 + \frac{\beta z}{1-z}\right]$$ $$> 1 - \alpha + \alpha\delta - \alpha\beta/2 \ge -1/2,$$ where the last inequality holds by our assumption $\alpha(\beta + 2(1 - \delta)) \leq 3$ . Therefore, by using [47, Theorem 1], one can conclude that $C_{\alpha\beta}[\varphi]$ is convex in one direction in $\mathbb{D}$ . The following result characterizes a function to be CHD. Lemma E ([46, Theorem 1]). Let $\varphi$ be a non-constant analytic function in $\mathbb{D}$ . The function $\varphi$ is CHD if and only if there are numbers $\mu$ and $\nu$ , $0 \le \mu < 2\pi$ and $0 \le \nu \le \pi$ , such that $$\operatorname{Re} \{ e^{i\mu} (1 - 2ze^{-i\mu} \cos \nu + z^2 e^{-2i\mu}) \varphi'(z) \} \ge 0, \quad z \in \mathbb{D}.$$ <span id="page-6-1"></span>Remark 3. By Theorem 3.3 we learn that the operator $C_{\alpha\beta}[\varphi]$ need not be CHD under the same assumptions. However, for all $\alpha, \beta \geq 0$ satisfying $\alpha(\beta + 2(1 - \delta)) \leq 3$ , the rotation $C_{\alpha\beta}^{\theta}[\varphi](z) := e^{-i\theta}C_{\alpha\beta}[\varphi](e^{i\theta}z)$ of $C_{\alpha\beta}[\varphi](z)$ will be CHD for a suitable choice of $\theta$ whenever $\varphi \in \mathcal{S}^*(\delta)$ . In particular, we write $J_{\alpha}^{\theta}[\varphi](z) := e^{-i\theta}J_{\alpha}[\varphi](e^{i\theta}z)$ and $C_{\alpha}^{\theta}[\varphi](z) := e^{-i\theta}C_{\alpha}[\varphi](e^{i\theta}z)$ . For instance, we here present an integral operator that is convex in one direction, but not in horizontal direction, which becomes CHD with a suitable rotation. For the function $\varphi(z)=z/(1-z^2)$ , one can show that by Theorem 3.3, the integral transform $J_{3/2}[\varphi](z)=\int_0^z (1-\zeta^2)^{-3/2} d\zeta$ is convex in one direction. At this moment we do not have any analytical proof for $J_{3/2}[\varphi](z)$ to be non-CHD; however the Mathematica graphics tool confirms it (see Figure 1). As a result, we now show that the rotation operator $J_{3/2}^{\pi/4}[\varphi](z)$ is CHD.   <span id="page-7-0"></span>Figure 1. The images $J_{3/2}[\varphi](\mathbb{D})$ and $J_{3/2}^{\pi/4}[\varphi](\mathbb{D})$ for $\varphi(z)=z(1-z^2)^{-1}$ Lemma E, for the choices $\mu = \pi/4, \nu = \pi/2$ , leads us in proving $$\operatorname{Re}\left\{e^{i\pi/4}(1-iz^2)(J_{3/2}^{\pi/4}[\varphi])'\right\} = \operatorname{Re}\left\{(1-iz^2)^{-1/2}\right\} > 0.$$ This is equivalent to proving $|\arg(1-iz^2)^{-1/2}| < \pi/2$ . For this, consider $$k(z) = \int_0^z (1 - i\zeta^2)^{-1} d\zeta$$ and we obtain $$1 + \text{Re}\left[\frac{zk''(z)}{k'(z)}\right] = 1 + 2\text{Re}\left[\frac{iz^2}{1 - iz^2}\right] > 0.$$ This shows that k(z) is a convex function and therefore, one can obtain $$|\arg(1-iz^2)^{-1/2}| = 1/2 \cdot |\arg(1-iz^2)^{-1}| < \pi/2.$$ Therefore, $J_{3/2}^{\pi/4}[\varphi](\mathbb{D})$ is CHD. We now define the corresponding harmonic mapping $F_{\alpha\beta}^{\theta}$ of the integral transform $C_{\alpha\beta}^{\theta}[\varphi]$ by using the shear construction algorithm as stated in Section 1. Theorem 3.3 and Remark 3 justify the validity of the following definition:

Theorem 3.6. Let $\varphi \in S^*(\delta)$ , and $F^{\theta}_{\alpha\beta} = H + \overline{G}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha\beta}$ . Then for all non-negative parameters $\alpha, \beta$ such that $\alpha(\beta + 2(1 - \delta)) \leq 2$ with $\alpha(1 + \beta)||w|| < 1/3$ , the corresponding $F^{\theta}_{\alpha\beta}$ is univalent in $\mathbb{D}$ . Proof. Let $F_{\alpha\beta}^{\theta} = H + \overline{G}$ be a sense-preserving harmonic mapping, which is a horizontal shear of $C_{\alpha\beta}^{\theta}[\varphi]$ . We have $$1 + \operatorname{Re}\left[\frac{z(C_{\alpha\beta}^{\theta}[\varphi])''(z)}{(C_{\alpha\beta}^{\theta}[\varphi])'(z)}\right] = 1 + \alpha \operatorname{Re}\left[\frac{ze^{i\theta}\varphi'(ze^{i\theta})}{\varphi(ze^{i\theta})} - 1 + \frac{\beta ze^{i\theta}}{1 - ze^{i\theta}}\right]$$ $$= 1 + \alpha \operatorname{Re}\left[\frac{\zeta\varphi'(\zeta)}{\varphi(\zeta)} - 1 + \frac{\beta\zeta}{1 - \zeta}\right], \quad \zeta = e^{i\theta}z$$ $$> 1 - \alpha + \alpha\delta - \alpha\beta/2 \ge 0,$$ where the last inequality holds by our assumption. Therefore, $C_{\alpha\beta}^{\theta}[\varphi]$ is a convex function and so $C_{\alpha\beta}^{\theta}[\varphi](\mathbb{D})$ is a 1-linearly connected domain; see for instance [14,40]. Using Lemma 7 of [3], we conclude that $F_{\alpha\beta}^{\theta}$ is univalent for $\alpha(1+\beta)||w|| < 1/3$ . $\square$ Remark 4. Since $\mathcal{K} \subset \mathcal{S}^*(1/2)$ , Theorem 3.6 is also valid whenever $\varphi$ is a convex function. We have a couple of immediate consequences of Theorem 3.6 which give the univalency of $\mathcal{G}^{\theta}_{\alpha}$ and $\mathcal{F}^{\theta}_{\alpha}$ .

Corollary 3.7. Let $\varphi \in \mathcal{K}$ , and $\mathcal{G}_{\alpha}^{\theta} = H + \overline{G}$ be a horizontal shear of $J_{\alpha}^{\theta}[\varphi]$ with dilatation $w_{\alpha 0}$ in $\mathbb{D}$ . Then for all $\alpha \in [0,2]$ with $\alpha ||w|| < 1/3$ , the mapping $\mathcal{G}_{\alpha}^{\theta}$ is univalent in $\mathbb{D}$ .

Corollary 3.8. Let $\varphi \in \mathcal{K}$ , and $\mathcal{F}_{\alpha}^{\theta} = H + \overline{G}$ be a horizontal shear of $C_{\alpha}^{\theta}[\varphi]$ with dilatation $w_{\alpha 1}$ in $\mathbb{D}$ . Then for all $\alpha \in [0,1]$ with $\alpha \|w\| < 1/6$ , the mapping $\mathcal{F}_{\alpha}^{\theta}$ is univalent in $\mathbb{D}$ . Next we focus on the univalence of $F_{\alpha\beta}^{\theta}$ in terms of harmonic pre-Schwarzian derivative, where Lemma A plays a crucial role. For this, a simplified version of the pre-Schwarzian derivative of $F_{\alpha\beta}^{\theta}$ is required. Indeed, by using (2), a direct calculation shows that the pre-Schwarzian derivative of $F_{\alpha\beta}^{\theta}$ is obtained as <span id="page-9-1"></span> $$P_{F_{\alpha\beta}^{\theta}}(z) = \alpha \left[ \frac{e^{i\theta} \varphi'(ze^{i\theta})}{\varphi(ze^{i\theta})} - \frac{1}{z} + \frac{\beta e^{i\theta}}{1 - e^{i\theta}z} + (1+\beta)w'(z) \left( \frac{1 - \overline{\alpha(1+\beta)w(z)}}{(1 - \alpha(1+\beta)w(z))(1 - |\alpha(1+\beta)|^2|w(z)|^2)} \right) \right].$$ $$(3)$$ For the sake of convenience, we define the following notation. Using the classical Schwarz-Pick lemma, we observe that <span id="page-10-2"></span> $$||w^*|| = \sup_{z \in \mathbb{D}} \frac{|w'(z)|(1-|z|^2)}{1-|w|^2} \le 1,$$ (4) where $||w^*||$ is called the hyperbolic norm of w(z). Thus, we have

Theorem 3.9. Let $F_{\alpha\beta}^{\theta} = H + \overline{G}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha\beta}$ . If $\varphi \in \mathcal{L}(\gamma)$ , then (i) for $\beta \geq 1$ , $F_{\alpha\beta}^{\theta} \in \mathcal{S}_{\mathbb{H}}$ for all non-negative values of $\alpha$ satisfying <span id="page-10-0"></span> $$\alpha \le \frac{1}{2\gamma + 2\beta + (1+\beta) \|w^*\| (1+\|w\|) ]}.$$ (5) - (ii) for $0 \le \beta < 1$ , two cases arise. - (a) If $(\beta+2(1+\beta) \|w^*\|(1+\|w\|)) \le 2(1-\beta)$ , then $F_{\alpha\beta}^{\theta} \in \mathcal{S}_{\mathbb{H}}$ for all non-negative values of $\alpha$ satisfying <span id="page-10-1"></span> $$\alpha \le \frac{4(1-\beta)}{\left[4(2\gamma+1)(1-\beta) + (\beta+(1+\beta)\|w^\|(1+\|w\|))^2 + 4(1-\beta^2)\|w^\|\right]}.$$ (6) (b) If $(\beta+2(1+\beta)\|w^*\|(1+\|w\|)) > 2(1-\beta)$ , then $F_{\alpha\beta}^{\theta} \in \mathcal{S}_{\mathbb{H}}$ for all non-negative values of $\alpha$ satisfying the inequality (5). Proof. Note that, by Lemma C and (3), for all $z \in \mathbb{D}$ we estimate $$(1 - |z|^{2})|zP_{F_{\alpha\beta}^{\theta}}(z)| = (1 - |z|^{2})\alpha \left| \frac{ze^{i\theta}\varphi'(ze^{i\theta})}{\varphi(ze^{i\theta})} - 1 + \frac{\beta ze^{i\theta}}{1 - ze^{i\theta}} \right| \\ + \frac{z(1 + \beta)w'(z)(1 - \overline{\alpha(1 + \beta)w(z)})}{(1 - \alpha(1 + \beta)w(z))(1 - (\alpha(1 + \beta))^{2}|w(z)|^{2})} \right| \\ \leq \alpha \left[ (1 - |z|^{2}) \left| \frac{ze^{i\theta}\varphi'(ze^{i\theta})}{\varphi(ze^{i\theta})} \right| + 1 - |z|^{2} + \beta|z|(1 + |z|) \right. \\ \left. + \frac{(1 - |z|^{2})(1 + \beta)|w'(z)||z|}{1 - (\alpha(1 + \beta))^{2}|w(z)|^{2}} \right] \\ \leq \alpha \left[ 2\gamma + 1 + (\beta - 1)|z|^{2} + (\beta + (1 + \beta)||w^{*}|||w||)|z| \right].$$ To find the supremum of the right-hand expression, we consider two cases: (i) The case $\beta \geq 1$ . In this case, the maximum value of the right-hand expression holds trivially for |z| = 1. This implies that $$(1 - |z|^2)|zP^{\theta}_{F_{\alpha\beta}}(z)| \le \alpha[2\gamma + 2\beta + (1+\beta) \|w^*\| \|w\|].$$ Thus, we compute $$(1-|z|^{2})|zP_{F_{\alpha\beta}^{\theta}}(z)| + \frac{|zw_{\alpha\beta}^{'}(z)|(1-|z|^{2})}{1-|w_{\alpha\beta}(z)|^{2}} \le \alpha[2\gamma+2\beta+(1+\beta)\|w^{*}\|(1+\|w\|)].$$ It follows from Lemma A that $F_{\alpha\beta}^{\theta}$ is univalent in $\mathbb{D}$ , if $\alpha$ and $\beta$ satisfy the bound given in (5). (ii) The case $\beta < 1$ . Clearly, the maximum value of the right-hand expression is attained for $$|z| = \frac{1}{2(1-\beta)}(\beta + (1+\beta) ||w^*|| ||w||).$$ The supremum quantity is discussed through two subcases, namely, (a) The subcase $(\beta + (1 + \beta) ||w^*|| ||w||) \le 2(1 - \beta)$ . For this case, we have $$(1-|z|^2)|zP_{F_{\alpha\beta}^{\theta}}(z)| \leq \frac{\alpha}{4(1-\beta)} \Big[ 4(2\gamma+1)(1-\beta) + (\beta+(1+\beta)\|w^*\|(1+\|w\|))^2 \Big],$$ and thus, $$(1 - |z|^{2})|zP_{F_{\alpha\beta}}^{\theta}| + \frac{|zw_{\alpha\beta}^{'}(z)|(1 - |z|^{2})}{1 - |w_{\alpha\beta}(z)|^{2}}$$ $$\leq \frac{\alpha}{4(1 - \beta)} \Big[ 4(2\gamma + 1)(1 - \beta) + (\beta + (1 + \beta) \|w^{}\|(1 + \|w\|))^{2} + 4(1 - \beta^{2}) \|w^{}\| \Big].$$ Again using Lemma A, we conclude that $F_{\alpha\beta}^{\theta}$ is univalent in $\mathbb{D}$ whenever $\alpha$ satisfies the inequality (6). (b) The subcase $(\beta + 2(1+\beta)||w^*||(1+||w||)) > 2(1-\beta)$ . Trivially, the maximum value of the right-hand expression holds for |z| = 1. Similarly, as an application of Lemma A, it then follows that $F_{\alpha\beta}^{\theta}$ is univalent in $\mathbb{D}$ whenever $\alpha$ and $\beta$ satisfy the inequality (5). This completes the proof. The concludes the univalence properties of $F_{\alpha\beta}^{\theta}$ for $\varphi$ that belong to specific subclasses of $\mathcal{S}$ .

Theorem 4.1. Let $F_{\alpha\beta}^{\theta}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha\beta}$ . If $\varphi \in \mathcal{S}^*(\delta)$ then $F_{\alpha\beta}^{\theta} \in \mathcal{SHU}$ for all non-negative $\alpha, \beta$ satisfying <span id="page-12-2"></span><span id="page-12-0"></span> $$\alpha \le \frac{1}{2(2+\beta+(1+\beta)\|w^*\|(1+\|w\|))}. (7)$$ Proof. Since $\varphi \in \mathcal{S}^*(\delta)$ , we have $\varphi(0) = 0$ which justifies the local univalence of $C^{\theta}_{\alpha\beta}[\varphi]$ and so $F^{\theta}_{\alpha\beta} = H + \overline{G}$ is well-defined. It is easy to see that for any $\lambda \in \mathbb{T}$ , the function $\Phi_{\lambda,\theta} = H + \lambda G$ satisfies <span id="page-12-3"></span> $$\Phi'_{\lambda,\theta}(z) = H'(z) \cdot [1 + \lambda w_{\alpha\beta}(z)] = (C^{\theta}_{\alpha\beta}[\varphi])'(z) \cdot \frac{1 + \lambda \alpha(1+\beta) w(z)}{1 - \alpha(1+\beta) w(z)}.$$ (8) Hence, for all $z \in \mathbb{D}$ , we have $$(1-|z|^2)\left|\frac{z\Phi_{\lambda,\theta}''(z)}{\Phi_{\lambda,\theta}'(z)}\right| = (1-|z|^2)\alpha\left|\frac{ze^{i\theta}\varphi'(ze^{i\theta})}{\varphi(ze^{i\theta})} - 1 + \frac{\beta ze^{i\theta}}{1-ze^{i\theta}} + \frac{\lambda(1+\beta)z\,w'(z)}{1+\lambda(1+\beta)\alpha\,w(z)}\right|$$ $$+ \frac{z(1+\beta)\,w'(z)}{1-\alpha(1+\beta)\,w(z)}.$$ (9) Since w(z) is a self-map of $\mathbb{D}$ and $|z\varphi'(z)/\varphi(z)-1| \leq 2/(1-|z|)$ , by the classical distortion theorem for $\mathcal{S}$ and (4), we find $$(1 - |z|^2) \left| z \frac{\Phi_{\lambda,\theta}''(z)}{\Phi_{\lambda,\theta}'(z)} \right| \le \alpha \left( 2(1 + |z|) + \beta(1 + |z|) + 2(1 + \beta) \|w^*\| (1 + \|w\|) |z| \right)$$ $$\le \alpha \left( 4 + 2\beta + 2(1 + \beta) \|w^*\| (1 + \|w\|) \right).$$ It follows that $\Phi_{\lambda,\theta}$ satisfies the Becker univalence criterion for all $\lambda \in \mathbb{T}$ (see [5] and also [20, Theorem 3.3.1, p. 130]), whenever $\alpha, \beta$ are related by (7). Therefore, by Lemma D, $F_{\alpha\beta}^{\theta}$ belongs to the class $\mathcal{SHU}$ under the restriction given by (7). For the choice $\beta = 1$ , Theorem 4.1 produces the stable harmonic univalence of $\mathcal{F}^{\theta}_{\alpha}$ as follows:

Corollary 4.2. Let $\mathcal{F}^{\theta}_{\alpha}$ be a horizontal shear of $C^{\theta}_{\alpha}[\varphi]$ with dilatation $w_{\alpha 1}$ in $\mathbb{D}$ . If $\varphi \in \mathcal{S}^*(\delta)$ , then $\mathcal{F}^{\theta}_{\alpha} \in \mathcal{SHU}$ for all non-negative $\alpha$ satisfying $$\alpha \le \frac{1}{2(3+2\|w^*\|(1+\|w\|))}.$$ Similarly, for the choice $\beta = 0$ , Theorem 4.1 produces the well-known fact about the stable harmonic univalency of $\mathcal{G}^{\theta}_{\alpha}$ (see [3, Theorem 2]), for $\alpha \geq 0$ , as follows:

Corollary 4.3. Let $\mathcal{G}^{\theta}_{\alpha}$ be a horizontal shear of $J^{\theta}_{\alpha}[\varphi]$ with dilatation $w_{\alpha 0}$ in $\mathbb{D}$ . If $\varphi \in \mathcal{S}^*(\delta)$ , then $\mathcal{G}^{\theta}_{\alpha} \in \mathcal{SHU}$ for all non-negative $\alpha$ satisfying $$\alpha \le \frac{1}{2(2 + \|w^*\|(1 + \|w\|))}.$$ Next we discuss the stable harmonic univalence of $F_{\alpha\beta}^{\theta}$ when $\varphi$ belongs to a class of linear invariant family.

Theorem 4.4. Let $\alpha \geq 0$ and $F_{\alpha\beta}^{\theta}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha\beta}$ . If $\varphi \in \mathcal{L}(\gamma)$ , $1 \leq \gamma < \infty$ , then we have (i) For $\beta \geq 1$ , $F_{\alpha\beta}^{\theta} \in \mathcal{SHU}$ for all values of $\alpha$ satisfying <span id="page-13-0"></span> $$\alpha \le \frac{1}{2(\gamma + \beta + (1+\beta) \|w^*\| (1+\|w\|))}.$$ (10) - (ii) For $0 \le \beta < 1$ , two cases arise. - (a) If $\beta + 2(1+\beta) \|w^*\|(1+\|w\|) \le 2(1-\beta)$ , then $F_{\alpha\beta}^{\theta} \in \mathcal{SHU}$ for all values of $\alpha$ satisfying <span id="page-13-1"></span> $$\alpha \le \frac{4(1-\beta)}{4(2\gamma+1)(1-\beta) + (\beta+2(1+\beta)\|w^*\|(1+\|w\|))^2}.$$ (11) (b) If $\beta + 2(1+\beta) \|w^*\| (1+\|w\|) > 2(1-\beta)$ , then $F_{\alpha\beta}^{\theta} \in \mathcal{SHU}$ for values of $\alpha$ satisfying the inequality (10). Proof. Using Lemma C and (9), we get $$(1-|z|^2)\left|\frac{z\Phi_{\lambda,\theta}''(z)}{\Phi_{\lambda,\theta}'(z)}\right| \le \alpha\left(2\gamma+1-|z|^2+\beta(1+|z|)|z|+2(1+\beta)\|w^*\|(1+\|w\|)|z|\right)$$ $$=\alpha\left(2\gamma+1+(\beta-1)|z|^2+(\beta+2(1+\beta)\|w^*\|(1+\|w\|))|z|\right).$$ To find the supremum of the right-hand expression, we consider two cases: (i) The case $\beta \geq 1$ . In this case, the maximum value of the right-hand expression holds trivially for |z|=1. Therefore, $\Phi_{\lambda,\theta}$ satisfies the Becker univalence criterion for all $\lambda\in\mathbb{T}$ whenever $\alpha$ satisfies the inequality (10). (ii) The case $0 \le \beta < 1$ . Clearly, the maximum value of the right-hand expression is attained for $$|z| = \frac{1}{2(1-\beta)} (\beta + 2(1+\beta) \|w^*\| (1+\|w\|)).$$ The supremum quantity is discussed through two subcases, namely, (a) The subcase $\beta + 2(1+\beta) \|w^*\| (1+\|w\|) \le 2(1-\beta)$ . In this case, $\Phi_{\lambda,\theta}$ satisfies the Becker univalence criterion for all $\lambda \in \mathbb{T}$ when $\alpha$ satisfies the inequality (11). (b) The subcase $\beta + 2(1+\beta) \|w^*\| (1+\|w\|) > 2(1-\beta)$ . Trivially, the maximum value of the right-hand expression holds for |z| = 1. It follows that $\Phi_{\lambda,\theta}$ satisfies the Becker univalence criterion for all $\lambda \in \mathbb{T}$ whenever $\alpha$ satisfies the inequality (10). This completes the proof. Until this point, whenever $\varphi$ is univalent, we have dealt with the stable harmonic univalence properties of $F^{\theta}_{\alpha\beta}$ . The features of $F^{\theta}_{\alpha\beta}$ that are close-to-convex are examined in the next section whenever $\varphi$ belongs to certain subclasses of $\mathcal{S}$ . Additionally, we offer bounds on $\alpha$ and $\beta$ under which $F^{\theta}_{\alpha\beta}$ is close-to-convex.

Theorem 5.1. Let $F_{\alpha\beta}^{\theta} = H + \overline{G}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha\beta}$ . If $\varphi \in \mathcal{S}^*(\delta)$ and $w(z) = \cos(\pi c)z/2$ , for some c, -1/2 < c < 0, then for all non-negative parameters $\alpha, \beta$ satisfying $\alpha(1+\beta) \leq 1$ with $\alpha(2(1-\delta)+\beta) \leq -2c$ , we have $F_{\alpha\beta}^{\theta} \in \mathcal{CC}_{\mathbb{H}}$ . Proof. Since $\varphi \in \mathcal{S}^*(\delta)$ , by Definition 3.4, the harmonic mapping $F_{\alpha\beta}^{\theta} = H + \overline{G}$ is well-defined. Clearly, for the given choice of w(z), we have $$|w_{\alpha\beta}(z)| = \alpha(1+\beta)|w(z)| < \frac{\cos(\pi|c|)}{2} < \cos(\pi|c|).$$ Since $C_{\alpha\beta}^{\theta}[\varphi] = H - G$ satisfies $(C_{\alpha\beta}^{\theta}[\varphi])'(z) = H'(z)(1 - w_{\alpha\beta}(z))$ , for all $z \in \mathbb{D}$ , it follows that $$1 + \operatorname{Re}\left[\frac{zH''(z)}{H'(z)}\right] = 1 + \alpha \operatorname{Re}\left[\frac{ze^{i\theta}\varphi'(ze^{i\theta})}{\varphi(ze^{i\theta})} - 1 + \frac{\beta ze^{i\theta}}{1 - ze^{i\theta}}\right] + \operatorname{Re}\left[\frac{zw'_{\alpha\beta}(z)}{1 - w_{\alpha\beta}(z)}\right]$$ $$= 1 + \alpha \operatorname{Re}\left[\frac{\zeta\varphi'(\zeta)}{\varphi(\zeta)} - 1 + \frac{\beta\zeta}{1 - \zeta}\right] - \operatorname{Re}\left[\frac{-\alpha(1 + \beta)zw'(z)}{1 - \alpha(1 + \beta)w(z)}\right]$$ $$> 1 + \alpha\delta - \alpha - \alpha\beta/2 - 1 \ge c,$$ with $\zeta = e^{i\theta}z$ , where the last inequality follows since $\alpha(2(1-\delta)+\beta) \leq -2c$ . Therefore according to Lemma B, $F_{\alpha\beta}^{\theta}$ is a close-to-convex mapping. Recall that the connection $\mathcal{K} \subset \mathcal{S}^*(1/2)$ is valid. Therefore, Theorem 5.1 offers the following univalence close-to-convexity of $\mathcal{G}^{\theta}_{\alpha}$ , if $\beta = 0$ is chosen.

Corollary 5.2. Let $\mathcal{G}^{\theta}_{\alpha}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha 0}$ . If $\varphi \in \mathcal{K}$ and $w(z) = \cos(\pi c)z/2$ , for some c, -1/2 < c < 0, then for all $\alpha \in [0, -2c]$ , the mapping $\mathcal{G}^{\theta}_{\alpha} \in \mathcal{CC}_{\mathbb{H}}$ . In the similar fashion, if one chooses $\beta=1$ in Theorem 5.1, then the close-to-convexity of $\mathcal{F}^{\theta}_{\alpha}$ follows.

Corollary 5.3. Let $\mathcal{F}^{\theta}_{\alpha}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha 1}$ . If $\varphi \in \mathcal{K}$ and $w(z) = \cos(\pi c)z/2$ , for some c, -1/2 < c < 0, then for all $\alpha \in [0, -c]$ , the mapping $\mathcal{F}^{\theta}_{\alpha} \in \mathcal{CC}_{\mathbb{H}}$ . The stable harmonic close-to-convexity of $F_{\alpha\beta}^{\theta}$ , whenever $\varphi \in \mathcal{S}^*(\delta)$ , is the subject of our next major finding. However, this is dependent on the next elementary lemma. In the remaining section we choose w(z) = z/2.

Lemma 5.4. Let $F_{\alpha\beta}^{\theta}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with dilatation $w_{\alpha\beta}$ . Then for all $\lambda \in \mathbb{T}$ and for all non-negative $\alpha, \beta$ with $\alpha(1+\beta) \leq 1$ , we have $$\left| \arg \left( \frac{2 + \lambda \alpha (1 + \beta)z}{2 - \alpha (1 + \beta)z} \right) \right| \le 2 \arcsin(r\alpha(1 + \beta)),$$ where r = |z| < 1. Proof. For any $\lambda \in \mathbb{T}$ , the relation (8) suggests us to consider the integral $$I(z) = \int_0^z \frac{\Phi'_{\lambda,\theta}(\zeta)}{(C^{\theta}_{\alpha\beta}[\varphi])'(\zeta)} d\zeta = \int_0^z \frac{2 + \lambda \alpha (1 + \beta)\zeta}{2 - \alpha (1 + \beta)\zeta} d\zeta.$$ Whence for all $z \in \mathbb{D}$ , the logarithmic derivative of I'(z) leads to <span id="page-15-0"></span> $$1 + \operatorname{Re}\left[\frac{zI''(z)}{I'(z)}\right] = 1 + \operatorname{Re}\left[\frac{z\lambda\alpha(1+\beta)}{2+\lambda\alpha(1+\beta)z}\right] - \operatorname{Re}\left[\frac{-z\alpha(1+\beta)}{2-\alpha(1+\beta)z}\right]. \tag{12}$$ It follows that $$\operatorname{Re}\left[\frac{z\lambda\alpha(1+\beta)}{2+\lambda\alpha(1+\beta)z}\right] = \frac{\partial}{\partial\theta}\left\{\operatorname{arg}(2+\lambda\alpha(1+\beta)re^{i\theta})\right\}, \quad z = re^{i\theta}.$$ Geometrically, the function $2 + \lambda \alpha (1 + \beta)z$ being a Möbius transformation, it maps each circle |z| = r < 1 onto another circle. It thus follows that $\arg(2 + \lambda \alpha (1 + \beta)z)$ increases as z moves around the circle |z| = r in the positive sense. That is, $$\frac{\partial}{\partial \theta} \{ \arg(2 + \lambda \alpha (1 + \beta) r e^{i\theta}) \} > 0, \quad z = r e^{i\theta}.$$ Equivalently, on the one hand, we have $$\operatorname{Re}\left[\frac{z\lambda\alpha(1+\beta)}{2+\lambda\alpha(1+\beta)z}\right] > 0.$$ On the other hand, one can easily see that $$\operatorname{Re}\left[\frac{-z\alpha(1+\beta)}{2-\alpha(1+\beta)z}\right] \le \frac{1}{2-|z|} < 1.$$ Thus, by (12), we obtain $$1 + \operatorname{Re}\left[\frac{zI''(z)}{I'(z)}\right] > 0,$$ leading to the convexity of I(z) in $\mathbb{D}$ . Now, the rotation theorem for convex functions [16, Page 103], yields $$|\arg(I'(z))| = \left|\arg\left(\frac{2+\lambda\alpha(1+\beta)z}{2-\alpha(1+\beta)z}\right)\right| \le 2\arcsin(r\alpha(1+\beta)), \quad |z| = r < 1,$$ <span id="page-16-0"></span>completing the

Theorem 5.5. Let $F_{\alpha\beta}^{\theta}$ be a sense-preserving harmonic mapping in $\mathbb{D}$ with its dilatation $w_{\alpha\beta}$ . If $\varphi \in \mathcal{S}^*(\delta)$ and $\alpha \in [0, 1/(1+\beta)\sqrt{2}]$ , then $F_{\alpha\beta}^{\theta} \in \mathcal{SHCC}$ . Proof. Let $\lambda \in \mathbb{T}$ be arbitrary. Consider $\Phi_{\lambda,\theta}$ as defined in the proof of Theorem 4.1. For $0 \leq t_2 - t_1 \leq 2\pi$ and $z = re^{it}$ , we first compute $$\int_{t_1}^{t_2} \operatorname{Re}\left[1 + \frac{z\Phi_{\lambda,\theta}''(z)}{\Phi_{\lambda,\theta}'(z)}\right] dt = \int_{t_1}^{t_2} \left(1 + \operatorname{Re}\left[\frac{\alpha z e^{i\theta}\varphi'(ze^{i\theta})}{\varphi(ze^{i\theta})} - \alpha + \frac{\alpha\beta z e^{i\theta}}{1 - ze^{i\theta}}\right] + \frac{z\lambda\alpha(1+\beta)}{2 + \lambda\alpha(1+\beta)z} + \frac{z\alpha(1+\beta)}{2 - \alpha(1+\beta)z}\right] dt$$ $$> \left(1 + (\delta - 1)\alpha - \frac{\alpha\beta}{2}\right)(t_2 - t_1)$$ $$+ \operatorname{arg}\left(\frac{2 + \lambda\alpha(1+\beta)re^{it_2}}{2 + \lambda\alpha(1+\beta)re^{it_1}} \cdot \frac{2 - \alpha(1+\beta)re^{it_1}}{2 - \alpha(1+\beta)re^{it_2}}\right).$$ Since $t_2 - t_1 \ge 0$ , it follows that $$\int_{t_1}^{t_2} \operatorname{Re}\left[1 + \frac{z\Phi_{\lambda,\theta}''(z)}{\Phi_{\lambda,\theta}'(z)}\right] dt > \operatorname{arg}\left(\frac{2 + \lambda\alpha(1+\beta)re^{it_2}}{2 - \alpha(1+\beta)re^{it_2}}\right) + \operatorname{arg}\left(\frac{2 - \alpha(1+\beta)re^{it_1}}{2 + \lambda\alpha(1+\beta)re^{it_1}}\right) \ge -4 \arcsin(r\alpha(1+\beta)) > -4 \arcsin(\alpha(1+\beta)),$$ where the second inequality holds by Lemma 5.4. Note that, if $\arcsin(\alpha(1+\beta)) \le \pi/4$ , or equivalently $0 \le \alpha(1+\beta) \le 1/\sqrt{2}$ , immediately give us $$\int_{t_1}^{t_2} \operatorname{Re}\left\{1 + z \frac{\Phi_{\lambda,\theta}''(z)}{\Phi_{\lambda,\theta}'(z)}\right\} dt > -\pi.$$ Hence $\Phi_{\lambda,\theta}$ is a close-to-convex mapping in the unit disk. This completes the proof. $\square$ As we recall $\mathcal{K} \subsetneq \mathcal{S}^*(1/2)$ , Theorem 5.5 provides the following immediate consequences, respectively for $\beta = 0$ and $\beta = 1$ :

Corollary 5.6. Let $\mathcal{G}_{\alpha}^{\theta}$ be a horizontal shear of $J_{\alpha}^{\theta}[\varphi]$ with dilatation $w_{\alpha 0}$ in $\mathbb{D}$ . If $\varphi \in \mathcal{K}$ , then for all $\alpha \in [0, 1/\sqrt{2}]$ , we have $\mathcal{G}_{\alpha}^{\theta} \in \mathcal{SHCC}$ .

Corollary 5.7. Let $\mathcal{F}_{\alpha}^{\theta}$ be a horizontal shear of $C_{\alpha}^{\theta}[\varphi]$ with dilatation $w_{\alpha 1}$ in $\mathbb{D}$ . If $\varphi \in \mathcal{K}$ , then for all $\alpha \in [0, 1/2\sqrt{2}]$ , we have $\mathcal{F}_{\alpha}^{\theta} \in \mathcal{SHCC}$ .

Definition 3.2. A domain $D \subset \mathbb{C}$ is called convex in the direction $\theta \ (0 \leq \theta < \pi)$ if every line parallel to the line through 0 and $e^{i\theta}$ has a connected or empty intersection with D. A univalent harmonic mapping f in D is said to be convex in the direction $\theta$ if f(D) is convex in the direction $\theta$ . The case $\theta = 0$ corresponds to CHD.

Definition 3.4. Let $\alpha, \beta \geq 0$ and $\alpha(\beta + 2(\delta - 1)) \leq 3$ . Then we define $F_{\alpha\beta}^{\theta}(z) = H(z) + \overline{G(z)}$ , with the usual normalization H(0) = G(0) = 0, H'(0) = 1 and G'(0) = 0, as a horizontal shear of $C_{\alpha\beta}^{\theta}[\varphi](z) = H(z) - G(z)$ having its dilatation $w_{\alpha\beta}(z) = \alpha(1+\beta)w(z)$ for some analytic function w(z) satisfying |w(z)| < 1. Note that one can choose w in such a way that the condition $|w_{\alpha\beta}(z)| < 1$ is satisfied. In particular, we also use the notations $\mathcal{F}^{\theta}_{\alpha}$ and $\mathcal{G}^{\theta}_{\alpha}$ for the horizontal shears of $C^{\theta}_{\alpha}[\varphi]$ and $J^{\theta}_{\alpha}[\varphi]$ with their dilatations $w_{\alpha 1}$ and $w_{\alpha 0}$ , respectively. One can take $F_{\alpha\beta}^{\theta} = H + \overline{G}$ as a vertical shear of the analytic function $C_{\alpha\beta}^{\theta}[\varphi] = H + G$ for some $\theta$ (0 $\leq \theta < \pi$ ) with the same normalization. However, this small change in the sign produces serious structural difference (see [17, Section 3.4, p. 40]). Next, we provide a counterexample to the statement that $F_{11}^{\theta} \in \mathcal{S}_H$ , a horizontal shear of $C^{\theta}[\varphi]$ , while $\varphi$ ranges over the class $\mathcal{S}^*(\delta)$ , $0 \leq \delta < 1$ . This motivates us to study the univalence property of $F_{\alpha\beta}^{\theta}$ under certain restrictions on the parameters $\alpha$ and $\beta$ . We begin our investigation with the counterexample followed by the main results. <span id="page-8-1"></span>Example 3.5. For $\lambda \in \mathbb{T}$ , consider a locally univalent analytic function $\Phi_{\lambda,\theta} = H + \lambda G$ in $\mathbb{D}$ . Now $F_{11}^{\theta} = H + \overline{G}$ is a well defined sense-preserving harmonic mapping, a horizontal shear of $C^{\theta}[\varphi] = H - G$ , with its dilatation $w_{11} = G'/H'$ . Adhering to our counterexample, we take $\varphi(z) = z/(1-z)^2$ with $\theta = 0$ and w(z) = z/2. For any $\lambda \in \mathbb{T}$ , it is easy to see that the function $\Phi_{\lambda,0} = H + \lambda G$ satisfies $$\Phi'_{\lambda,0}(z) = H'(z) \cdot [1 + \lambda w_{11}(z)] = (C_{11}^0[\varphi])'(z) \cdot \frac{1 + \lambda z}{1 - z}.$$ Thus, for all $z \in \mathbb{D}$ and for all $\lambda \in \mathbb{T}$ , we compute $$(1 - |z|^2) \left| \frac{\Phi_{\lambda,0}''(z)}{\Phi_{\lambda,0}'(z)} \right| = (1 - |z|^2) \left| \frac{4}{1 - z} + \frac{\lambda}{1 + \lambda z} \right|.$$ By choosing z = 1/2 and $\lambda = 1$ , we notice that $$\sup_{z \in \mathbb{D}} (1 - |z|^2) \left| \frac{\Phi_{\lambda,0}''(z)}{\Phi_{\lambda,0}'(z)} \right| \ge \frac{26}{4} > 6,$$ which contradicts the well-known univalence criteria (an immediate consequence of [16, Theorem 2.4]). Therefore, $\Phi_{1,0} = H + G$ is not univalent. It follows by Lemma D that $F_{11}^{\theta} \notin \mathcal{S}_{\mathbb{H}}$ . The graph in relation to the non-univalency of $F_{11}^{\theta}$ for $\varphi(z) = z/(1-z)^2$ is also shown in Figure 2.  <span id="page-8-0"></span>Figure 2. Image of $\mathbb{D}$ under $F_{11}$ In what follows, our first main result provides conditions on $\alpha$ and $\beta$ for which $F_{\alpha\beta}^{\theta}$ , with its dilatation $w_{\alpha\beta}$ , is univalent whenever $\varphi$ is a starlike function of order $\delta$ , $0 \le \delta < 1$ . For this purpose, we use the idea of linearly connected domains.