Region of variability for certain subclass of univalent functions

Lemma 1.2. [7] For $\theta \in \mathbb{R}$ and $0 < \lambda < 1$ , the function $$G(z) = \int_0^z \frac{e^{i\theta}z}{\left(1 + \left(e^{i\theta} - 1\right)\lambda z - e^{i\theta}z\right)^2} dz, \ z \in \mathbb{D}$$ has a double zero at the origin and no zeros elsewhere in $\mathbb{D}$ . Furthermore, these exists a starlike univalent function $G_0$ in $\mathbb{D}$ such that $G = 2^{-1}e^{i\theta}G_0^2$ and $G_0(0) = G_0'(0) - 1$ . For a positive integer p, let $$(\mathcal{S}^)^p = \{ f = f_0^p : f_0 \in \mathcal{S}^ \}.$$

Lemma 1.3. [15] Let f be an analytic function in $\mathbb{D}$ with $f(z) = z^p + \cdots$ . If Re $$\left(1 + z \frac{f''(z)}{f'(z)}\right) > 0, \ z \in \mathbb{D},$$ then $f \in (\mathcal{S}^*)^p$ .

Proposition 3.3. For $f \in \mathcal{F}(\lambda)$ where $0 \le \lambda < 1$ , we have $$\left| \frac{f''(z)}{f'(z)} - c(z, \lambda, \beta) \right| \le r(z, \lambda, \beta), \ z \in \mathbb{D}.$$ where $$c(z,\lambda,\beta) = \frac{2(1-\beta)\left(\lambda(1-|z|^2) + (|z|^2 - \lambda^2)\right)\bar{z}}{(1-|z|^2)(1-2\lambda(\text{Re}z) + |z|^2)}$$ and $$r(z,\lambda,\beta) = \frac{2(1-\beta)(1-\lambda^2)|z|}{(1-|z|^2)(1-2\lambda(\text{Re}z)+|z|^2)}$$ For each $z \in \mathbb{D} \setminus \{0\}$ , equality hold if, and only if, $f = F_{e^{i\theta} \lambda}$ for some $\theta \in \mathbb{R}$ .

Corollary 3.4. Let $\gamma := z(t)$ , where $0 \le t \le 1$ be a $C^1$ curve in $\mathbb{D}$ with z(0) = 0 and $z(1) = z_0$ . Then, we have $$V(z_0, \lambda, \beta) \subset \left\{ w \in \mathbb{C} : |\omega - C(\lambda, \gamma, \beta)| \right\} \leq R(\lambda, \gamma, \beta),$$ $$where \ C(\lambda,\gamma,\beta) = \int_0^1 c(z(t),\lambda,\beta)z'(t)dt \ \ and \ \ R(\lambda,\gamma,\beta) = \int_0^1 r(z(t),\lambda,\beta)|z'(t)|dt.$$ Now we state the following result which is vital to prove our main result.

Proposition 3.5. (Uniqueness of the curve) Let $z_0 \in \mathbb{D} \setminus \{0\}$ . Then, for $\theta \in (-\pi, \pi]$ , we have $$\log F'_{e^{i\theta}}(z_0) \in \partial V(z_0, \lambda, \beta).$$ Furthermore, if $\log f'(z_0) = \log F'_{e^{i\theta},\lambda}(z_0)$ for some $f \in \mathcal{F}(\lambda)$ and $\theta \in (-\pi, \pi]$ , then $f = F_{e^{i\theta},\lambda}$ . Finally, we state our main result to obtain the region of variability of $\log f'(z_0)$ where f runs over the class $\mathcal{F}(\lambda)$ .

Theorem 3.6. For $0 \le \lambda < 1$ and $z_0 \in \mathbb{D} \setminus \{0\}$ , the boundary $\partial V(z_0, \lambda, \beta)$ is the Jordan curve given by $$(3.7) (-\pi, \pi] \ni \theta \mapsto \log F'_{e^{i\theta}, \lambda}(z_0) = \int_0^{z_0} \frac{\delta(e^{i\theta}z, \lambda)}{z\delta(e^{i\theta}z, \lambda) - 1} dz, \quad z \in \mathbb{D}.$$ If $\log f'(z_0) = \log F'_{e^{i\theta},\lambda}(z_0)$ for some $f \in \mathcal{F}(\lambda)$ and $\theta \in (-\pi,\pi]$ , then $f(z) = F_{e^{i\theta},\lambda}(z)$ . Here $F_{e^{i\theta},\lambda}(z)$ is given by (3.8) $$F_{e^{i\theta},\lambda}(z) = f_{\alpha}(z) = \int_0^z \exp\left(\int_0^{\xi} \frac{2(1-\beta)\delta(e^{i\theta}\xi,\lambda)}{1-\delta(e^{i\theta}\xi,\lambda)}d\xi\right)d\xi.$$