On a subclass of starlike functions associated with a vertical strip domain

Lemma 1. (see [2]) Let $f \in A$ . Then $f \in \mathcal{MS}(\alpha)$ $(\pi/2 \le \alpha < \pi)$ if and only if $$\left(\frac{zf'(z)}{f(z)} - 1\right) \prec F_{\alpha}(z) = \frac{1}{2i\sin\alpha}\log\left(\frac{1 + e^{i\alpha}z}{1 + e^{-i\alpha}z}\right) \quad (z \in \mathbb{U}). \tag{2.4}$$

Lemma 2. (see [6]) Let h be analytic and convex univalent in $\mathbb{U}$ , and $\beta, \gamma \in \mathbb{R}$ with $\Re(\beta h(z) + \gamma) \geq 0$ . If q is analytic in $\mathbb{U}$ , with q(0) = h(0), then $$q(z) + \frac{zq'(z)}{\beta q(z) + \gamma} \prec h(z) \Longrightarrow q(z) \prec h(z) \quad (z \in \mathbb{U}).$$

Lemma 3. (see [7]) Let the function r(z) given by $$r(z) = \sum_{n=1}^{\infty} C_n z^n$$ be analytic and univalent in $\mathbb{U}$ , and suppose that r(z) maps $\mathbb{U}$ onto a convex domain. If the function q(z) given by $$q(z) = \sum_{n=1}^{\infty} A_n z^n$$ is analytic in $\mathbb{U}$ and satisfies the following subordination relation: $$q(z) \prec r(z) \quad (z \in \mathbb{U}),$$ then $$|A_n| \le |C_1| \quad (n \in \mathbb{N}).$$

Theorem 1. A function $f \in \mathcal{MS}(\alpha)$ $(\pi/2 \le \alpha < \pi)$ if and only if $$f(z) = z \cdot \exp\left[\frac{1}{2i\sin\alpha} \int_0^z \frac{1}{t} \log\left(\frac{1 + e^{i\alpha}w(t)}{1 + e^{-i\alpha}w(t)}\right) dt\right] \quad (z \in \mathbb{U}), \tag{3.1}$$ where w(z) is a Schwarz function.

Theorem 2. A function $f \in \mathcal{MS}(\alpha)$ $(\pi/2 \le \alpha < \pi)$ if and only if $$f(z) * \left\{ \frac{z^2}{(1-z)^2} - \frac{z}{1-z} \cdot \frac{1}{2i\sin\alpha} \log\left(\frac{1+e^{i(\theta+\alpha)}}{1+e^{i(\theta-\alpha)}}\right) \right\} \neq 0 \quad (z \in \mathbb{U}), \tag{3.4}$$ where \* denotes the Hadamard product, $0 < \theta < 2\pi$ and $\theta - \alpha \neq \pi$ .

Theorem 3. Let $f \in A$ and satisfy the following subordination $$\left(1 + \frac{zf''(z)}{f'(z)}\right) \prec 1 + F_{\alpha}(z) \quad (z \in \mathbb{U}). \tag{3.8}$$ Then $$\frac{zf'(z)}{f(z)} \prec 1 + F_{\alpha}(z) \quad (z \in \mathbb{U}), \tag{3.9}$$ that is, $f \in \mathcal{MS}(\alpha)$ , where $F_{\alpha}$ is given by (2.1).

Theorem 4. Let $$f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{MS}(\alpha)$$ . Then $|a_n| \leq 1 \quad (n \in \mathbb{N})$ .

Theorem 5. Let $f \in \mathcal{MS}(\alpha)$ . Then, for each z (|z| = r < 1), $$1 + \frac{1}{2\sin\alpha} \left[ M_1(r,\alpha) - M_2(r,\alpha) \right] \le \Re\left( \frac{zf'(z)}{f(z)} \right) \le 1 + \frac{1}{2\sin\alpha} \left[ M_1(r,\alpha) + M_2(r,\alpha) \right] \tag{4.1}$$ and $$\left|\Im\left(\frac{zf'(z)}{f(z)}\right)\right| \le \frac{1}{2\sin\alpha}\log\left[N(r,\alpha)\right],\tag{4.2}$$ where $$M_1(r,\alpha) = \arcsin\left(\frac{-r^2\sin 2\alpha}{\sqrt{1 - 2r^2\cos 2\alpha + r^4}}\right),\tag{4.3}$$ $$M_2(r,\alpha) = \arcsin\left(\frac{2r\sin\alpha}{\sqrt{1 - 2r^2\cos 2\alpha + r^4}}\right),\tag{4.4}$$ $$N(r,\alpha) = \frac{\sqrt{1 - 2r^2 \cos 2\alpha + r^4 + 2r \sin \alpha}}{1 - r^2}.$$ (4.5)

Theorem 6. Assume that $$\frac{\pi}{2} \le \alpha < \pi \text{ and } 0 \le \gamma < 1.$$ Then $$\mathcal{MS}(\alpha) \subset \mathcal{SS}(\gamma) \quad (|z| < r_1),$$ where $r_1 \in (0,1)$ is the least positive root of the following equation: $$\arctan\left(\frac{\log\left[N(r,\alpha)\right]}{2\sin\alpha + M_1(r,\alpha) - M_2(r,\alpha)}\right) - \frac{\pi}{2}\gamma = 0 \quad \left(0 \le r < 1\right),$$ where $M_1(r,\alpha)$ , $M_2(r,\alpha)$ and $N(r,\alpha)$ are given by (4.3), (4.4) and (4.5), respectively.

Theorem 7. Let $\pi/2 \leq \alpha < \pi$ . Then $$\mathcal{MS}(\alpha) \subset \mathcal{PS} \quad (|z| \leq r_2),$$ where $r_2 \in (0,1)$ is the least positive root of the equation: $$\frac{1}{4\sin^2\alpha} \left\{ \log\left[N(r,\alpha)\right] \right\}^2 - \frac{1}{\sin\alpha} \left[M_1(r,\alpha) - M_2(r,\alpha)\right] - 1 = 0 \quad \left(0 \le r < 1\right),$$ where $M_1(r,\alpha)$ , $M_2(r,\alpha)$ and $N(r,\alpha)$ are given by (4.3), (4.4) and (4.5), respectively.

Theorem 8. Let $\pi/2 \leq \alpha < \pi$ . Then $$\mathcal{MS}(\alpha) \subset \mathcal{SL} \qquad (|z| \le r_0),$$ where $r_0 := \min\{r_3, r_4\}, r_3, r_4 \in (0,1)$ are the least positive root of the equations: $$\left\{ \left( 1 + \frac{1}{2\sin\alpha} \left[ M_1(r,\alpha) + M_2(r,\alpha) \right] \right)^2 + \frac{1}{4\sin^2\alpha} \left( \log \left[ N(r,\alpha) \right] \right)^2 \right\}^2 \\ = 2 \left( 1 + \frac{1}{2\sin\alpha} \left[ M_1(r,\alpha) + M_2(r,\alpha) \right] \right)^2 - \frac{1}{2\sin^2\alpha} \left( \log \left[ N(r,\alpha) \right] \right)^2, \tag{4.15}$$ and $$\left\{ \left( 1 + \frac{1}{2\sin\alpha} \left[ M_1(r,\alpha) - M_2(r,\alpha) \right] \right]^2 + \frac{1}{4\sin^2\alpha} \left( \log \left[ N(r,\alpha) \right] \right)^2 \right\}^2 \\ = 2 \left( 1 + \frac{1}{2\sin\alpha} \left[ M_1(r,\alpha) - M_2(r,\alpha) \right] \right)^2 - \frac{1}{2\sin^2\alpha} \left( \log \left[ N(r,\alpha) \right] \right)^2, \tag{4.16}$$ respectively, where $M_1(r,\alpha)$ , $M_2(r,\alpha)$ and $N(r,\alpha)$ are given by (4.3), (4.4) and (4.5), respectively.

Definition 1. A function $f \in \mathcal{A}$ is said to belong to the class $\mathcal{MS}(\alpha)$ $(\pi/2 \le \alpha < \pi)$ , if it satisfies the following conditions: $$1 + \frac{\alpha - \pi}{2\sin\alpha} < \Re\left(\frac{zf'(z)}{f(z)}\right) < 1 + \frac{\alpha}{2\sin\alpha} \quad (z \in \mathbb{U}). \tag{1.2}$$ Remark 1. We note that the inequalities (see [2]) $$1 - \frac{\pi}{4} \le 1 + \frac{\alpha - \pi}{2\sin\alpha} < \frac{1}{2}, \ 1 + \frac{\alpha}{2\sin\alpha} \ge 1 + \frac{\pi}{4} \quad (\pi/2 \le \alpha < \pi). \tag{1.3}$$ It is clear that $$\mathcal{MS}(\alpha) \subset \mathcal{S}^* \ (\pi/2 \leq \alpha < \pi) \text{ and } \mathcal{MS}(\pi/2) \subset \mathcal{S}(1 - \pi/4, 1 + \pi/4),$$ where the class $S(\beta, \gamma)$ , $0 \le \beta < 1 < \gamma$ , was considered recently by Kwon et al. in [4]. This paper is organized as follows. In Section 2, we recall certain preliminary lemmas, which are useful in the study of the above classes of functions. In Section 3, we consider some basic properties of the class $\mathcal{MS}(\alpha)$ , such as integral representation, property of convolution, sufficient condition and coefficient inequalities. In Section 4, we consider radius problems and inclusion relations for certain classes of strongly starlike functions, parabolic starlike functions and $\mathcal{SL} \subset \mathcal{S}$ , which are closely related to the class $\mathcal{MS}(\alpha)$ , and the derivations are similar to those used earlier by Sun et al. [13] and Kwon et al. [4]. Our results are essential improvements of the corresponding results obtained by Kargar et al.* [2].