On A Subclass of Close-to-Convex Functions

Lemma 1.1. [16] If $g(z) = z + \sum_{n=2}^{\infty} b_n z^n \in \mathcal{S}^*(\frac{k-1}{k})$ , then $$G_k(z) = \frac{g_k(z)}{z^{k-1}} = z + \sum_{n=2}^{\infty} B_n z^n \in \mathcal{S}^* \subset \mathcal{S}.$$ $$(1.3)$$

Lemma 1.2. [12] Let $f(z) = 1 + \sum_{k=1}^{\infty} c_k z^k$ be analytic in $\mathcal{U}$ and $g(z) = 1 + \sum_{k=1}^{\infty} d_k z^k$ be analytic and convex in $\mathcal{U}$ . If $f \prec g$ , then $$|c_k| \le |d_1|$$ where $k \in \mathbb{N} := \{1, 2, 3, \ldots\}.$

Lemma 1.3. [17] Let $\gamma \geq 0$ and $f \in \mathcal{K}$ . Then $$F(z) = \frac{1+\gamma}{z^{\gamma}} \int_0^z t^{\gamma-1} f(t) dt \in \mathcal{K}.$$

Theorem 2.1. Let $0 \le \mu \le \lambda \le 1$ . Then we have $$\mathcal{K}_{s}^{(k)}(\lambda,\mu,\varphi)\subset\mathcal{K}\subset\mathcal{S}.$$

Theorem 2.2. Let $0 \le \mu \le \lambda \le 1$ . If $f \in K_s^{(k)}(\lambda, \mu, \varphi)$ , then $$|a_n| \le \frac{1}{1 + (n-1)(\lambda - \mu + n\lambda\mu)} \left(1 + \frac{|\varphi'(0)|(n-1)}{2}\right) \quad (n \in \mathbb{N}).$$

Corollary 2.1. If $f \in \mathcal{K}_s^{(k)}(\lambda, \varphi)$ , then $$|a_n| \le \frac{1}{1 + \lambda(n-1)} \left( 1 + \frac{|\varphi'(0)|(n-1)}{2} \right) \quad (n \in \mathbb{N}).$$ Furthermore, let $\lambda = 0$ in Corollary 2.1, we have

Corollary 2.2. If $f \in \mathcal{K}_s^{(k)}(\varphi)$ , then $$|a_n| \le \left(1 + \frac{|\varphi'(0)|(n-1)}{2}\right) \quad (n \in \mathbb{N}).$$ In this section, we obtain the Fekete-Szegö inequality. To prove our result, we need the following lemmas:

Lemma 2.1. [8] If $p(z) = 1 + c_1 z + c_2 z^2 + c_3 z^3 + ...$ is a function with positive real part, then for any complex number $\mu$ $$|c_2 - \mu c_1^2| \le 2 \max\{1, |2\mu - 1|\}$$ and the result is sharp for the functions given by $p(z) = \frac{1+z^2}{1-z^2}$ and $p(z) = \frac{1+z}{1-z}$ .

Lemma 2.2. [8] Let $G(z) = z + b_2 z^2 + \cdots$ is in $S^*$ . Then, $$|b_3 - \lambda b_2^2| \le \max\{1 - |3 - 4\lambda|\}$$ which is sharp for the Koebe function, k if $|\lambda - \frac{3}{4}| \ge \frac{1}{4}$ and for $(k(z^2))^{\frac{1}{2}} = \frac{z}{1-z^2}$ if $|\lambda - \frac{3}{4}| \le \frac{1}{4}$ .

Theorem 2.3. Let $\varphi(z) = 1 + Q_1 z + Q_2 z^2 + Q_3 z^3 + ...$ where $\varphi(z) \in \mathcal{A}$ and $\varphi'(0) > 0$ . For a function $f(z) = z + a_2 z^2 + a_3 z^3 + ...$ belonging to the class $\mathcal{K}_s^{(k)}(\lambda, \mu, \varphi)$ and $\mu \in \mathbb{C}$ , the following sharp estimate holds $$|a_{3}-\mu a_{2}^{2}| \leq \frac{1}{3(1+2\lambda-2\mu+6\lambda\mu)} \max\{1, |3-4\alpha|\} + \frac{Q_{1}}{3(1+2\lambda-2\mu+6\lambda\mu)} \max\{1, |2\beta-1|\} + 2Q_{1}\left(\frac{1}{3(1+2\lambda-2\mu+6\lambda\mu)} - \frac{\mu}{2(1+\lambda-\mu+2\lambda\mu)^{2}}\right). \quad (2.7)$$ where $$\alpha = \frac{3\delta(1 + 2\lambda - 2\mu + 6\lambda\mu)}{4(1 + \lambda - \mu + 2\lambda\mu)}$$ and $$\beta = \frac{1}{2} \left( 1 - \frac{Q_2}{Q_1} - \frac{3\delta Q_2^2 d_1^2 (1 + 2\lambda - 2\mu + 6\lambda\mu)}{4(1 + \lambda - \mu + 2\lambda\mu)^2} \right)$$

Theorem 2.4. Let $g(z) = z + \sum_{n=2}^{\infty} b_n z^n$ be analytic in $\mathcal{U}$ and $-1 \leq B < A \leq 1$ . If $f(z) \in A$ defined by (1.1) satisfies the inequality $$(1+|B|)\sum_{n=2}^{\infty}n[1+(n-1)(\lambda-\mu+n\lambda\mu)]|a_n|+(1+|A|)\sum_{n=2}^{\infty}|B_n| \le A-B \quad (2.11)$$ and for n = 2, 3, ... the coefficients of $B_n$ given by (1.4), then $f(z) \in \mathcal{K}_s^{(k)}(\lambda, \mu, A, B)$ .

Corollary 2.3. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ and $g(z) = z + \sum_{n=2}^{\infty} b_n z^n$ be analytic in $\mathcal{U}$ and $-1 \leq B < A \leq 1$ . If $$(1+|B|)\sum_{n=2}^{\infty}n[1+\lambda(n-1)]|a_n|+(1+|A|)\sum_{n=2}^{\infty}|B_n|\leq A-B,$$ where $B_n$ given by (1.4), then $f(z) \in \mathcal{K}_s^{(k)}(\lambda, A, B)$ . Further setting $\lambda = 0$ in Corollary 2.3, we obtain

Corollary 2.4. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ and $g(z) = z + \sum_{n=2}^{\infty} b_n z^n$ be analytic in $\mathcal{U}$ and $-1 \leq B < A \leq 1$ . If $$(1+|B|)\sum_{n=2}^{\infty}n|a_n|+(1+|A|)\sum_{n=2}^{\infty}|B_n| \le A-B,$$ where $B_n$ given by (1.4), then $f(z) \in \mathcal{K}_s^{(k)}(A, B)$ . Remark 2.1. By taking $A=\beta, B=-\alpha\beta$ in Corollary 2.4, we get the result obtained in [15, Theorem 5]. In addition, by taking $A=1-2\gamma, B=-1$ , we get the result obtained in [13,Theorem 2].

Definition 1.1. [4] Let f(z) be analytic in U. We say f ∈ K^s if there exists a function g(z) ∈ S^∗ (1/2) such that $$\Re\Big(-\frac{z^2f'(z)}{g(z)g(-z)}\Big)>0.$$ Remark 1.1. Note that if g(z) ∈ S^∗ (1/2), then (−g(z)g(−z))/z ∈ S^∗ [3]. Here, we recall the concept of subordination between analytic functions. Given two functions f(z) and g(z), which are analytic in U. The function f(z) is subordinate to g(z), written as f(z) ≺ g(z), if there exists an analytic function w(z) defined in U with $$w(0) = 0$$ and $|w(z)| < 1$ such that $$f(z) = g(w(z)).$$ In particular, if g is univalent in $\mathcal{U}$ , then we have the following equivalence $$f(0) = g(0)$$ and $f(\mathcal{U}) \subset g(\mathcal{U})$ . Using the concept of subordination, Wang et al.[15] introduced a general class $\mathcal{K}_s(\varphi)$ .

Definition 1.2. [15] For a function $\varphi$ with positive real part, the class $\mathcal{K}_s(\varphi)$ consists of function $f \in \mathcal{A}$ satisfying $$-\frac{z^2f'(z)}{g(z)g(-z)} \prec \varphi(z)$$ for some function $g(z) \in \mathcal{S}^*(1/2)$ . Recently, Goyal and Singh[6] introduced and studied the following subclass of analytic functions:

Definition 1.3. [6] For a function $\varphi$ with positive real part, a function $f \in \mathcal{A}$ is said to be in the class $\mathcal{K}_s(\lambda, \mu, \varphi)$ if it satisfies the following subordination condition: $$\frac{z^2 f'(z) + z^3 f''(z)(\lambda - \mu + 2\lambda\mu) + \lambda\mu z^4 f'''(z)}{-q(z)g(-z)} \prec \varphi(z)$$ where $0 \le \mu \le \lambda \le 1$ and $g(z) \in \mathcal{S}^*(1/2)$ . Motivated by aforementioned works, we now introduce the following subclass of analytic functions:

Definition 1.4. Suppose $\varphi \in \mathcal{P}$ . A function $f \in \mathcal{A}$ is said to be in the class $K_s^{(k)}(\lambda, \mu, \varphi)$ if it satisfies the following subordination condition: $$\frac{z^k f'(z) + z^{k+1} f''(z)(\lambda - \mu + 2\lambda\mu) + \lambda\mu z^{k+2} f'''(z)}{g_k(z)} \prec \varphi(z)$$ where $0 \le \mu \le \lambda \le 1$ , $g(z) = z + \sum_{n=2}^{\infty} b_n z^n \in \mathcal{S}^*(\frac{k-1}{k})$ , $k \ge 1$ is a fixed positive integer and $g_k(z)$ is defined by the following equality $$g_k(z) = \prod_{v=0}^{k-1} \varepsilon^{-v} g(\varepsilon^v z)$$ (1.1) with $\varepsilon = e^{2\pi i/k}$ . For $\varphi(z) = (1 + Az)/(1 + Bz)$ , we get the class

Definition 1.5. A function $f \in \mathcal{A}$ is said to be in the class $K_s^{(k)}(\lambda, \mu, A, B)$ if it satisfies the following subordination condition: $$\frac{z^k f'(z) + z^{k+1} f''(z)(\lambda - \mu + 2\lambda\mu) + \lambda\mu z^{k+2} f'''(z)}{g_k(z)} \prec \frac{1 + Az}{1 + Bz}$$ (1.2) where $0 \le \mu \le \lambda \le 1$ , $g(z) = z + \sum_{n=2}^{\infty} b_n z^n \in \mathcal{S}^*(\frac{k-1}{k})$ , $k \ge 1$ is a fixed positive integer and $g_k(z)$ is defined by the following equality $$g_k(z) = \prod_{v=0}^{k-1} \varepsilon^{-v} g(\varepsilon^v z)$$ with $\varepsilon = e^{2\pi i/k}$ . The condition in (1.2) is equivalent to $$\left| \frac{z^{k}f'(z) + z^{k+1}f''(z)(\lambda - \mu + 2\lambda\mu) + \lambda\mu z^{k+2}f'''(z)}{g_{k}(z)} - 1 \right| < \left| A + \frac{B(z^{k}f'(z) + z^{k+1}f''(z)(\lambda - \mu + 2\lambda\mu) + \lambda\mu z^{k+2}f'''(z))}{g_{k}(z)} \right|.$$ Remark 1.2. (a) For $\mu = 0$ , and k = 2, we have the class $\mathcal{K}_s(\lambda, A, B)[17]$ . - (b) When $A = 1 2\gamma$ , B = -1 and $\lambda = \mu = 0$ , we obtain the class $\mathcal{K}_s^{(k)}(\gamma)$ [15]. In addition, if k = 2, then we obtain the class $\mathcal{K}_s(\gamma)$ [11]. - (c) When $A = \beta$ , $B = -\alpha\beta$ and $\lambda = \mu = 0$ , then we obtain the class $\mathcal{K}_s^{(k)}(\alpha, \beta)$ in [18]. In addition, if k = 2, then we obtain the class $\mathcal{K}_s(\alpha, \beta)$ [16]. The following lemmas are needed in order to prove our main results: