On a class of analytic functions closely related to starlike functions with respect to a boundary point

Theorem 2.1. Let $0 < c \le 2$ . Furthermore, let $$\beta_c(z) = -\frac{1}{2c} \int_0^z \left( \frac{\left(\frac{1+t}{1-t}\right)^c - 1}{t} \right) dt.$$ (2.1) A function g is in $G_c$ if and only if there exists a starlike function $s \in ST$ such that <span id="page-3-0"></span> $$g(z) = \left(\frac{s(z)}{z}\right)^{\frac{1}{2}} \exp\{c \,\beta_c(z)\}.$$ (2.2)

Corollary 2.1. [16] A function g is in $G_1$ if and only if there exists a function $s \in ST$ such that $(g(z))^2 = \left(\frac{s(z)}{\tau}\right)(1-z)^2$ .

Corollary 2.2. A function g is in $G_2$ if and only if there exists a function $s \in ST$ such that $g(z) = \left(\frac{s(z)}{z}\right)^{\frac{1}{2}} \exp\left(\frac{-2}{1-z}\right)$ .

Theorem 2.2. (Herglotz representation theorem) Let $0 < c \le 2$ and let g be an analytic function in $\mathbb{D}$ such that g(0) = 1. Then, $g \in \mathcal{G}_c$ if and only if <span id="page-4-0"></span> $$g(z) = \exp\left[-\int_{-\pi}^{\pi} \log(1 - ze^{-it}) d\mu(t) - \frac{1}{2} \int_{0}^{z} \left(\frac{\left(\frac{1+t}{1-t}\right)^{c} - 1}{t}\right) dt\right],\tag{2.3}$$ where $\mu(t)$ is a probability measure on $[-\pi, \pi]$ .

Corollary 2.5. Let $g \in \mathcal{G}_1$ if and only if there exists a function $p \in \mathcal{P}$ such that $$g(z) = \frac{1-z}{\sqrt{z}} \exp\left(\frac{1}{2} \int_{0}^{z} \frac{p(\zeta)}{\zeta} d\zeta\right).$$

Corollary 2.6. Let $g \in \mathcal{G}_2$ if and only if there exists a function $p \in \mathcal{P}$ such that $$g(z) = \frac{1}{\sqrt{z}} \exp\left(\frac{1}{2} \int_{0}^{z} \frac{p(\zeta)}{\zeta} d\zeta - \frac{2z}{1-z}\right).$$

Theorem 2.4. Let $0 < c \le 2$ . A function $g \in \mathcal{G}_c$ if and only if there exists a function $p \in \mathcal{H}$ such that $p < \frac{1+z}{1-z}$ and for $z \in \mathbb{D}$ , $$g(z) = \frac{1}{\sqrt{z}} \exp\left(\frac{1}{2} \left( \int_{0}^{z} \frac{p(\zeta)}{\zeta} d\zeta - \int_{0}^{z} \left( \frac{\left(\frac{1+t}{1-t}\right)^{c} - 1}{t} \right) dt \right)\right).$$

Theorem 3.1. Let $0 < c \le 2$ and $z \in \mathbb{D}$ . If $g \in \mathcal{G}_c$ , we have the following sharp inequalities. <span id="page-6-0"></span> $$|c + d_1| \le 1,$$ (3.1) $$\left| c^2 + 2d_2 - d_1^2 \right| \le 1,\tag{3.2}$$ and <span id="page-6-1"></span> $$\left|2c^3 + c + 9d_3 - 9d_1d_2 + 3d_1^3\right| \le 3. \tag{3.3}$$ Further, for $\alpha \in \mathbb{R}$ , let $$\mathcal{H}(\alpha, c) = 4d_2 - 8\alpha c d_1 - 2d_1^2 (1 + 2\alpha) + 2c^2 (1 - 2\alpha). \tag{3.4}$$ Then, <span id="page-6-3"></span> $$|\mathcal{H}(\alpha,c)| \le \begin{cases} 2\left(1 - 2\alpha|c + d_1|^2\right), & \text{if } \alpha \le \frac{1}{2}, \\ 2\left(1 - 2(1 - \alpha)|c + d_1|^2\right), & \text{if } \alpha \ge \frac{1}{2}. \end{cases}$$ (3.5) Proof. Let $$p(z) = 2z \frac{g'(z)}{g(z)} + \left(\frac{1+z}{1-z}\right)^c, \ z \in \mathbb{D}.$$ On expanding the right hand side of the above function p, we get <span id="page-6-2"></span> $$p(z) = 1 + 2(c + d_1)z + 2(c^2 + 2d_2 - d_1^2)z^2 + \frac{2}{3}(2c^3 + c + 9d_3 - 9d_1d_2 + 3d_1^3)z^3 + \cdots$$ (3.6) By making use of the known inequality $|p_i| \le 2$ for all $p \in \mathcal{P}$ , we can get the sharp inequalities given in (3.1)–(3.3). From (1.1) and (3.6) and from the known fact that $$|p_2 - \alpha p_1^2| \le \begin{cases} 2 - \alpha |p_1|^2, & \text{if } \alpha \le \frac{1}{2}, \\ 2 - (1 - \alpha) |p_1|^2, & \text{if } \alpha \ge \frac{1}{2}, \end{cases}$$ we can obtain (3.5) For c = 1 and c = 2, we have the following corollaries as stated below.

Corollary 3.1. [2] Let $z \in \mathbb{D}$ . If $g \in \mathcal{G}_1$ , we have the following inequalities. $$|1+d_1| \le 1$$ , $|1+2d_2-d_1^2| \le 1$ , $|1+3d_3-3d_1d_2+d_1^3| \le 1$ . Further, $$|\mathcal{H}(\alpha, 1)| \le \begin{cases} 2\left(1 - 2\alpha|1 + d_1|^2\right), & \text{if } \alpha \le \frac{1}{2}, \\ 2\left(1 - 2(1 - \alpha)|1 + d_1|^2\right), & \text{if } \alpha \ge \frac{1}{2}. \end{cases}$$ All of these inequalities are sharp.

Corollary 3.2. Let $z \in \mathbb{D}$ . If $g \in \mathcal{G}_2$ , the following inequalities hold. $$\left|1 + \frac{d_1}{2}\right| \le \frac{1}{2}, \left|4 + 2d_2 - d_1^2\right| \le 1, \left|6 + 3d_3 - 3d_1d_2 + d_1^3\right| \le 1.$$ Also, $$|\mathcal{H}(\alpha,2)| \le \begin{cases} 2\left(1-2\alpha|2+d_1|^2\right), & \text{if } \alpha \le \frac{1}{2}, \\ 2\left(1-2(1-\alpha)|2+d_1|^2\right), & \text{if } \alpha \ge \frac{1}{2}. \end{cases}$$ All of these inequalities are sharp.

Theorem 3.2. Let $0 < c \le 2$ and let the function g(z) be of the form (1.3) belong to the class $G_c$ . Then, for $n = 2, 3, \dots$ , the following estimates $$\left| nd_n - c(n-2)d_{n-1} + \dots + \left[ 1 + (-1)^{n-1} \right] \frac{c(c-1) \dots (c-n+2)}{2(n-1)!} d_1 + \left[ 1 - (-1)^n \right] \frac{c(c-1)(c-2) \dots (c-n+1)}{2n!} \right|^2$$ $$\leq 1 + \sum_{k=1}^{n-1} \left| (k+1)d_k - c(k-1)d_{k-1} + \dots + \left[ 1 + 3(-1)^{k-1} \right] \frac{c(c-1)(c-2)\cdots(c-k+2)}{2(k-1)!} d_1 + \left[ 1 + (-1)^k \right] \frac{c(c-1)\cdots(c-k+1)}{2k!} \right|^2$$ hold.

Lemma 3.1. [5] If $$\omega \in \mathcal{B}_0$$ is of the form $\omega(z) = \sum_{n=1}^{\infty} \omega_n z^n$ , $z \in \mathbb{D}$ , then for $v \in \mathbb{C}$ , $$\left|\omega_2 - \nu \omega_1^2\right| \le \max\left\{1, |\nu|\right\}.$$

Lemma 3.2. If $\omega \in \mathcal{B}_0$ is of the form $\omega(z) = \sum_{i=1}^{\infty} \omega_n z^n$ , $z \in \mathbb{D}$ , then for any real numbers $q_1$ and $q_2$ , the following sharp estimates holds: $$\left|\omega_3 + q_1 \ \omega_1 \ \omega_2 + q_2 \ \omega_1^3\right| \le H(q_1, q_2),$$ (3.13) where $$H(q_{1},q_{2}) := \begin{cases} 1 & \text{if } (q_{1},q_{2}) \in D_{1} \cup D_{2} \\ |q_{2}| & \text{if } (q_{1},q_{2}) \in \cup_{k=3}^{7} D_{k} \\ \frac{2}{3} & (|q_{1}|+1) \left(\frac{|q_{1}|+1}{3(|q_{1}|+1+q_{2})}\right)^{\frac{1}{2}} & \text{if } (q_{1},q_{2}) \in D_{8} \cup D_{9} \\ \frac{q_{2}}{3} & \left(\frac{q_{1}^{2}-4}{q_{1}^{2}-4q_{2}}\right) \left(\frac{q_{1}^{2}-4}{3(q_{2}-1)}\right)^{\frac{1}{2}} & \text{if } (q_{1},q_{2}) \in D_{10} \cup D_{11}/\left\{\pm 2,1\right\} \\ \frac{2}{3} & (|q_{1}|-1) \left(\frac{|q_{1}|-1}{3(|q_{1}|-1-q_{2})}\right)^{\frac{1}{2}} & \text{if } (q_{1},q_{2}) \in D_{12} \end{cases}$$ and the sets $D_{k}, k = 1, 2, \cdots$ are defined in [15]. Now we obtain a few upper bounds for early coefficients and for the Fekete-Szegö functional in the class $\mathcal{G}_c$ .

Theorem 3.3. Let $g \in \mathcal{G}_c$ , $0 < c \le 2$ . Then, $$|c + d_1| \le 1,\tag{3.14}$$ $$|d_1| \le 1 + c,\tag{3.15}$$ $$|c^2 + 2d_2 - d_1^2| \le 1, (3.16)$$ $$|d_2| \le 1 + c,\tag{3.17}$$ $$|3d_3 - 3d_1d_2 + d_1^3| \le \frac{4c^3 + 2c + 3}{6} \tag{3.18}$$ and $$|d_3| \le \frac{3 + 2c + 18c^2 + 10c^3}{18}. (3.19)$$ Furthermore, for $\delta \in \mathbb{R}$ <span id="page-10-3"></span> $$|d_2 - \delta d_1^2| \le \frac{1}{2} \max\{1, 2|1 - \delta|\} + c|2\delta - 1| + c^2| - \delta|. \tag{3.20}$$

Lemma 3.3. If $\omega \in \mathcal{B}_0$ is of the form $\omega(z) = \sum_{n=1}^{\infty} \omega_n z^n$ , $z \in \mathbb{D}$ , then $$\left|\omega_{2} - \nu \omega_{1}^{2}\right| \leq \begin{cases} -\nu, \ \nu \leq -1, \\ 1, \ -1 \leq \nu \leq 1, \\ \nu, \ \nu \geq 1. \end{cases}$$ (3.27) For v < -1 or v > 1, equality holds if and only if $\omega(z) = z$ or one of its rotations. For -1 < v < 1, equality holds if and only if $\omega(z) = z^2$ , $z \in \mathbb{D}$ or one of its rotations. For v = -1 equality holds if and only if $\omega(z) = \frac{z(\lambda+z)}{(1+\lambda z)}$ , $z \in \mathbb{D}$ or one of its rotations, while for v = 1 equality holds if and only if $w(z) = \frac{-z(\lambda+z)}{(1+\lambda z)}$ , $0 \le \lambda \le 1$ , $z \in \mathbb{D}$ or one of its rotations. We can improve the results obtained in (3.20), in view of Lemma 3.3 as follows: For $\delta \in \mathbb{R}$ , we get, $$\left| d_{2} - \delta d_{1}^{2} \right| \leq \begin{cases} (1+c) - 2(1+c)^{2} \delta, \ \delta \leq 0, \\ \frac{1 - 2c + 2(c^{2} - 2c)\delta}{2}, \ 0 \leq \delta \leq \frac{1}{2}, \\ \frac{1 - 2c + 2(c^{2} + 2c)\delta}{2}, \ \frac{1}{2} \leq \delta \leq \frac{3}{2}, \\ (1+c)^{2} \delta - 2(1+c), \ \delta \geq \frac{3}{2}. \end{cases}$$ (3.28)

Theorem 3.4. Let 0 < r < 1. If $g \in \mathcal{G}_c$ then for |z| = r < 1, <span id="page-12-2"></span> $$\sqrt{\frac{-f_{\alpha}(-r)}{r}} \le |g(z)| e^{-2c\beta_c'(r)} \le \sqrt{\frac{f_{\alpha}(-r)}{r}}.$$ (3.29)

Corollary 3.3. For 0 < r < 1, if $g \in \mathcal{G}_1$ then we have for |z| = r < 1 $$\sqrt{\frac{-f_{\alpha}(-r)}{r}}(1-r) \le |g(z)| \le \sqrt{\frac{f_{\alpha}(-r)}{r}}(1+r). \tag{3.34}$$

Corollary 3.4. For 0 < r < 1, if $g \in \mathcal{G}_2$ then we have for |z| = r < 1, $$\sqrt{\frac{-f_{\alpha}(-r)}{r}} \exp \frac{2}{1-r} \le |g(z)| \le \sqrt{\frac{f_{\alpha}(-r)}{r}} \exp \frac{2}{1+r}.$$ (3.35)

Theorem 3.5. Let $g \in \mathcal{G}_c$ . Then, <span id="page-13-0"></span> $$|d_1| < 1, |d_2| < 1, |d_3| < 1, |d_4| < 1$$ (3.36) and <span id="page-13-1"></span> $$|d_2^2 - d_3| \le 1. (3.37)$$

Lemma 4.1. [13] Let $\tau$ be univalent in $\mathbb{D}$ , $\psi$ and $\phi$ be analytic in a domain D containing $\tau(\mathbb{D})$ with $\phi(\omega) \neq 0$ when $\omega \in \tau(\mathbb{D})$ . Let $T(z) = z\tau'\phi(\tau(z))$ and $\kappa(z) = \psi(\tau(z)) + T(z)$ for $z \in \mathbb{D}$ and satisfy either T is starlike univalent in $\mathbb{D}$ or $\kappa$ is convex univalent in $\mathbb{D}$ . Also, assume that $\Re\left\{\frac{z\kappa'(z)}{T(z)}\right\} > 0, z \in \mathbb{D}$ . If $p \in \mathcal{H}$ with $p(0) = \tau(0), p(\mathbb{D}) \subset D$ , and $$\psi(p(z)) + zp'(z)\phi(p(z)) < \psi(\tau(z)) + z\tau'(z)\phi(\tau(z)), \ z \in \mathbb{D}$$ then $p < \tau$ and $\tau$ is the best dominant.

Theorem 4.1. Let g be an analytic function with g(0) = 1 and let $0 < c \le 2$ . If g satisfies <span id="page-14-1"></span> $$2z\frac{g'(z)}{g(z)} + \left(\frac{1+z}{1-z}\right)^c < 1 + \frac{2z}{1-z^2}, \ z \in \mathbb{D}$$ (4.2) then <span id="page-14-2"></span> $$p(z) := (g(z))^2 \exp\left\{-2z\beta_c'(z)\right\} < L_0(z), \ z \in \mathbb{D}.$$ (4.3)

Theorem 4.2. Let g(z) be an analytic function with g(0) = 1 and let $0 < c \le 2$ . If <span id="page-15-0"></span> $$2z\frac{g'(z)}{g(z)} + \left(\frac{1+z}{1-z}\right)^c < \frac{1+z}{1-z} + \frac{2z}{1-z^2}, \ z \in \mathbb{D}$$ (4.6) then <span id="page-15-1"></span> $$p(z) := z (g(z))^{2} \exp \left\{-2z\beta'_{c}(z)\right\} \left(\int_{0}^{z} (g(\zeta))^{2} \exp \left\{-2\zeta\beta'_{c}(\zeta)\right\} d\zeta\right)^{-1} < L_{0}(z), \ z \in \mathbb{D}.$$ (4.7)

Definition 1.1. Let $G_c$ be the class consisting of all functions of the form (1.3) satisfying <span id="page-2-0"></span> $$\Re\left\{2z\,\frac{g'(z)}{g(z)} + \left(\frac{1+z}{1-z}\right)^c\right\} > 0, \ z \in \mathbb{D},\tag{1.5}$$ where $0 < c \le 2$ . If c = 1, the class $\mathcal{G}_1 = \mathcal{G}$ was introduced and investigated by Robertson [16]. For this new class of functions, we obtain representation theorem, interesting coefficient estimates and also certain differential subordination implications involving this new class.