Starlikeness Using Special Functions and Subordination

Lemma 1.3. [23, Theorem 3.4h, p.132] Let $q : \mathbb{D} \to \mathbb{C}$ be analytic and $\psi$ and v be analytic in a domain $U \supseteq q(\mathbb{D})$ with $\psi(w) \neq 0$ whenever $w \in q(\mathbb{D})$ . Set $$Q(z) := zq'(z)\psi(q(z))$$ and $h(z) := v(q(z)) + Q(z), z \in \mathbb{D}$ . Suppose that - (i) either h(z) is convex, or Q(z) is starlike univalent in $\mathbb D$ and - (ii) $\operatorname{Re}\left(\frac{zh'(z)}{Q(z)}\right) > 0, z \in \mathbb{D}.$ If p is analytic in $\mathbb{D}$ , with p(0) = q(0), $p(\mathbb{D}) \subset U$ and $$v(p(z)) + zp'(z)\psi(p(z)) \prec v(q(z)) + zq'(z)\psi(q(z))$$ <span id="page-2-1"></span>then $p \prec q$ , and q is the best dominant.

Theorem 1.4. [15, Theorem 1(a)] Let a, b and c be non-zero real numbers such that $0 < a \le b \le c$ . Then, $$1 - \frac{ab}{b+c} \le \sigma(zF(a,b;c;z)) \le 1 - \frac{ab}{2c}.$$

Theorem 2.9. [20] Let $p \in \mathcal{H}[1, n]$ and $\Omega$ be a set in $\mathbb{C}$ . If $\psi \in \Psi[\Omega; e^z]$ and $$\psi(p(z), zp'(z), z^2p''(z); z) \subset \Omega.$$ Then, p(z) is subordinate to $e^z$ . Motivated by the above mentioned definitions and theorems, the following subordination implication is established. In the following two results, we have taken $q(z) = e^z$ .

Theorem 2.10. Let p(z) denote an analytic function with p(0) = 1 and $\alpha \ge e(e-1) + 1$ . Then, $$|1 + \alpha z p'(z) + z^2 p''(z)| < e \implies p(z) \prec e^z.$$

Theorem 2.12. [6] Assume $p \in \mathcal{H}[a, n]$ with $n \geq 2$ and $q \in \mathcal{Q}(a)$ satisfying the inequalities $$\operatorname{Re} \frac{wq''(z)}{q'(z)} \ge 0$$ and $\left| \frac{zp'(z)}{q'(w)} \right| \le k$ , for $k \geq n$ , $z \in \mathbb{D}$ and $w \in \partial \mathbb{D} \setminus \mathbf{E}(q)$ . If $\Omega \subset \mathbb{C}$ , $\psi \in \Psi_n[\Omega; q]$ and $$\psi(p(z), zp'(z), z^2p''(z), z^3p'''(z); z) \subset \Omega,$$ then p is subordinate to q. <span id="page-12-1"></span>For $q(z) = e^z$ , the following result is a particular case of above theorem.

Theorem 2.13. Assume $p \in \mathcal{H}[1, n]$ with $n \in \mathbb{N}$ satisfy the inequality $$|zp'(z)| \le n,$$ for $z \in \mathbb{D}$ and $w \in \partial \mathbb{D} \setminus \mathbf{E}(q)$ . If $\Omega \subset \mathbb{C}$ , $\psi \in \Psi_n[\Omega; q]$ and $$\psi(p(z), zp'(z), z^2p''(z), z^3p'''(z); z) \subset \Omega,$$ then $p(z) \prec e^z$ .

Theorem 2.14. Let $0 < A \le 1$ and let $\alpha, \beta, \gamma$ be non-negative real numbers. Then, the following relations are sufficient for $p(z) \prec e^z$ . (i) $$1 + \alpha z^3 p'''(z) + \beta z^2 p''(z) + \gamma z p'(z) \prec 1 + Az$$ , where $$\gamma \ge \begin{cases} Ae + \alpha + \beta^2/8\alpha, & when \ \beta \le 4\alpha \\ Ae - \alpha + \beta, & when \ \beta \ge 4\alpha. \end{cases}$$ (ii) $$1 + \alpha (z^3 p'''(z)/zp'(z)) + \beta (zp'(z)/p(z) + 1) \prec 1 + Az$$ , where $$\begin{cases} \beta - \alpha - \beta^2 / 8\alpha \ge A, & \text{when } \beta \le 4\alpha \\ \alpha \ge A, & \text{when } \beta \ge 4\alpha. \end{cases}$$

Theorem 2.15. Let $\alpha, \beta, \gamma$ be non-negative real numbers. Then, the following relations are sufficient for $p(z) \prec e^z$ . (i) $1 + \alpha z^3 p'''(z) + \beta z^2 p''(z) + \gamma z p'(z) \prec e^z$ , where $$\gamma \geq \begin{cases} e(e-1) + \alpha + \beta^2/8\alpha, & when \ \beta \leq 4\alpha \\ e(e-1) - \alpha + \beta, & when \ \beta \geq 4\alpha. \end{cases}$$ (ii) $1 + \alpha (z^3 p'''(z)/z p'(z)) + \beta (z p'(z)/p(z) + 1) \prec e^z$ , where $$\begin{cases} \beta \ge \alpha + \beta^2/8\alpha + e - 1, & \text{when } \beta \le 4\alpha \\ \alpha \ge e - 1, & \text{when } \beta \ge 4\alpha. \end{cases}$$

Theorem 2.16. Let $\alpha, \beta, \gamma$ be non-negative real numbers and $\beta_0 \approx 0.475319$ . Then, the following relations are sufficient for $p(z) \prec e^z$ . (i) $\alpha z^3 p'''(z) + \beta z^2 p''(z) + \gamma z p'(z) \prec \phi_{SG}(z)$ , where $$\gamma \ge \begin{cases} \beta_0 e + \alpha + \beta^2 / 8\alpha, & when \ \beta \le 4\alpha \\ \beta_0 e - \alpha + \beta, & when \ \beta \ge 4\alpha. \end{cases}$$ (ii) $\alpha(z^3p'''(z)/zp'(z)) + \beta(zp'(z)/p(z) + 1) \prec \phi_{SG}(z)$ , where $$\begin{cases} \beta \ge \beta_0 + \alpha + \beta^2 / 8\alpha, & \text{when } \beta \le 4\alpha \\ \alpha \ge \beta_0, & \text{when } \beta \ge 4\alpha. \end{cases}$$

Corollary 2.17. As an application, we obtain sufficient conditions for the function $f \in \mathcal{A}$ to be exponential starlike in $\mathbb{D}$ by substituting p(z) = zf'(z)/f(z).

Definition 1.1. (Gauss Hypergeometric Function): For |z| < 1 and parameters $a,b \in \mathbb{C}$ and $c \notin \{0 \cup \mathbb{Z}_-\}$ , the hypergeometric function ${}_2F_1(a,b;c;z)$ is defined by the convergent power series <span id="page-2-0"></span> $$F(a,b;c;z) = {}_{2}F_{1}(a,b;c;z) = \sum_{k=0}^{\infty} \frac{(a)_{k}(b)_{k}}{(c)_{k}k!} z^{k}.$$ (1.1) The function F(a, b; c; z) is analytic in $\mathbb{C}$ and is one of the solutions of the differential equation $$z(1-z)y'' + [c - (a+b+1)z]y' - aby = 0$$ at z=0. The differential of the function F(a,b;c;z) satisfies the relation $$\frac{\partial F(a,b;c;z)}{\partial z} = \frac{ab}{c}F(a+1,b+1;c+1;z).$$

Definition 1.2. (Order of Starlikeness): For an analytic function f, its order of starlikeness with respect to zero is defined as follows: $$\sigma(f) := \inf_{z \in \mathbb{D}} \operatorname{Re} \left( \frac{zf'(z)}{f(z)} \right) \in [-\infty, 1]$$ <span id="page-2-2"></span>In the proof of our first result, the following results are used:

Definition 2.4. Consider the analytic functions having Taylor series expansion $f(z) = a + a_n z^n + a_{n+1} z^{n+1} + \cdots$ . Then, $\mathcal{H}[a, n]$ denote the class of all such functions for some $a \in \mathbb{C}$ and fixed integer n.

Definition 2.5. [6, Definition 1, p.440] Let $\psi(r, s, t, u; z) : \mathbb{C}^4 \times \mathbb{D} \to \mathbb{D}$ be analytic and $h \in \mathcal{U}$ . Then, the function $p \in \mathcal{A}$ , satisfying the third order differential subordination relation $$\psi(p(z), zp'(z), z^2p''(z), z^3p'''(z); z) \prec h(z)$$ is called its solution. Analogous result for the second order differential subordination is stated as below:

Definition 2.6. [23] Let $\psi(r, s, t; z) : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{D}$ be analytic and h be a univalent function. For $p \in \mathcal{A}$ , the subordination relation $$\psi(p(z), zp'(z), z^2p''(z); z) \prec h(z)$$ is known as the second order differential subordination. Using the concept of admissibility, authors in [20] derived the admissibility conditions for the class associated with exponential function as follows:

Definition 2.7. Let $\Omega \subset \mathbb{C}$ be the domain. The class $\Psi_n[\Omega; e^z]$ is defined as the class of all those functions $\psi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ such that $$\psi(r, s, t; z) \notin \Omega$$ whenever $(r, s, t; z) \in \text{Dom } \psi$ , $$r = e^{e^{i\theta}}, s = me^{i\theta}r, \operatorname{Re}\left(1 + \frac{t}{s}\right) \ge m(1 + \cos\theta),$$ for $z \in \mathbb{D}$ , $\theta \in (0, 2\pi)$ and $m \ge 1$ . Extending the existing work done and using the results given in [6], we formulated an additional condition involving third order differential subordination parameter. Henceforth, the admissibility conditions for the exponential function are restated as follows:

Definition 2.8. Let $\Omega \subset \mathbb{C}$ and $n \geq 1$ . For $q(z) = e^z$ , the admissibility conditions are given as follows: $$\psi(r, s, t, u; z) \notin \Omega$$ whenever $(r, s, t, u; z) \in \text{Dom } \psi$ , $r = e^{e^{i\theta}}, s = me^{i\theta}r, \text{Re}\left(1 + \frac{t}{s}\right) \ge m(1 + \cos\theta)$ , and $\text{Re}\left(\frac{u}{s}\right) \ge m\cos 2\theta$ for $z \in \mathbb{D}$ , $\theta \in (0, 2\pi)$ , $\cos 2\theta \ge 0$ and $m \ge 1$ .

Definition 2.11. [23] Let $\mathbf{E}(q)$ be the collection of all those points $\zeta \in \partial \mathbb{D}$ such that $q(z) \to \infty$ as $z \to \zeta$ . Then, $\mathcal{Q}$ denote the class of all functions q, which are analytic and univalent on $\overline{\mathbb{D}} \setminus \mathbf{E}(q)$ .