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 $$v(p(z)) + zp'(z)\psi(p(z)) \prec v(q(z)) + zq'(z)\psi(q(z))$$ 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 2.1. We assume $$\chi(\beta, A, B) := -\frac{A - B}{\beta + 1} \sum_{j=0}^{\infty} \frac{\Gamma(j)}{(j-1)!(1+\beta+j\beta)} B^j + \frac{\beta}{\beta+1} + \frac{1}{\beta+1}$$ (2.1) and $$\xi(\beta, A, B) := \frac{(A - B)}{\beta + 1} \sum_{j=0}^{\infty} \frac{\Gamma(j)}{(j - 1)!(1 + \beta + j\beta)} (-B)^j - \frac{\beta}{\beta + 1} - \frac{1}{\beta + 1}$$ (2.2) where $-1 \le B < A \le 1$ . Let p be an analytic function defined on $\mathbb D$ with p(0) = 1 and satisfying $$p(z) + \beta z p'(z) \prec \frac{1 + Az}{1 + Bz}$$ . If $\beta \geq \max\{\beta_1, \beta_2\}$ , then (a) $p(z) \prec e^z$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$e\chi(\beta, A, B) = 1$$ and $\xi(\beta, A, B) = e$ respectively. (2.3) (b) $p(z) \prec \phi_s(z)$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$\chi(\beta, A, B) + \sin 1 = 1 \quad and \quad \xi(\beta, A, B) - \sin 1 = 1 \quad respectively. \tag{2.4}$$ (c) $p(z) \prec \phi_0(z)$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$\chi(\beta, A, B) = 2(\sqrt{2} - 1)$$ and $\xi(\beta, A, B) = 2$ respectively. (2.5) (d) $p(z) \prec \phi_c(z)$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$3\chi(\beta, A, B) = 1$$ and $\xi(\beta, A, B) = 3$ respectively. (2.6) (e) $p(z) \prec \phi_{Ne}(z)$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$3\chi(\beta, A, B) = 1$$ and $3\xi(\beta, A, B) = 5$ respectively. (2.7) (f) $p(z) \prec \phi_q(z)$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$\chi(\beta, A, B) = \sqrt{2} - 1$$ and $\xi(\beta, A, B) = \sqrt{2} + 1$ respectively. (2.8) (g) $p(z) \prec \sqrt{1+z}$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$\chi(\beta, A, B) = 0 \text{ and } \xi(\beta, A, B) = \sqrt{2} \text{ respectively.}$$ (2.9) (h) $p(z) \prec e^{e^z-1}$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$\chi(\beta, A, B) = e^{e^{-1}-1}$$ and $\xi(\beta, A, B) = e^{e^{-1}}$ respectively. (2.10) (i) $p(z) \prec 1 + \tanh z$ , where $\beta_1$ and $\beta_2$ are positive roots of equations $$\chi(\beta, A, B) + \tanh 1 = 1 \quad and \quad \xi(\beta, A, B) - \tanh 1 = 1 \quad respectively. \tag{2.11}$$ All the estimates on $\beta$ are sharp.

Corollary 2.3. Let $p(z) = \frac{zf'(z)}{f(z)}$ , where $f \in A$ . Assume $$\frac{z}{f(z)}\left((1+\beta)f'(z) + \beta z \left(f''(z) - \frac{f'(z)^2}{f(z)}\right)\right) \prec \frac{1+Az}{1+Bz}$$ Then. - (a) $f \in S_e^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.3). - (b) $f \in S_s^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.4). - (c) $f \in S_R^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.5). - (d) $f \in S_c^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.6). - (e) $f \in S_{Ne}^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.7). - (f) $f \in S_q^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.8). - (g) $f \in S_L^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.9). - (h) $f \in S_B^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.10). - (i) $f \in S_{tanh}^*$ if $\beta \ge \max\{\beta_1, \beta_2\}$ , where $\beta_1$ and $\beta_2$ are positive roots of the equations given in (2.11). The study of univalent funtions and differential subordination is a very vast domain and it is difficult to obtain all the subordination implications following the same methods. A great deal of research is being done in this direction to explore different ways to solve higher order differential subordination relations. Miller and Mocanu in their monograph mentioned the concept of admissible functions. Recently, authors in [29] studied first and second order differential subordination using this approach.

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. 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 $$F(a,b;c;z) = {}_{2}F_{1}(a,b;c;z) = \sum_{k=0}^{\infty} \frac{(a)_{k}(b)_{k}}{(c)_{k}k!} z^{k}.$$ (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: $$\sigma(f) := \inf_{z \in \mathbb{D}} \operatorname{Re} \left( \frac{zf'(z)}{f(z)} \right) \in [-\infty, 1]$$ 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 $\overline{\mathbb{D}} \setminus \mathbf{E}(q)$ .