Starlike Functions Associated with a Non-Convex Domain

Theorem 1. A function $f \in \mathcal{S}_{nc}^*$ if and only if there exists an analytic function $q(z) \prec \varphi_{nc}(z) = (1+z)/\cos z$ , such that <span id="page-2-0"></span> $$f(z) = z \exp \int_0^z \frac{q(t) - 1}{t} dt. \tag{4}$$

Theorem 2. Let $f \in \mathcal{S}_{nc}^*$ and $\tilde{f}$ be the extremal function given by (6). Then the following holds: 1. Growth theorem: For $|z_0| = r < 1$ , we have $$-\tilde{f}(-r) \le |f(z_0)| \le \tilde{f}(r).$$ Equality holds for some $z_0 \neq 0$ if and only if f is a rotation of $\tilde{f}$ . 2. Rotation theorem: For $|z_0| = r < 1$ , we have $$|\arg\{f(z_0)/z_0\}| \le \max_{|z|=r} \arg\{\tilde{f}(z)/z\}.$$ 3. Distortion theorem: For $|z_0| = r < 1$ , we have $$\tilde{f}'(-r) \le |f'(z_0)| \le \tilde{f}'(r).$$ Equality holds for some $z_0 \neq 0$ if and only if f is a rotation of $\tilde{f}$

Theorem 3. For |z| = r, min $\operatorname{Re}\{\varphi_{nc}(z)\} = \varphi(-r)$ and $\max \operatorname{Re}\{\varphi_{nc}(z)\} = \varphi(r)$ .

Theorem 4. (Function's Bounds) Let $\Phi_R$ and $\Phi_I$ denote the real and imaginary part of $\Phi_{nc}(z) = (1+z)/\cos z$ respectively. Then we have - 1. $0 \le \Phi_R \le (4\cos 1)/(1+\cos 2)$ . - 2. $-\gamma_0 \leq \Phi_I \leq \gamma_0$ , where $\gamma_0 \approx 1.6471$ . - <span id="page-3-0"></span>3. $|\arg \varphi_{nc}(z)| \leq \pi/2$ .  Figure 2: Bounds of $\varphi_{nc}$

Theorem 6. If $f \in \mathcal{S}_{nc}^*$ , then $\det T_{3,1}(f) \geq -1/15$ . The bound is sharp.

Lemma 7. [19] If $p(z) = 1 + \sum_{n=2}^{\infty} p_n z^n \in \mathcal{P}$ , then for a complex number $\nu$ $$|p_2 - \nu p_1^2| \le 2 \max(1, |2\nu - 1|).$$ This result is sharp for the function $p(z) = (1+z^2)/(1-z^2)$ and p(z) = (1+z)/(1-z).

Lemma 8. [27] [5] Let $p(z) = 1 + \sum_{n=2}^{\infty} p_n z^n \in \mathcal{P}$ , then for $n, m \in \mathbb{N}$ $$|p_{n+m} - \nu p_n p_m| \le \begin{cases} 2, & 0 \le \nu \le 1\\ 2|2\nu - 1|, & elsewhere. \end{cases}$$ The next lemma gives the representations of $p_2$ , $p_3$ and $p_4$ in terms of $p_1$ . The expressions for $p_2$ and $p_3$ were given in [14, 15] and for $p_4$ in [13].

Lemma 9. Let $p(z) = 1 + \sum_{n=2}^{\infty} p_n z^n \in \mathcal{P}$ , then $$2p_{2} = p_{1}^{2} + \gamma(4 - p_{1}^{2}),$$ $$4p_{3} = p_{1}^{3} + 2(4 - p_{1}^{2})p_{1}\gamma - (4 - p_{1}^{2})p_{1}\gamma^{2} + 2(4 - p_{1}^{2})(1 - |\gamma|^{2})\eta \quad and$$ $$8p_{4} = p_{1}^{4} + (4 - p_{1}^{2})\gamma(p_{1}^{2}(\gamma^{2} - 3\gamma + 3) + 4\gamma) - 4(4 - p_{1}^{2})(1 - |\gamma|^{2})(p_{1}(\gamma - 1)\eta + \bar{\gamma}\eta^{2} - (1 - |\gamma|^{2})\rho)$$ for some complex numbers $\gamma, \eta, \rho$ with $|\gamma| \leq 1$ , $|\eta| \leq 1$ and $|\rho| \leq 1$ .

Theorem 10. If $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}_{nc}^*$ , then $$\sum_{n=2}^{\infty} (n^2 k_1 - 4) |a_n|^2 \le (4 - k_1),$$ where $k_1 = \cos^2 1$ .

Theorem 12. If the function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ belongs to the class $\mathcal{S}_{nc}^*$ , then $$|a_2| \le 1$$ , $|a_3| \le \frac{3}{4}$ , $|a_4| \le \frac{7}{12}$ , $|a_5| \le \frac{1}{3}$ . The bounds of $|a_n|$ for n = 2, 3, 4 are sharp.

Theorem 15. $k - \mathcal{ST} \subset \mathcal{S}_{nc}^*$ for $k \ge (4\cos 1)/(4\cos 1 - \cos 2 - 1)$ .

Theorem 16. $S_{nc}^* \subset ST_p(a)$ whenever $a \ge a_0 \approx 0.402301$ .

Theorem 19. If $f \in \mathcal{S}_{nc}^*$ , then f is convex function of order $\alpha$ for $|z| < r_c$ , where $r_c$ is the root of <span id="page-13-2"></span> $$(1-r)^2 - (r+\alpha(1-r))\cos r - r(1-r)\sin r = 0.$$ (16)

Theorem 20. If $f \in \mathcal{S}_{nc}^*$ then f is M-starlike in $|z| < r_3(M)$ , where $r_3(M)$ is the root of the equation $$1 - r - 2M\cos r = 0.$$ when $0 < M \le 1/2$ and $r_3(M) = 1$ when M > 1/2.

Lemma 21. [21, p.24] Let $q \in \mathcal{Q}$ with q(0) = a, and let $p(z) = a + a_n z^n + \cdots$ be analytic in $\mathbb{D}$ with $p(z) \neq a$ and $n \geq 1$ . If p is not subordinate to q, then there exist points $z_0 = r_0 e^{i\theta_0} \in \mathbb{D}$ and $\zeta_0 \in \partial \mathbb{D} \setminus E(q)$ and an $m \geq n \geq 1$ for which $p(\mathbb{D}_{r_0}) \subset q(\mathbb{D})$ , - 1. $p(z_0) = q(\zeta_0)$ , - 2. $z_0 p'(z_0) = m\zeta_0 q'(\zeta_0),$ - 3. $\operatorname{Re}\left\{\frac{z_0 p''(z_0)}{p'(z_0)} + 1\right\} \ge m \operatorname{Re}\left\{\frac{\zeta_0 q''(\zeta_0)}{q'(\zeta_0)} + 1\right\}$

Lemma 22. [21, Corollary 3.4h, p.132] Let q be univalent in $\mathbb{D}$ , and let $\theta$ and $\psi$ be analytic in a domain D containing $q(\mathbb{D})$ with $\psi(w) \neq 0$ , when $w \in q(\mathbb{D})$ . Set $Q(z) := zq'(z)\psi(q(z))$ and $h(z) := \theta(q(z)) + Q(z)$ . Suppose that either h is convex or Q(z) is starlike. In addition, assume that Re(zh'(z)/Q(z)) > 0 $(z \in \mathbb{D})$ . If p is analytic in $\mathbb{D}$ with p(0) = q(0), $p(\mathbb{D}) \subset D$ and $$\theta(p(z)) + zp'(z)\psi(p(z)) \prec \theta(q(z)) + zq'(z)\psi(q(z)),$$ then $p \prec q$ and q is the best dominant.

Theorem 23. If $p \in \mathcal{H}$ such that p(0) = 1 and <span id="page-15-1"></span> $$\operatorname{Re}\left\{\frac{zp'(z)}{p(z)}\right\} < \frac{1}{2} + k_2,\tag{18}$$ where $k_2 = 1/\cosh 2$ , then $$p(z) \prec \frac{1+z}{\cos z}$$ .

Theorem 24. If $f \in A$ and satisfies $$\operatorname{Re}\left(1 + \frac{zf''(z)}{f'(z)}\right) > \frac{1}{2} + \frac{2 + \sinh 1}{\cos 1} \quad (z \in \mathbb{D}),$$ then $f \in \mathcal{S}_{nc}^*$ .

Theorem 26. If $p \in \mathcal{H}$ such that $$p(z) + \frac{zp'(z)}{p(z)} \in \Omega_b, \tag{25}$$ then $p(z) \prec \varphi_{nc}(z)$ for $b < b_0 \approx -0.005796$ .