Sufficient conditions for strong starlikeness

Lemma 1.1. [16, Theorem 2.3i, p.35] Let $\Omega \subset \mathbb{C}$ and suppose that $\psi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ satisfies the condition $\psi(i\rho, \sigma, \mu + i\nu; z) \notin \Omega$ whenever $\rho, \sigma, \mu$ and $\nu$ are real numbers, $\sigma \leq -n(1+\rho^2)/2, \mu + \sigma \leq 0$ . If $p \in \mathcal{H}[1,n]$ and $\psi(p(z), zp'(z), z^2p''(z); z) \in \Omega$ for $z \in \mathbb{D}$ , then $\operatorname{Re} p(z) > 0$ in $\mathbb{D}$ .

Lemma 1.2. [16, Theorem 3.4i, p.134] Let q be univalent in $\mathbb{D}$ and let $\varphi$ and $\nu$ be analytic in a domain D containing $q(\mathbb{D})$ with $\varphi(w) \neq 0$ when $w \in q(\mathbb{D})$ . Set $Q(z) := zq'(z)\varphi(q(z))$ , $h(z) := \nu(q(z)) + Q(z)$ . Suppose that (i) either h is convex or Q(z) is starlike univalent in $\mathbb{D}$ and (ii) $\operatorname{Re}(zh'(z)/Q(z)) > 0$ for $z \in \mathbb{D}$ . If p is analytic in $\mathbb{D}$ , p(0) = q(0) and satisfies (1.1) $$\nu(p(z)) + zp'(z)\varphi(p(z)) \prec \nu(q(z)) + zq'(z)\varphi(q(z)),$$ then $p \prec q$ and q is the best dominant.

Lemma 1.3. [16, Theorem 3.4a, p.120] Let q be analytic in $\mathbb{D}$ and $\phi$ be analytic in a domain D containing $q(\mathbb{D})$ and suppose that (i) $\operatorname{Re} \phi(q(z)) > 0$ and either (ii) q is convex, or (iii) $Q(z) = zq'(z)\phi(q(z))$ is starlike. If p is analytic in $\mathbb{D}$ , p(0) = q(0), $p(\mathbb{D}) \subset D$ and $p(z) + zp'(z)\phi(p(z)) \prec q(z)$ , then $p \prec q$ .

Theorem 2.1. Let n be a positive integer, $0 < \alpha \le 1, C(z) = C \ge 0$ . Suppose that the functions $D, E : \mathbb{D} \to \mathbb{C}$ satisfy $$(2.1) |\operatorname{Im} E(z)| < n\alpha(\operatorname{Re} D(z) - C).$$ If $p \in \mathcal{H}[1, n]$ satisfies the equation <span id="page-2-4"></span>(2.2) $$Cz^{2}p''(z) + D(z)zp'(z) + E(z)p(z) = 0$$ and $p(z) \neq 0$ , then $p(z) \prec ((1+z)/(1-z))^{\alpha}$ .

Corollary 2.2. Let n be a positive integer, $C(z) = C \ge 0$ . Suppose that the functions $D, E : \mathbb{D} \to \mathbb{C}$ satisfy $$|\operatorname{Im} E(z)| < n(\operatorname{Re} D(z) - C).$$ If $p \in \mathcal{H}[1,n]$ satisfies $Cz^2p''(z) + D(z)zp'(z) + E(z)p(z) = 0$ and $p(z) \neq 0$ , then $\operatorname{Re} p(z) > 0$ . By taking C(z) = 0 in Theorem 2.1, we get the following result for first order differential subordination.

Corollary 2.3. Let n be a positive integer, $0 < \alpha \le 1$ . Suppose that the functions $D, E : \mathbb{D} \to \mathbb{C}$ satisfy $$|\operatorname{Im} E(z)| < n\alpha \operatorname{Re} D(z).$$ If $p \in \mathcal{H}[1,n]$ satisfies D(z)zp'(z) + E(z)p(z) = 0 and $p(z) \neq 0$ , then $p(z) \prec ((1+z)/(1-z))^{\alpha}$ . Remark 2.4. The Corollary 2.3 for $\alpha = 1$ should be compared with [16, Corollary 4.1a.1, p. 189] The confluent (Kummer) hypergeometric function $\Phi(a, c; z)$ is given by (2.10) $$\Phi(a,c;z) = \frac{\Gamma(c)}{\Gamma(a)} \sum_{n=0}^{\infty} \frac{\Gamma(a+n)}{\Gamma(c+n)} \frac{z^n}{n!} = \sum_{n=0}^{\infty} \frac{(a)_n}{(c)_n} \frac{z^n}{n!},$$ where $a, c \in \mathbb{C}$ , $c \neq 0, -1, -2, \cdots$ , and $(\lambda)_n$ denotes the Pochhammer symbol defined by $(\lambda)_0 = 1$ , $(\lambda)_n = \lambda(\lambda + 1)_{n-1}$ . The function $\Phi \in \mathcal{H}[1, 1]$ is a solution of the differential equation <span id="page-4-0"></span>(2.11) $$z\Phi''(a,c;z) + (c-z)\Phi'(a,c;z) - a\Phi(a,c;z) = 0$$ introduced by Kummer in 1837 [29]. The function $\Phi(a, c; z)$ satisfies the following recursive relation $$c\Phi'(a; c; z) = a\Phi(a+1; c+1; z).$$ When $\operatorname{Re} c > \operatorname{Re} a > 0$ , the function $\Phi$ can be expressed in the integral form <span id="page-4-1"></span> $$\Phi(a; c; z) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int_0^1 t^{a-1} (1-t)^{c-a-1} e^{tz} dt.$$ There has been several works [1, 4, 14, 22] studying geometric properties of the function $\Phi(a; c; z)$ , such as on its close-to-convexity, starlikeness and convexity. By the use of Theorem 2.1, we obtain the following sufficient conditions for $\Phi(a, c; z) \prec ((1+z)/(1-z))^{\alpha}$ , $(0 < \alpha \le 1)$ . <span id="page-4-2"></span>Corollary 2.5. Let $0 < \alpha \le 1$ . If $a, c \in \mathbb{R}$ satisfy $$|c - 1| > \sqrt{1 + a^2/\alpha^2}$$ and $\Phi(a; c; z) \neq 0$ , then $\Phi(a; c; z) \prec ((1+z)/(1-z))^{\alpha}$ .

Corollary 2.8. Suppose that $0 < \alpha \le 1$ and if $b, p, c \in \mathbb{R}$ satisfy the following condition $$|c| < 4\alpha(k-1)$$ and $u_p(z) \neq 0$ , then $u_p(z) \prec ((1+z)/(1-z))^{\alpha}$ .

Corollary 2.11. Let n be a positive integer, $0 < \alpha \le 1$ and $C(z) = C \ge 0$ . Suppose the functions $D, E : \mathbb{D} \to \mathbb{C}$ satisfy $$|\operatorname{Im} E(z)| < n\alpha(\operatorname{Re} D(z) - C).$$ If $h \in \mathcal{A}_n$ satisfy $$C\left(2\left(\frac{zh'(z)}{h(z)}\right)^{3} - \frac{3z^{3}h'(z)h''(z)}{h^{2}(z)} + \frac{z^{3}h'''(z)}{h(z)}\right) + (2C + D(z))\left(\frac{z^{2}h''(z)}{h(z)} - \left(\frac{zh'(z)}{h(z)}\right)^{2}\right) + (D(z) + E(z))\frac{zh'(z)}{h(z)} = 0,$$ then $h \in \mathcal{S}^*[\alpha]$ .

Theorem 3.1. Let p be an analytic function defined on $\mathbb{D}$ with p(0) = 1 satisfying $$1 + \beta \frac{zp'(z)}{p(z)} \prec \varphi_{CAR}(z).$$ If $$|\beta| \ge (\sqrt{(4\sqrt{3}+8)/(3\sqrt{3})})/\alpha \simeq 1.6947/\alpha \ (0 < \alpha \le 1)$$ , then $p(z) \prec ((1+z)/(1-z))^{\alpha}$ .

Theorem 3.2. Let p be an analytic function defined on $\mathbb{D}$ with p(0) = 1 satisfying $$1 + \beta \frac{zp'(z)}{p^2(z)} \prec \varphi_{CAR}(z),$$ then the following results hold: (a) If $\beta \ge 4$ or $\beta \le -4/3$ , then p(z) < (1+z)/(1-z). (b) If $\beta > 0$ and $0 < \alpha < 1$ satisfy <span id="page-9-2"></span> $$(3.5) \qquad 9\alpha^2\beta^2 \left(\frac{1-\alpha}{1+\alpha}\right)^{\alpha} (1-\alpha^2)^{-2} \left(-8+\alpha^2\left(8+3\beta^2\left(\frac{1-\alpha}{1+\alpha}\right)^{\alpha}\right)\right) \\ -64\alpha\beta(1-\alpha^2)^{-\frac{1}{2}} \left(\sqrt{\frac{1-\alpha}{1+\alpha}}\right)^{\alpha} \sin\left(\frac{\alpha\pi}{2}\right) \ge 16\beta^2$$ then $$p(z) \prec ((1+z)/(1-z))^{\alpha}$$ .

Theorem 3.7. Let p be an analytic function defined on $\mathbb{D}$ with p(0) = 1 satisfying $$p(z) + \beta z p'(z) \prec \varphi_{CAR}(z)$$ for $\beta \geq 0$ . Then $p(z) \prec (1+z)/(1-z)$ .

Theorem 3.9. Let p be an analytic function defined on $\mathbb{D}$ with p(0) = 1 satisfying $$p(z) + \beta \frac{zp'(z)}{p^2(z)} \prec \varphi_{CAR}(z) \quad for \quad \beta \leq 0.$$ Then $p(z) \prec (1+z)/(1-z)$ .