Differential Subordinations Involving Generalized Bessel Functions

Lemma 1.1. [7, p. 28] Let $\psi \in \Psi_n[\Omega, q]$ with q(0) = a. If the analytic function $p \in \mathcal{H}[a, n]$ satisfies the following inclusion relationship $$\psi\left(p(z), zp'(z), z^2p''(z); z\right) \in \Omega,$$ for all $z \in \mathbb{D}$ , then $p \prec q$ .

Lemma 1.2. [8, p. 818] Let $\psi \in \Psi'_n[\Omega, q]$ with q(0) = a. If $p \in \mathcal{Q}(a)$ and $\psi(p(z), zp'(z), z^2p''(z); z)$ is univalent in $\mathbb{D}$ , then $$\Omega \subset \left\{ \psi \left( p(z), zp'(z), z^2p''(z); z \right) : z \in \mathbb{D} \right\}$$ implies that $p \prec q$ .

Theorem 2.1. Let $\phi \in \Phi_H[\Omega, q]$ . If $f \in \mathcal{A}$ satisfies the following inclusion relationship $$\{\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right) : z \in \mathbb{D}\} \subset \Omega,$$ then $$B_{\kappa+1}^c f(z) \prec q(z) \quad (z \in \mathbb{D}).$$

Corollary 2.2. Let $\phi \in \Phi_H[h,q]$ . If $f \in \mathcal{A}$ satisfies $$\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right) \prec h(z),$$ then $$B_{\kappa+1}^c f(z) \prec q(z) \quad (z \in \mathbb{D}).$$ Our next result is an extension of Theorem 2.1 to the case when the behavior of q on $\partial \mathbb{D}$ is not known.

Corollary 2.3. Let $\Omega \subset \mathbb{C}$ and let q be univalent in $\mathbb{D}$ with q(0) = 0. Let $\phi \in \Phi_H[\Omega, q_\rho]$ for some $\rho \in (0, 1)$ , where $q_\rho(z) = q(\rho z)$ . If $f \in \mathcal{A}$ satisfies the following inclusion relationship $$\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right) \in \Omega,$$ then $$B_{\kappa+1}^c f(z) \prec q(z) \quad (z \in \mathbb{D}).$$ Proof. We note from Theorem 2.1 that $$B_{\kappa+1}^c f(z) \prec q_{\rho}(z) \quad (z \in \mathbb{D}).$$ The result asserted by Corollary 2.3 is now deduced from the following subordination relationship $$q_{\rho}(z) \prec q(z) \quad (z \in \mathbb{D}).$$

Theorem 2.4. Let h and q be univalent in $\mathbb{D}$ with q(0) = 0 and set $q_{\rho}(z) = q(\rho z)$ and $h_{\rho}(z) = h(\rho z)$ . Let $\phi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ satisfy one of the following conditions - (1) $\phi \in \Phi_H[\Omega, q_\rho]$ for some $\rho \in (0, 1)$ , or - (2) there exists $\rho_0 \in (0,1)$ such that $\phi \in \Phi_H[h_\rho, q_\rho]$ for all $\rho \in (\rho_0, 1)$ . If $f \in A$ satisfies (2.7), then $$B_{z+1}^c f(z) \prec q(z) \quad (z \in \mathbb{D}).$$

Theorem 2.5. Let h be univalent in $\mathbb{D}$ and $\phi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ . Suppose that the differential equation $$(2.8) \qquad \phi\left(q(z), \frac{zq'(z) + (\kappa - 1)q(z)}{\kappa}, \frac{z^2q''(z) + 2(\kappa - 1)zq'(z) + (\kappa - 1)(\kappa - 2)q(z)}{\kappa(\kappa - 1)}; z\right) = h(z)$$ has a solution q with q(0) = 0 and satisfies one of the following conditions - (1) $q \in \mathcal{Q}_0$ and $\phi \in \Phi_H[h, q]$ - (2) q is univalent in $\mathbb{D}$ and $\phi \in \Phi_H[h, q_\rho]$ for some $\rho \in (0, 1)$ - (3) q is univalent in $\mathbb{D}$ and there exists $\rho_0 \in (0,1)$ such that $\phi \in \Phi_H[h_\rho, q_\rho]$ for all $\rho \in (\rho_0, 1)$ . If $f \in \mathcal{A}$ satisfies (2.7), then $$B_{\kappa+1}^c f(z) \prec q(z) \quad (z \in \mathbb{D})$$ and q is the best dominant.

Corollary 2.6. Let $\phi \in \Phi_H[\Omega, M]$ . If $f \in \mathcal{A}$ satisfies the following inclusion relationship $$\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right) \in \Omega,$$ then $$B_{\kappa+1}^c f(z) \prec Mz \quad (z \in \mathbb{D}).$$ In the special case when $\Omega = \{w : |w| < M\} = q(\mathbb{D})$ , the class $\Phi_H[\Omega, M]$ is simply denoted by $\Phi_H[M]$ . Corollary 2.6 can now be written in the following form.

Corollary 2.7. Let $\phi \in \Phi_H[M]$ . If $f \in \mathcal{A}$ satisfies the following inequality $$\left|\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right)\right| < M,$$ then $$\left| B_{\kappa+1}^c f(z) \right| < M.$$

Corollary 2.8. If M > 0, $Re(\kappa) \ge \frac{1-k}{2}$ , where $k \ge 1$ , and $f \in \mathcal{A}$ satisfies the following inequality $$|B_{\kappa}^{c} f(z)| < M,$$ then $$\left| B_{\kappa+1}^c f(z) \right| < M.$$ П

Corollary 2.9. Let $\kappa \in \mathbb{C} \setminus \{0\}$ and M > 0. If $f \in \mathcal{A}$ satisfies the following inequality $$\left|B_{\kappa}^{c}f(z)-B_{\kappa+1}^{c}f(z)\right|<\frac{M}{|\kappa|},$$ then $$\left| B_{\kappa+1}^c f(z) \right| < M.$$

Theorem 2.10. Let $\phi \in \Phi_{H,1}[\Omega, q]$ . If $f \in \mathcal{A}$ satisfies the following inclusion relationship $$\left\{\phi\left(\frac{B_{\kappa}^{c}f(z)}{B_{\kappa+1}^{c}f(z)}, \frac{B_{\kappa-1}^{c}f(z)}{B_{\kappa}^{c}f(z)}, \frac{B_{\kappa-2}^{c}f(z)}{B_{\kappa-1}^{c}f(z)}; z\right) : z \in \mathbb{D}\right\} \subset \Omega,$$ then $$\frac{B_{\kappa}^{c}f(z)}{B_{\kappa+1}^{c}f(z)} \prec q(z) \quad (z \in \mathbb{D}).$$

Corollary 2.11. Let $\phi \in \Phi_{H,1}[h,q]$ . If $f \in \mathcal{A}$ satisfies $$\phi\left(\frac{B_{\kappa}^{c}f(z)}{B_{\kappa+1}^{c}f(z)}, \frac{B_{\kappa-1}^{c}f(z)}{B_{\kappa}^{c}f(z)}, \frac{B_{\kappa-2}^{c}f(z)}{B_{\kappa-1}^{c}f(z)}; z\right) \prec h(z),$$ then $$\frac{B_{\kappa}^{c}f(z)}{B_{\kappa+1}^{c}f(z)} \prec q(z) \quad (z \in \mathbb{D}).$$ In the particular case when q(z) = 1 + Mz; M > 0, the class of admissible functions $\Phi_{H,1}[\Omega, q]$ is simply denoted by $\Phi_{H,1}[\Omega, M]$ .

Corollary 2.12. Let $\phi \in \Phi_{H,1}[\Omega, M]$ . If $f \in \mathcal{A}$ satisfies the following inclusion relationship $$\phi\left(\frac{B_{\kappa}^cf(z)}{B_{\kappa+1}^cf(z)},\frac{B_{\kappa-1}^cf(z)}{B_{\kappa}^cf(z)},\frac{B_{\kappa-2}^cf(z)}{B_{\kappa-1}^cf(z)};z\right)\in\Omega,$$ then $$\frac{B_{\kappa}^{c}f(z)}{B_{\kappa+1}^{c}f(z)} - 1 \prec Mz \quad (z \in \mathbb{D}).$$ In the special case $\Omega = \{w : |w-1| < M\} = q(\mathbb{D})$ , the class $\Phi_{H,1}[\Omega, M]$ is simply denoted by $\Phi_{H,1}[M]$ , and Corollary 2.12 takes the following form.

Corollary 2.13. Let $\phi \in \Phi_{H,1}[M]$ . If $f \in \mathcal{A}$ satisfies the following inequality $$\left| \phi \left( \frac{B_{\kappa}^{c} f(z)}{B_{\kappa+1}^{c} f(z)}, \frac{B_{\kappa-1}^{c} f(z)}{B_{\kappa}^{c} f(z)}, \frac{B_{\kappa-2}^{c} f(z)}{B_{\kappa-1}^{c} f(z)}; z \right) - 1 \right| < M,$$ then for all $z \in \mathbb{D}$ we have $$\left| \frac{B_{\kappa}^{c} f(z)}{B_{\kappa+1}^{c} f(z)} - 1 \right| < M.$$

Corollary 2.14. Let $\kappa \in \mathbb{C}$ ( $\kappa \neq 1$ ) and M > 0. If $f \in \mathcal{A}$ satisfies the following inequality $$\left| \frac{B_{\kappa-1}^{c} f(z)}{B_{\kappa}^{c} f(z)} - \frac{B_{\kappa}^{c} f(z)}{B_{\kappa+1}^{c} f(z)} \right| < \frac{M^{2}}{|\kappa - 1| (1 + M)},$$ then for all $z \in \mathbb{D}$ we have $$\left| \frac{B_{\kappa}^c f(z)}{B_{\kappa+1}^c f(z)} - 1 \right| < M.$$

Theorem 2.15. Let $\phi \in \Phi_{H,2}[\Omega,q]$ . If $f \in \mathcal{A}$ satisfies the following inclusion relationship (2.19) $$\left\{\phi\left(\frac{B_{\kappa+1}^c f(z)}{z}, \frac{B_{\kappa}^c f(z)}{z}, \frac{B_{\kappa-1}^c f(z)}{z}; z\right) : z \in \mathbb{D}\right\} \subset \Omega,$$ then $$\frac{B_{\kappa+1}^c f(z)}{z} \prec q(z) \quad (z \in \mathbb{D}).$$

Corollary 2.16. Let $\phi \in \Phi_{H,2}[h,q]$ . If $f \in \mathcal{A}$ satisfies (2.26) $$\phi\left(\frac{B_{\kappa+1}^c f(z)}{z}, \frac{B_{\kappa}^c f(z)}{z}, \frac{B_{\kappa-1}^c f(z)}{z}; z\right) \prec h(z),$$ then $$\frac{B_{\kappa+1}^c f(z)}{z} \prec q(z) \quad (z \in \mathbb{D}).$$ In the particular case when q(z) = 1 + Mz; M > 0, the class of admissible functions $\Phi_{H,2}[\Omega, q]$ , denoted by $\Phi_{H,2}[\Omega, M]$ .

Corollary 2.17. Let $\phi \in \Phi_{H,2}[\Omega, M]$ . If $f \in \mathcal{A}$ satisfies the following inclusion relationship $$\phi\left(\frac{B_{\kappa+1}^cf(z)}{z},\frac{B_{\kappa}^cf(z)}{z},\frac{B_{\kappa-1}^cf(z)}{z};z\right)\in\Omega,$$ then $$\frac{B_{\kappa+1}^c f(z)}{z} - 1 \prec Mz \quad (z \in \mathbb{D}).$$ In the special case when $\Omega = \{w : |w-1| < M\} = q(\mathbb{D})$ , the class $\Phi_{H,2}[\Omega, M]$ is simply denoted by $\Phi_{H,2}[M]$ , and Corollary 2.17 takes the following form.

Corollary 2.18. Let $\phi \in \Phi_{H,2}[M]$ . If $f \in \mathcal{A}$ satisfies the following inequality $$\left| \phi\left(\frac{B_{\kappa+1}^c f(z)}{z}, \frac{B_{\kappa}^c f(z)}{z}, \frac{B_{\kappa-1}^c f(z)}{z}; z\right) - 1 \right| < M,$$ then for all $z \in \mathbb{D}$ we have $$\left| \frac{B_{\kappa+1}^c f(z)}{z} - 1 \right| < M.$$

Corollary 2.19. Let $\kappa \neq 0$ and M > 0. If $f \in \mathcal{A}$ satisfies the following inequality $$\left| \frac{B_{\kappa}^{c} f(z)}{z} - \frac{B_{\kappa+1}^{c} f(z)}{z} \right| < \frac{M}{|\kappa|}$$ then for all $z \in \mathbb{D}$ we have $$\left| \frac{B_{\kappa+1}^c f(z)}{z} - 1 \right| < M.$$

Corollary 2.20. Let $\operatorname{Re}(\kappa) \geq -\frac{k}{2}$ , $\kappa \neq 0$ , $k \geq 1$ and M > 0. If $f \in \mathcal{A}$ satisfy the following inequality $$\left| \frac{B_{\kappa}^{c} f(z)}{z} - 1 \right| < M,$$ then for all $z \in \mathbb{D}$ we have $$\left| \frac{B_{\kappa+1}^c f(z)}{z} - 1 \right| < M.$$

Theorem 3.1. Let $\phi \in \Phi'_H[\Omega, q]$ . If $f \in \mathcal{A}$ , $B^c_{\kappa+1}f \in \mathcal{Q}_0$ and $$\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right)$$ is univalent in $\mathbb{D}$ , then (3.1) $$\Omega \subset \left\{ \phi \left( B_{\varepsilon+1}^c f(z), B_{\varepsilon}^c f(z), B_{\varepsilon-1}^c f(z); z \right) : z \in \mathbb{D} \right\}$$ implies $$q(z) \prec B_{\kappa+1}^c f(z) \quad (z \in \mathbb{D}).$$

Corollary 3.2. Let $q \in \mathcal{H}_0$ , h be analytic in $\mathbb{D}$ and $\phi \in \Phi'_H[h,q]$ . If $f \in \mathcal{A}$ , $B^c_{\kappa+1}f \in \mathcal{Q}_0$ and $$\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right)$$ is univalent in $\mathbb{D}$ , then $$(3.2) h(z) \prec \phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right)$$ implies $$q(z) \prec B_{\kappa+1}^c f(z) \quad (z \in \mathbb{D}).$$ Theorem 3.1 and Corollary 3.2 can only be used to obtain subordinants of differential superordination of the form (3.1) or (3.2). The following theorem proves the existence of the best subordinant of (3.2) for an appropriate $\phi$ .

Theorem 3.3. Let h be univalent in $\mathbb{D}$ and $\phi: \mathbb{C}^3 \times \overline{\mathbb{D}} \to \mathbb{C}$ . Suppose that the differential equation $$\phi\left(q(z),\frac{zq'(z)+(\kappa-1)q(z)}{\kappa},\frac{z^2q''(z)+2(\kappa-1)zq'(z)+(\kappa-1)(\kappa-2)q(z)}{\kappa(\kappa-1)};z\right)=h(z)$$ has a solution $q \in \mathcal{Q}_0$ . If $\phi \in \Phi'_H[h,q], f \in \mathcal{A}, B^c_{\kappa+1}f \in \mathcal{Q}_0$ and $$\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right)$$ is univalent in $\mathbb{D}$ , then $$h(z) \prec \phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right)$$ which implies $$q(z) \prec B_{\kappa+1}^c f(z) \quad (z \in \mathbb{D})$$ П and q is best subordinant.

Corollary 3.4. Let $h_1$ and $q_1$ be analytic functions in $\mathbb{D}$ , $h_2$ be a univalent function in $\mathbb{D}$ , $q_2 \in \mathcal{Q}_0$ with $q_1(0) = q_2(0) = 1$ and $\phi \in \Phi_H[h_2, q_2] \cap \Phi'_H[h_1, q_1]$ . If $f \in \mathcal{A}$ , $B^c_{\kappa+1}f \in \mathcal{Q}_0 \cap \mathcal{H}_0$ and $$\phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right)$$ is univalent in $\mathbb{D}$ , then $$h_1(z) \prec \phi\left(B_{\kappa+1}^c f(z), B_{\kappa}^c f(z), B_{\kappa-1}^c f(z); z\right) \prec h_2(z)$$ which implies $$q_1(z) \prec B_{\kappa+1}^c f(z) \prec q_2(z) \quad (z \in \mathbb{D}).$$

Theorem 3.5. Let $\phi \in \Phi'_{H,1}[\Omega,q]$ . If $f \in \mathcal{A}$ , $B^c_{\kappa}f/B^c_{\kappa+1}f \in \mathcal{Q}_1$ and $$\phi\left(\frac{B_{\kappa}^c f(z)}{B_{\kappa+1}^c f(z)}, \frac{B_{\kappa-1}^c f(z)}{B_{\kappa}^c f(z)}, \frac{B_{\kappa-2}^c f(z)}{B_{\kappa-1}^c f(z)}; z\right)$$ is univalent in $\mathbb{D}$ , then (3.3) $$\Omega \subset \left\{ \phi \left( \frac{B_{\kappa}^c f(z)}{B_{\kappa+1}^c f(z)}, \frac{B_{\kappa-1}^c f(z)}{B_{\kappa}^c f(z)}, \frac{B_{\kappa-2}^c f(z)}{B_{\kappa-1}^c f(z)}; z \right) : z \in \mathbb{D} \right\}$$ which implies $$q(z) \prec \frac{B_{\kappa}^{c} f(z)}{B_{\kappa+1}^{c} f(z)} \quad (z \in \mathbb{D}).$$

Theorem 3.6. Let $q \in \mathcal{H}_1$ , h be analytic in $\mathbb{D}$ and $\phi \in \Phi'_{H_1}[h,q]$ . If $f \in \mathcal{A}$ , $B^c_{\kappa}f/B^c_{\kappa+1}f \in \mathcal{Q}_1$ and $$\phi\left(\frac{B_{\kappa}^c f(z)}{B_{\kappa+1}^c f(z)}, \frac{B_{\kappa-1}^c f(z)}{B_{\kappa}^c f(z)}, \frac{B_{\kappa-2}^c f(z)}{B_{\kappa-1}^c f(z)}; z\right)$$ is univalent in $\mathbb{D}$ , then (3.4) $$h(z) \prec \phi \left( \frac{B_{\kappa}^{c} f(z)}{B_{\kappa+1}^{c} f(z)}, \frac{B_{\kappa-1}^{c} f(z)}{B_{\kappa}^{c} f(z)}, \frac{B_{\kappa-2}^{c} f(z)}{B_{\kappa-1}^{c} f(z)}; z \right)$$ which implies $$q(z) \prec \frac{B_{\kappa}^{c} f(z)}{B_{\kappa+1}^{c} f(z)} \quad (z \in \mathbb{D}).$$ Combining Theorems 2.11 and 3.6, we obtain the following sandwich-type theorem.

Corollary 3.7. Let $h_1$ and $q_1$ be analytic functions in $\mathbb{D}$ , $h_2$ be a univalent function in $\mathbb{D}$ , $q_2 \in \mathcal{Q}_1$ with $q_1(0) = q_2(0) = 1$ and $\phi \in \Phi_{H,1}[h_2, q_2] \cap \Phi'_{H,1}[h_1, q_1]$ . If $f \in \mathcal{A}$ , $B_{\kappa}^c f/B_{\kappa+1}^c f \in \mathcal{Q}_1 \cap \mathcal{H}_1$ and $$\phi\left(\frac{B_{\kappa}^c f(z)}{B_{\kappa+1}^c f(z)}, \frac{B_{\kappa-1}^c f(z)}{B_{\kappa}^c f(z)}, \frac{B_{\kappa-2}^c f(z)}{B_{\kappa-1}^c f(z)}; z\right)$$ is univalent in $\mathbb{D}$ , then $$h_1(z) \prec \phi\left(\frac{B_{\kappa}^c f(z)}{B_{\kappa+1}^c f(z)}, \frac{B_{\kappa-1}^c f(z)}{B_{\kappa}^c f(z)}, \frac{B_{\kappa-2}^c f(z)}{B_{\kappa-1}^c f(z)}; z\right) \prec h_2(z)$$ which implies $$q_1(z) \prec \frac{B_{\kappa}^c f(z)}{B_{\kappa+1}^c f(z)} \prec q_2(z) \quad (z \in \mathbb{D}).$$

Theorem 3.8. Let $\phi \in \Phi'_{H,2}[\Omega,q]$ . If $f \in \mathcal{A}$ , $\frac{B^c_{\kappa+1}f(z)}{z} \in \mathcal{Q}_1$ and $$\phi\left(\frac{B_{\kappa+1}^cf(z)}{z},\frac{B_{\kappa}^cf(z)}{z},\frac{B_{\kappa-1}^cf(z)}{z};z\right)$$ is univalent in $\mathbb{D}$ , then (3.5) $$\Omega \subset \left\{ \phi\left(\frac{B_{\kappa+1}^c f(z)}{z}, \frac{B_{\kappa}^c f(z)}{z}, \frac{B_{\kappa-1}^c f(z)}{z}; z\right) : z \in \mathbb{D} \right\}$$ which implies $$q(z) \prec \frac{B_{\kappa+1}^c f(z)}{z} \quad (z \in \mathbb{D}).$$

Corollary 3.9. Let $q \in \mathcal{H}_1$ , h be analytic in $\mathbb{D}$ and $\phi \in \Phi'_{H,2}[h,q]$ . If $f \in \mathcal{A}$ , $\frac{B_{\kappa+1}^c f(z)}{z} \in \mathcal{Q}_1$ and $$\phi\left(\frac{B_{\kappa+1}^c f(z)}{z}, \frac{B_{\kappa}^c f(z)}{z}, \frac{B_{\kappa-1}^c f(z)}{z}; z\right)$$ is univalent in $\mathbb{D}$ , then (3.6) $$h(z) \prec \phi\left(\frac{B_{\kappa+1}^c f(z)}{z}, \frac{B_{\kappa}^c f(z)}{z}, \frac{B_{\kappa-1}^c f(z)}{z}; z\right)$$ which implies $$q(z) \prec \frac{B_{\kappa+1}^c f(z)}{z} \quad (z \in \mathbb{D}).$$ Combining Corollaries 2.16 and 3.9, we obtain the following sandwich-type theorem.

Corollary 3.10. Let $h_1$ and $q_1$ be analytic functions in $\mathbb{D}$ , $h_2$ be a univalent function in $\mathbb{D}$ , $q_2 \in \mathcal{Q}_1$ with $q_1(0) = q_2(0) = 1$ and $\phi \in \Phi_{H,2}[h_2, q_2] \cap \Phi'_{H,2}[h_1, q_1]$ . If $f \in \mathcal{A}$ , $\frac{B_{\kappa+1}^c f(z)}{z} \in \mathcal{Q}_1 \cap \mathcal{H}_1$ and $$\phi\left(\frac{B^c_{\kappa+1}f(z)}{z},\frac{B^c_{\kappa}f(z)}{z},\frac{B^c_{\kappa-1}f(z)}{z};z\right)$$ is univalent in $\mathbb{D}$ , then $$h_1(z) \prec \phi\left(\frac{B_{\kappa+1}^c f(z)}{z}, \frac{B_{\kappa}^c f(z)}{z}, \frac{B_{\kappa-1}^c f(z)}{z}; z\right) \prec h_2(z)$$ which implies $$q_1(z) \prec \frac{B_{\kappa+1}^c f(z)}{z} \prec q_2(z) \quad (z \in \mathbb{D}).$$ We note that in particular the above main results reduce to results for the operators $\mathcal{J}_p f$ , $\mathcal{I}_p f$ and $\mathcal{S}_p f$ , which are defined by (1.11), (1.12) and (1.13), respectively. Acknowledgement. The research of A. Baricz was supported by a research grant of the Romanian ´ National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-RU-TE-2012-3- 0190/2014. The research of E. Deniz was supported by the Commission for the Scientific Research Projects of Kafkas Univertsity, project number 2012-FEF-30. The research of M. C¸ a˘glar and H. Orhan was supported by the Atat¨urk University Rectorship under The Scientific and Research Project of Atat¨urk University, project number 2012/173.

Definition 1.1. [7, p. 21] Let us denote by Q the class of functions q, which are analytic and injective on $\overline{\mathbb{D}} \setminus E(q)$ , where $$E(q) = \left\{ \zeta : \zeta \in \partial \mathbb{D} \text{ and } \lim_{z \to \zeta} q(z) = \infty \right\},$$ and are such that $q'(\zeta) \neq 0$ ( $\zeta \in \partial \mathbb{D} \setminus E(q)$ ). Further let the subclass of $\mathcal{Q}$ for which q(0) = a be denoted by $\mathcal{Q}(a)$ , $\mathcal{Q}(0) \equiv \mathcal{Q}_0$ and $\mathcal{Q}(1) \equiv \mathcal{Q}_1$ .

Definition 1.2. [7, p. 27] Let $\Omega$ be a set in $\mathbb{C}$ , $q \in \mathcal{Q}$ and $n \in \mathbb{N}$ . The class of admissible functions $\Psi_n[\Omega, q]$ consists of those functions $\psi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ that satisfy the following admissibility condition $$\psi(r, s, t; z) \notin \Omega$$ whenever $$r = q(\zeta), \quad s = k\zeta q'(\zeta) \quad and \quad \operatorname{Re}\left(\frac{t}{s} + 1\right) \ge k\operatorname{Re}\left(\frac{\zeta q''(\zeta)}{q'(\zeta)} + 1\right)$$ $(z \in \mathbb{D}; \ \zeta \in \partial \mathbb{D} \setminus E(q); \ k \ge n).$ We write $\Psi_1 [\Omega, q]$ simply as $\Psi [\Omega, q]$ . In particular when $$q(z) = M \frac{Mz + a}{M + \overline{a}z},$$ with M > 0 and |a| < M, then $q(\mathbb{D}) = \mathbb{D}_M = \{w : |w| < M\}$ , q(0) = a, $E(q) = \emptyset$ and $q \in \mathcal{Q}$ . In this case, we set $\Psi_n[\Omega, M, a] = \Psi_n[\Omega, q]$ , and in the special case when the set $\Omega = \mathbb{D}_M$ , the class is simply denoted by $\mathbb{D}_M[M, a]$ .

Definition 1.3. [8, p. 817] Let $\Omega$ be a set in $\mathbb{C}$ and $q \in \mathcal{H}[a, n]$ with $q'(z) \neq 0$ . The class of admissible functions $\Psi'_n[\Omega, q]$ consist of those functions $$\psi: \mathbb{C}^3 \times \overline{\mathbb{D}} \to \mathbb{C}$$ that satisfy the following admissibility condition $$\psi(r, s, t; \varsigma) \in \Omega$$ whenever $$r = q(z), \quad s = \frac{zq'(z)}{m} \quad and \quad \operatorname{Re}\left(\frac{t}{s} + 1\right) \le \frac{1}{m}\operatorname{Re}\left(\frac{zq''(z)}{q'(z)} + 1\right)$$ $(z \in \mathbb{D}; \ \varsigma \in \partial \mathbb{D}; \ m \ge n \ge 1).$ We write $\Psi'_1[\Omega,q]$ simply as $\Psi'[\Omega,q]$ . For the above two classes of admissible functions, Miller and Mocanu [7, 8] proved the following results.

Definition 2.1. Let $\Omega$ be a set in $\mathbb{C}$ and $q \in \mathcal{Q}_0 \cap \mathcal{H}_0$ . The class of admissible functions $\Phi_H[\Omega, q]$ consists of those functions $\phi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ that satisfy the admissibility condition $$\phi(u, v, w; z) \notin \Omega$$ whenever $$u = q(\zeta), \ v = \frac{k\zeta q'(\zeta) + (\kappa - 1)q(\zeta)}{\kappa} \quad (\kappa \in \mathbb{C}, \ \kappa \neq 0, 1)$$ an.d $$\operatorname{Re}\left(\frac{\kappa(\kappa-1)w - (\kappa-2)(\kappa-1)u}{v\kappa - (\kappa-1)u} - (2\kappa-3)\right) \ge k\operatorname{Re}\left(\frac{\zeta q''(\zeta)}{q'(\zeta)} + 1\right)$$ $$(z \in \mathbb{D}; \ \zeta \in \partial \mathbb{D} \setminus E(q); \ k \ge 1).$$

Definition 2.2. Let $\Omega$ be a set in $\mathbb{C}$ and M > 0. The class of admissible functions $\Phi_H[\Omega, M]$ consists of those functions $\phi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ such that (2.9) $$\phi\left(Me^{i\theta}, \frac{k + (\kappa - 1)}{\kappa}Me^{i\theta}, \frac{L + (\kappa - 1)(2k + \kappa - 2)Me^{i\theta}}{\kappa(\kappa - 1)}; z\right) \notin \Omega,$$ whenever $z \in \mathbb{D}$ , $\kappa \in \mathbb{C}$ ( $\kappa \neq 0, 1$ ) and $\text{Re}(Le^{-i\theta}) \geq (k-1)kM$ for all $\theta \in \mathbb{R}$ , $k \geq 1$ .

Definition 2.3. Let $\Omega$ be a set in $\mathbb{C}$ and $q \in \mathcal{Q}_1 \cap \mathcal{H}_1$ . The class of admissible functions $\Phi_{H,1}[\Omega,q]$ consists of those functions $\phi: \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ that satisfy the admissibility condition $$\phi(u, v, w; z) \notin \Omega$$ whenever $$u = q(\zeta), \ v = \frac{1}{\kappa - 1} \left( \frac{k\zeta q'(\zeta)}{q(\zeta)} + \kappa q(\zeta) - 1 \right) \quad (\kappa \in \mathbb{C}, \ \kappa \neq 0, 1, 2, \ q(\zeta) \neq 0),$$ and $$\operatorname{Re}\left(\frac{(\kappa-1)v\left[(\kappa-1)(w-v)+1-w\right]}{(\kappa-1)v+1-\kappa u}-\left[2\kappa u-1-(\kappa-1)v\right]\right)\geq k\operatorname{Re}\left(\frac{\zeta q''(\zeta)}{q'(\zeta)}+1\right)$$ $$(z\in\mathbb{D};\;\zeta\in\partial\mathbb{D}\setminus E(q);\;k\geq1).$$

Definition 2.4. Let $\Omega$ be a set in $\mathbb{C}$ , and M > 0. The class of admissible functions $\Phi_{H,1}[\Omega, M]$ consists of those functions $\phi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ such that $$(2.18) \qquad \phi \left( 1 + Me^{i\theta}, 1 + \frac{k + \kappa(1 + Me^{i\theta})}{(\kappa - 1)(1 + Me^{i\theta})} Me^{i\theta}, 1 + \frac{k + \kappa(1 + Me^{i\theta})}{(\kappa - 2)(1 + Me^{i\theta})} Me^{i\theta} + \frac{(M + e^{-i\theta}) \left[ Le^{-i\theta} + (1 + \kappa)kM + \kappa kM^{2}e^{i\theta} \right] - k^{2}M^{2}}{(\kappa - 2)(M + e^{-i\theta}) \left[ (\kappa - 1)e^{-i\theta} + \kappa M^{2}e^{i\theta} + (1 + k + 2\kappa)M \right]} \right) \notin \Omega$$ whenever $z \in \mathbb{D}$ , $\kappa \in \mathbb{C}$ ( $\kappa \neq 0, 1, 2$ ) and $\text{Re}(Le^{-i\theta}) \geq (k-1)kM$ for all $\theta \in \mathbb{R}$ , $k \geq 1$ .

Definition 2.5. Let $\Omega$ be a set in $\mathbb{C}$ and $q \in \mathcal{Q}_1 \cap \mathcal{H}_1$ . The class of admissible functions $\Phi_{H,2}[\Omega,q]$ consists of those functions $\phi: \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ that satisfy the admissibility condition $$\phi(u, v, w; z) \notin \Omega$$ where $$u = q(\zeta), \ v = \frac{k\zeta q'(\zeta) + \kappa q(\zeta)}{\kappa} \quad (\kappa \in \mathbb{C}, \ \kappa \neq 0, 1)$$ and $$\operatorname{Re}\left(\frac{(\kappa-1)(w-u)}{v-u} + (1-2\kappa)\right) \ge k \operatorname{Re}\left(\frac{\zeta q''(\zeta)}{q'(\zeta)} + 1\right)$$ $$(z \in \mathbb{D}; \ \zeta \in \partial \mathbb{D} \setminus E(q); \ k \ge 1).$$

Definition 2.6. Let $\Omega$ be a set in $\mathbb{C}$ and M > 0. The class of admissible functions $\Phi_{H,2}[\Omega, M]$ consists of those functions $\phi : \mathbb{C}^3 \times \mathbb{D} \to \mathbb{C}$ such that (2.27) $$\phi\left(1 + Me^{i\theta}, 1 + \frac{k + \kappa}{\kappa}Me^{i\theta}, 1 + \frac{L + \kappa(2k + \kappa - 1)Me^{i\theta}}{\kappa(\kappa - 1)}; z\right) \notin \Omega$$ whenever $z \in \mathbb{D}$ , $\kappa \in \mathbb{C}$ $(\kappa \neq 0, 1)$ and $\text{Re}(Le^{-i\theta}) \geq (k-1)kM$ for all $\theta \in \mathbb{R}$ , $k \geq 1$ .

Definition 3.1. Let $\Omega$ be a set in $\mathbb{C}$ , $q \in \mathcal{H}_0$ with $zq'(z) \neq 0$ . The class of admissible functions $\Phi'_H[\Omega, q]$ consists of those functions $\phi : \mathbb{C}^3 \times \overline{\mathbb{D}} \to \mathbb{C}$ that satisfy the admissibility condition $$\phi(u, v, w; \varsigma) \notin \Omega$$ whenever $$u = q(z), \ v = \frac{zq'(z) + m(\kappa - 1)q(z)}{m\kappa}, \quad (\kappa \in \mathbb{C}, \ \kappa \neq 0, 1)$$ and $$\operatorname{Re}\left(\frac{\kappa(\kappa-1)w - (\kappa-2)(\kappa-1)u}{v\kappa - (\kappa-1)u} - (2\kappa-3)\right) \le \frac{1}{m}\operatorname{Re}\left(\frac{zq''(z)}{q'(z)} + 1\right)$$ $$(z \in \mathbb{D}; \ \varsigma \in \partial\mathbb{D}; \ m > 1).$$

Definition 3.2. Let $\Omega$ be a set in $\mathbb{C}$ , $q \in \mathcal{H}_1$ with $q(z) \neq 0$ , $zq'(z) \neq 0$ . The class of admissible functions $\Phi'_{H,1}[\Omega,q]$ consists of those functions $\phi: \mathbb{C}^3 \times \overline{\mathbb{D}} \to \mathbb{C}$ that satisfy the admissibility condition $$\phi(u, v, w; \varsigma) \in \Omega$$ whenever $$u = q(z), \ v = \frac{1}{\kappa - 1} \left( \frac{zq'(z)}{mq(z)} + \kappa q(z) - 1 \right) \quad (\kappa \in \mathbb{C}, \ \kappa \neq 0, 1, 2)$$ ana $$\operatorname{Re}\left(\frac{(\kappa-1)v\left[(\kappa-1)(w-v)+1-w\right]}{(\kappa-1)v+1-\kappa u}-\left[2\kappa u-1-(\kappa-1)v\right]\right)\leq \frac{1}{m}\operatorname{Re}\left(\frac{zq''(z)}{q'(z)}+1\right)$$ $$(z\in\mathbb{D};\ \varsigma\in\partial\mathbb{D};\ m\geq1).$$

Definition 3.3. Let $\Omega$ be a set in $\mathbb{C}$ , $q \in \mathcal{H}_1$ with $q(z) \neq 0$ , $zq'(z) \neq 0$ . The class of admissible functions $\Phi'_{H,2}[\Omega,q]$ consists of those functions $\phi: \mathbb{C}^3 \times \overline{\mathbb{D}} \to \mathbb{C}$ that satisfy the admissibility condition $$\phi(u, v, w; \varsigma) \in \Omega$$ whenever $$u = q(z), \ v = \frac{\zeta q'(\zeta) + \kappa m q(\zeta)}{\kappa m} \quad (\kappa \in \mathbb{C}, \ \kappa \neq 0, 1)$$ and $$\operatorname{Re}\left(\frac{(\kappa-1)(w-u)}{v-u} + (1-2\kappa)\right) \le \frac{1}{m}\operatorname{Re}\left(\frac{zq''(z)}{q'(z)} + 1\right)$$ $$(z \in \mathbb{D}; \ \varsigma \in \partial \mathbb{D}; \ m > 1).$$