Subordination and Superordination Properties for Certain Family of Analytic Functions Associated with Mittag–Leffler Function

Abstract

We obtain new outcomes of analytic functions linked with operator ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f)$ defined by Mittag–Leffler function. Moreover, new theorems of differential sandwich-type are obtained.

1. Basic Definitions and Preliminaries

Let $\mathbb{A}$ define the class of analytic functions in the open unit disk $\mathbb{U}=\{z\in \mathbb{C}:|z|<1\}$ and let $\mathcal{H}[a,n]$ be the subclass of $\mathbb{A}$, which is

$$f(z)=a+a_n{z}^n+a_{n+1}{z}^{n+1}+a_{n+2}{z}^{n+2}+a_{n+3}{z}^{n+3}+\dots (a\in \mathbb{C}),$$

Furthermore, let $\mathcal{H}$ be the subclass of $\mathbb{A}$ of all the functions $f(z)\in \mathcal{H}$ normalized by

$$f(z)=z+\sum _{n=2}^{\infty }{a}_n{z}^n.$$

(1)

Attiya [1] introduced and investigated the operator ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z)):\mathcal{H}\to \mathcal{H}$, which ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))$ is defined by

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))=Q_{\alpha ,\beta }^{\eta ,k}(z)\ast f(z),(z\in \mathbb{U}),$$

for $f(z)\in \mathcal{H}$ given by (1), the symbol ∗ denotes the Hadamard product, and

$$Q_{\alpha ,\beta }^{\eta ,k}(z)=\frac{\Gamma (\alpha +\beta )}{{(\eta )}_k}\left(E_{\alpha ,\beta }^{\eta ,k}(z)-\frac{1}{\Gamma (\beta )}\right),(z\in \mathbb{U}).$$

Moreover, the function $E_{\alpha ,\beta }^{\eta ,k}(z)$ is called the general Mittag–Leffler function defined by

$$E_{\alpha ,\beta }^{\eta ,k}(z)=\sum _{n=0}^{\infty }\frac{{(\eta )}_{nk}{z}^n}{\Gamma (\alpha n+\beta )n!},(\alpha ,\beta ,\eta \in \mathbb{C};Re(\alpha )>max\{0,Re(k)-1\};Re(k)>0),$$

where

$${(\eta )}_n=\frac{\Gamma (\eta +n)}{\Gamma (\eta )}=\left\{\begin{array}{cc}1,\hfill & n=0,\hfill \\ \eta (\eta +1)(\eta +2)\dots (\eta +n-1),\hfill & n\in \mathbb{N}.\hfill \end{array}\right.$$

The function $E_{\alpha ,\beta }^{\eta ,k}(z)$ was investigated by Srivastava and Tomovski [2]. Many authors studied and investigated Mittag–Leffler function; for more details on Mittag–Leffler function and general Mittag–Leffler function see, e.g., [1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19].

Moreover, Attiya [1] deduced that ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))$ can be put in

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))=z+\sum _{n=2}^{\infty }\frac{\Gamma (\eta +nk)\Gamma (\alpha +\beta )}{\Gamma (\eta +k)\Gamma (n\alpha +\beta )}{a}_n{z}^n(z\in \mathbb{U}).$$

(2)

It follows from (2) that (see [1])

$$kz{\left({\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right)}^{\prime }=(\eta +k)\left({\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))\right)-\eta \left({\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right),$$

(3)

and

$$\alpha z{\left({\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right)}^{\prime }=(\alpha +\beta )\left({\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right)-\beta \left({\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right).$$

(4)

It should be remarked that the operator ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))$ for some special cases of $\alpha ,\beta ,\eta$, and k provides many special functions, e.g.,

$${\mathcal{H}}_{0,\beta }^{1,1}(f)(z)=f(z).$$

$${\mathcal{H}}_{0,\beta }^{2,1}(f)(z)=\frac{1}{2}(f(z)+z{f}^{\prime }(z)).$$

$${\mathcal{H}}_{0,\beta }^{0,1}(f)(z)=\underset{0}{\overset{z}{\int }}\frac{1}{t}f(t)dt.$$

$${\mathcal{H}}_{1,0}^{1,1}(\frac{z}{1-z})=z{e}^z.$$

$${\mathcal{H}}_{1,1}^{1,1}(\frac{z}{1-z})=e^z-1.$$

$${\mathcal{H}}_{2,1}^{1,1}(\frac{z}{1-z})=-2+\cosh (\sqrt{z}).$$

Definition 1.

Let functions $f(z)$ and $g(z)$ be analytic in the open unit disk $\mathbb{U}$. Then $f(z)$ is subordinate to $g(z)$ if there exists a Schwarz function $\omega (z)$, analytic in $\mathbb{U}$ with $\omega (0)=0$ and $|\omega (z)|<1,(z\in \mathbb{U})$, such that $f(z)=g(\omega (z)),(z\in \mathbb{U})$, we denote this subordination by $f(z)\prec g(z)$. In particular, if $g(z)$ is univalent in $\mathbb{U}$, then subordination is equivalent to $f(z)\prec g(z)\iff f(0)=g(0)$ and $f(\mathbb{U})\subset g(\mathbb{U})$.

Definition 2.

If $\mathbb{Q}$ the set of all functions $q(z)$ that are analytic and univalent on $\overline{\mathbb{U}}\backslash E(q)$, where

$$E(q)=\left\{\xi \in \partial \mathbb{U}:\underset{z\to \xi }{lim}q(z)=\infty \right\},$$

and $min|q^{\prime }(\xi )|=\rho >0$ for $\xi \in \partial \mathbb{U}\backslash E(q)$. Further, let $\mathbb{Q}(a)=\left\{q(z)\in \mathbb{U}:q(0)=a\right\}$ and ${\mathbb{Q}}_1=\mathbb{Q}(1)$.

Definition 3.

If $\psi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ and $h(z)$ be univalent in $\mathbb{U}$. If $p(z)$ is analytic in $\mathbb{U}$, and satisfies the third-order differential subordination

$$\psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z)\prec h(z),$$

(5)

then $p(z)$ is called a solution of the differential subordination and $q(z)$ is called a dominant of the solutions of the differential subordination as well as a dominant if $p(z)\prec q(z)$ for all $p(z)$ satisfying (5). $\tilde{q}(z)$ that satisfies $\tilde{q}(z)\prec q(z)$ for all dominants of (5) is called the best dominant of (5).

Definition 4.

Let $\Omega \subseteq \mathbb{C}$, $q(z)\in \mathbb{Q}$ and $n\in \mathbb{N}\backslash \left\{1\right\}$. The class of admissible functions ${\Psi }_n[\Omega ,q(z)]$ consists of those functions $\psi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ that satisfy the admissibility condition:

$$\psi (r,s,t,u;z)\notin \Omega ,$$

whenever

$$r=q(\zeta ),s=\ell \zeta q^{\prime }(\zeta ),Re\left(\frac{t}{s}+1\right)\ge \ell Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}+1\right),$$

and

$$Re\left(\frac{u}{s}\right)\ge {\ell }^{2}Re\left(\frac{{\zeta }^2{q}^{‴}(\zeta )}{q^{\prime }(\zeta )}\right),$$

where $z\in \mathbb{U}$; $\zeta \in \partial \mathbb{U}\backslash E(q)$ and $\ell \ge n$.

Analogous to the second order differential super-ordinations introduced by Miller and Mocanu [20], Tang et al. [21] defined the differential super-ordinations as follows:

Definition 5.

Let $\psi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ and the function $h(z)$ be analytic in $\mathbb{U}$. If functions $p(z)$ and $\psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z))$ are univalent in $\mathbb{U}$, and satisfy the following third-order differential super-ordination

$$h(z)\prec \psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z)),$$

(6)

then $p(z)$ is called a solution of the differential superordination and $q(z)$ is called a subordinant of the solutions of the differential super-ordinations as well as a subordinant if $p(z)\prec q(z)$ for all $p(z)$ satisfying Equation (6). A univalent subordinant $\tilde{q}(z)$ that satisfies $\tilde{q}(z)\prec q(z)$ for all super-ordinations of (6) is the best superordinant.

Definition 6.

Let $\Omega \subseteq \mathbb{C}$, $q(z)\in \mathcal{H}[a,n]$ with $n\in \mathbb{N}\backslash \left\{1\right\}$ and $q^{\prime }(z)\ne 0$. The class of admissible functions ${\Psi }_n^{\prime }[\Omega ,q(z)]$ consists of those functions $\psi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ that satisfy the admissibility condition:

$$\psi (r,s,t,u;\zeta )\in \Omega ,$$

whenever

$$r=q(z),s=\frac{z{q}^{\prime }(z)}{m},Re\left(\frac{t}{s}+1\right)\le \frac{1}{m}Re\left(\frac{z{q}^{″}(z)}{q^{\prime }(z)}+1\right),$$

and

$$Re\left(\frac{u}{s}\right)\le \frac{1}{m^2}Re\left(\frac{z^2{q}^{‴}(z)}{q^{\prime }(z)}\right),$$

where $z\in \mathbb{U}$; $\zeta \in \partial \mathbb{U}$ and $m\ge n\ge 2$.

Here, we use the following theorems given by Antonino and Miller [22]:

Theorem 1

([22]). Let $p(z)\in \mathcal{H}[a,n]$ with $n\in \mathbb{N}\backslash \left\{1\right\}$. Also, let $q(z)\in \mathbb{Q}(a)$ and satisfy the following conditions:

$$\begin{array}{cc}\hfill & Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}\right)>0,\left|\frac{z{p}^{\prime }(z)}{q^{\prime }(\zeta )}\right|\le \ell ,(z\in \mathbb{U};\zeta \in \partial \mathbb{U}\backslash E(q);\ell \ge n),\hfill \\ \hfill & \psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z)\in \Omega ,\hfill \end{array}$$

and, if $\Omega \subseteq \mathbb{C}$, $\psi \in {\Psi }_n[\Omega ,q(z)]$, then $p(z)\prec q(z)$.

Theorem 2.

Let $q(z)\in \mathcal{H}[a,n]$ and $\psi \in {\Psi }_n^{\prime }[\Omega ,q(z)]$. If $p(z)\in \mathbb{Q}(a)$ and $\psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z)$ is univalent in $\mathbb{U}$ and

$$\begin{array}{cc}\hfill & Re\left(\frac{z{q}^{″}(z)}{q^{\prime }(z)}\right)\ge 0,\left|\frac{\zeta p^{\prime }(\zeta )}{q^{\prime }(z)}\right|\le m,(z\in \mathbb{U};\zeta \in \partial \mathbb{U};m\ge n\ge 2)\hfill \\ \hfill & \Omega \subset \left\{\psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z):z\in \mathbb{U}\right\},\hfill \end{array}$$

then $q(z)\prec p(z)$.

Here, we study a certain family of admissible functions by using the third-order differential subordination and superordination given by Antonino and Miller [22] and Tang et al. [21]—see also Attiya et al. [23]—we obtain new results of subordination and superordination properties of analytic functions linked with the operator ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f)$.

2. Main Results

Definition 7.

Let $\Omega \subseteq \mathbb{C}$ and $q(z)\in \mathbb{Q}$. The class of admissible functions ${\Psi }_{\Gamma }[\Omega ,q(z)]$ consists of those functions $\varphi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ that satisfy the admissibility condition

$$\varphi (a_1,a_2,a_3,a_4;z)\notin \Omega ,$$

whenever

$$a_1=q(\zeta ),a_2=\frac{\ell k\zeta q^{\prime }(\zeta )+bq(\zeta )}{b},$$

$$Re\left(\frac{(b+1)(a_3-a_1)}{k(a_2-a_1)}-\frac{2b+1}{k}\right)\ge \ell Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}+1\right),$$

$$\begin{array}{cc}\hfill Re(\frac{(b+1)(b+2)(a_4-a_1)-3(b+1)(b+k+1)(a_3-a_1)}{k^2(a_2-a_1)}& +\frac{3b(b+1)+1}{k^2}\hfill \\ \hfill +\frac{6b+3}{k}+2& )\ge {\ell }^{2}Re\left(\frac{{\zeta }^2{q}^{‴}(\zeta )}{q^{\prime }(\zeta )}\right),\hfill \end{array}$$

where $z\in \mathbb{U}$; $\zeta \in \partial \mathbb{U}\backslash E(q)$, $\ell \in \mathbb{N}\backslash \left\{1\right\}$ and $b=\eta +k$.

Theorem 3.

Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,q(z)]$. If $f(z)\in \mathcal{H}$ and $q(z)\in {\mathbb{Q}}_1$ satisfy:

$$Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}\right)\ge 0,\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z{q}^{\prime }(\zeta )}\right|\le |\frac{k}{b}|\ell ,$$

(7)

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\}\subset \Omega ,$$

(8)

then

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\prec q(z).$$

Proof. 

Let

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))=zp(z)z\in \mathbb{U},$$

(9)

From (3), we have

$$\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z}=(\frac{k}{b})\left(z{p}^{\prime }(z)+\frac{b}{k}p(z)\right),$$

(10)

which implies

$$\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z}=\frac{k^2}{b(b+1)}\left(z^2{p}^{″}(z)+(\frac{2b+1}{k}+1)z{p}^{\prime }(z)+\frac{b(b+1)}{k^2}p(z)\right).$$

(11)

Furthermore, we have

$$\begin{array}{cc}\hfill \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z}& =\frac{k^3}{b(b+1)(b+2)}(z^3{p}^{‴}(z)+3(\frac{b+1}{k}+1)z^2{p}^{″}(z)\hfill \\ \hfill & +(\frac{3b^2+6b+2}{k^2}+\frac{3(b+1)}{k}+1)z{p}^{\prime }(z)+\frac{b(b+1)(b+2)}{k^3}p(z)).\hfill \end{array}$$

(12)

Now, we define the parameters $a_1,a_2,a_3$, and $a_4$ as

$$\begin{array}{cc}\hfill a_1& =r,a_2=(\frac{k}{b})\left(s+\frac{b}{k}r\right),a_3=\frac{k^2}{b(b+1)}\left(t+(\frac{2b+1}{k}+1)s+\frac{b(b+1)}{k^2}r\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill a_4=\frac{k^3}{b(b+1)(b+2)}(u+3(\frac{b+1}{k}+1)t+(\frac{3b^2+6b+2}{k^2}& +\frac{3(b+1)}{k}+1)s\hfill \\ \hfill & +\frac{b(b+1)(b+2)}{k^3}r).\hfill \end{array}$$

Then, transformation $\psi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ as

$$\psi (r,s,t,u;z)=\varphi (a_1,a_2,a_3,a_4;z),$$

(13)

by using the relations from (9) to (12), we have

$$\begin{array}{cc}\hfill \psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z)& =\hfill \\ \hfill \varphi (\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},& \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z),\hfill \end{array}$$

(14)

therefore, we recompute (8) as

$$\psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z)\in \Omega ,$$

then, the proof is completed by showing that the admissibility condition for $\varphi \in {\Psi }_{\Gamma }[\Omega ,q(z)]$ is equivalent to the admissibility condition for $\psi$ as given in Definition 3, since

$$\frac{t}{s}+1=\frac{(b+1)(a_3-a_1)}{k(a_2-a_1)}-\frac{2b+1}{k},$$

and

$$\begin{array}{cc}\hfill \frac{u}{s}=\frac{(b+1)(b+2)(a_4-a_1)-3(b+1)(b+k+1)(a_3-a_1)}{k^2(a_2-a_1)}& \hfill \\ \hfill +\frac{3b(b+1)+1}{k^2}& +\frac{6b+3}{k}+2,\hfill \end{array}$$

we also note that

$$\left|\frac{z{p}^{\prime }(z)}{q^{\prime }(\zeta )}\right|=|\frac{(\frac{b}{zk})\left({\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right)}{q^{\prime }(\zeta )}|\le \ell ,$$

therefore, $\psi \in {\Psi }_{\Gamma }[\Omega ,q(z)]$ and by Theorem 1, $p(z)\prec q(z)$. □

In a similar way, we define the parameters $a_1,a_2,a_3$, and $a_4$ as follows:

Definition 8.

Let $\Omega \subseteq \mathbb{C}$ and $q(z)\in \mathbb{Q}$. The class of admissible functions ${\Psi }_{\Gamma }[\Omega ,q(z)]$ consists of those functions $\varphi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ that satisfy the admissibility condition

$$\varphi (a_1,a_2,a_3,a_4;z)\notin \Omega ,$$

whenever

$$a_1=q(\zeta ),a_2=\frac{\ell \alpha \zeta q^{\prime }(\zeta )+cq(\zeta )}{c},$$

$$Re\left(\frac{(c-1)(a_3-a_1)}{\alpha (a_2-a_1)}-\frac{2c-1}{\alpha }\right)\ge \ell Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}+1\right),$$

$$\begin{array}{cc}\hfill Re(\frac{(c-1)(c-2)(a_4-a_1)-3(c-1)(c+\alpha -1)(a_3-a_1)}{{\alpha }^2(a_2-a_1)}& +\frac{3c(c-1)+1}{{\alpha }^2}\hfill \\ \hfill +\frac{6c-3}{\alpha }+2& )\ge {\ell }^{2}Re\left(\frac{{\zeta }^2{q}^{‴}(\zeta )}{q^{\prime }(\zeta )}\right),\hfill \end{array}$$

where $z\in \mathbb{U}$; $\zeta \in \partial \mathbb{U}\backslash E(q)$, $\ell \in \mathbb{N}\backslash \left\{1\right\}$ and $c=\alpha +\beta$.

Theorem 4.

Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,q(z)]$. If $f(z)\in \mathcal{H}$ and $q(z)\in {\mathbb{Q}}_1$ satisfy the following conditions:

$$Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right]}{q^{\prime }(\zeta )}\right|\le |\frac{\alpha }{c}|\ell ,$$

(15)

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\}\subset \Omega ,$$

(16)

then

$${\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\prec q(z).$$

Proof. 

Let

$${\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))=zp(z)z\in \mathbb{U},$$

(17)

From (4), we have

$$\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}=(\frac{\alpha }{c})\left(z{p}^{\prime }(z)+\frac{c}{\alpha }p(z)\right),$$

(18)

which implies

$$\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z}=\frac{{\alpha }^2}{c(c-1)}\left(z^2{p}^{″}(z)+(\frac{2c-1}{\alpha }+1)z{p}^{\prime }(z)+\frac{c(c-1)}{{\alpha }^2}p(z)\right).$$

(19)

Moreover, we have

$$\begin{array}{cc}\hfill \frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z}& =\frac{{\alpha }^3}{c(c-1)(c-2)}(z^3{p}^{‴}(z)+3(\frac{c-1}{\alpha }+1)z^2{p}^{″}(z)\hfill \\ \hfill & +(\frac{3c^2-6c+2}{{\alpha }^2}+\frac{3(c-1)}{\alpha }+1)z{p}^{\prime }(z)+\frac{c(c-1)(c-2)}{{\alpha }^3}p(z)).\hfill \end{array}$$

(20)

Parameters $a_1,a_2,a_3$ and, $a_4$ as

$$a_1=r,a_2=(\frac{\alpha }{c})\left(s+\frac{c}{\alpha }r\right),a_3=\frac{{\alpha }^2}{c(c-1)}\left(t+(\frac{2c-1}{\alpha }+1)s+\frac{c(c-1)}{{\alpha }^2}r\right),$$

and

$$\begin{array}{cc}\hfill a_4=\frac{{\alpha }^3}{c(c-1)(c-2)}(u+3(\frac{c-1}{\alpha }+1)t+(\frac{3c^2-6c+2}{{\alpha }^2}+& \frac{3(c-1)}{\alpha }+1)s\hfill \\ \hfill & +\frac{c(c-1)(c-2)}{{\alpha }^3}r).\hfill \end{array}$$

The transformation $\psi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$

$$\psi (r,s,t,u;z)=\varphi (a_1,a_2,a_3,a_4;z),$$

(21)

by using the relations from (17) to (20), we have

$$\begin{array}{cc}\hfill \psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z)& =\hfill \\ \hfill \varphi (\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},& \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z),\hfill \end{array}$$

we recompute (16) as

$$\psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z)\in \Omega ,$$

This completes the proof by showing that the admissibility condition for $\varphi \in {\Psi }_{\Gamma }[\Omega ,q(z)]$ is equivalent to the admissibility condition for $\psi$ as given in Definition 3, since

$$\frac{t}{s}+1=\frac{(c-1)(a_3-a_1)}{\alpha (a_2-a_1)}-\frac{2c-1}{\alpha },$$

and

$$\begin{array}{cc}\hfill \frac{u}{s}=\frac{(c-1)(c-2)(a_4-a_1)-3(c-1)(c+\alpha -1)(a_3-a_1)}{{\alpha }^2(a_2-a_1)}& \\ \hfill +\frac{3c(c-1)+1}{{\alpha }^2}& +\frac{6c-3}{\alpha }+2,\hfill \end{array}$$

we also note that

$$\left|\frac{z{p}^{\prime }(z)}{q^{\prime }(\zeta )}\right|=\left|\frac{(\frac{c}{z\alpha })\left({\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right)}{q^{\prime }(\zeta )}\right|\le \ell ,$$

therefore, $\psi \in {\Psi }_{\Gamma }[\Omega ,q(z)]$ and hence by Theorem 1, $p(z)\prec q(z)$. □

If $\Omega \ne \mathbb{C}$ is simply connected to the domain, then $\Omega =h(\mathbb{U})$ for some conformal mapping $h(z)$ of $\mathbb{U}$ onto $\Omega$. In this case, the class ${\Psi }_{\Gamma }[h(\mathbb{U}),q(z)]$ is written as ${\Psi }_{\Gamma }[h,q]$; the following theorem is a direct consequence of Theorems 3 and 4.

Theorem 5.

Let $\varphi \in {\Psi }_{\Gamma }[h,q]$. If $f(z)\in \mathcal{H}$ and $q(z)\in {\mathbb{Q}}_1$ satisfy the following conditions:

$$(i)Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right]}{q^{\prime }(\zeta )}\right|\le |\frac{k}{b}|\ell ,$$

(22)

$$(ii)\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right)\prec h(z),$$

then

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\prec q(z).$$

Theorem 6.

Let $\varphi \in {\Psi }_{\Gamma }[h,q]$. If $f(z)\in \mathcal{H}$ and $q(z)\in {\mathbb{Q}}_1$ satisfy the following conditions:

$$Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right]}{q^{\prime }(\zeta )}\right|\le |\frac{\alpha }{c}|\ell ,$$

(23)

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right)\prec h(z),$$

then

$${\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\prec q(z).$$

The next corollaries extend Theorems 3 and 4, when the behavior of $q(z)$ on $\partial \mathbb{U}$ is not known.

Corollary 1.

Let $\Omega \subset \mathbb{C}$ and let $q(z)$ be univalent in $\mathbb{U}$; $q(0)=1$. Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,q_{\sigma }]$ for some $\sigma \in (0,1)$ where $q_{\sigma }(z)=q(\sigma z)$. If $f(z)\in \mathcal{H}$ satisfies the following conditions:

$$Re\left(\frac{\zeta q_{\sigma }^{″}(\zeta )}{q_{\sigma }^{\prime }(\zeta )}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right]}{q_{\sigma }^{\prime }(\zeta )}\right|\le |\frac{k}{b}|\ell ,$$

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right)\in \Omega ,$$

then

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\prec q(z),$$

where $z\in \mathbb{U}$ and $\zeta \in \partial \mathbb{U}\backslash E(q_{\sigma }).$

Proof. 

by using Theorem 3, we have ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\prec q_{\sigma }(z)$. Then we obtain the result from $q_{\sigma }(z)\prec q(z)$. □

Corollary 2.

Let $\Omega \subset \mathbb{C}$ and let $q(z)$ be univalent in $\mathbb{U}$; $q(0)=1$. Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,q_{\sigma }]$ for some $\sigma \in (0,1)$, where $q_{\sigma }(z)=q(\sigma z)$. If $f(z)\in \mathcal{H}$ satisfy the following conditions:

$$Re\left(\frac{\zeta q_{\sigma }^{″}(\zeta )}{q_{\sigma }^{\prime }(\zeta )}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right]}{q_{\sigma }^{\prime }(\zeta )}\right|\le |\frac{\alpha }{c}|\ell ,$$

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right)\in \Omega ,$$

then

$${\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\prec q(z),$$

where $z\in \mathbb{U}$ and $\zeta \in \partial \mathbb{U}\backslash E(q_{\sigma }).$

Proof. 

By using Theorem 4, we have ${\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\prec q_{\sigma }(z)$. Then we obtain the result from $q_{\sigma }(z)\prec q(z)$. □

Corollary 3.

Let $\Omega \subset \mathbb{C}$ and let $q(z)$ be univalent in $\mathbb{U}$; $q(0)=1$. Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,q_{\sigma }]$ for some $\sigma \in (0,1)$, where $q_{\sigma }(z)=q(\sigma z)$. If $f(z)\in \mathcal{H}$ satisfy the following conditions:

$$Re\left(\frac{\zeta q_{\sigma }^{″}(\zeta )}{q_{\sigma }^{\prime }(\zeta )}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right]}{q_{\sigma }^{\prime }(\zeta )}\right|\le |\frac{k}{b}|\ell ,$$

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right)\prec h(z),$$

(24)

then

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\prec q(z),$$

where $z\in \mathbb{U}$ and $\zeta \in \partial \mathbb{U}\backslash E(q_{\sigma }).$

Corollary 4.

Let $\Omega \subset \mathbb{C}$ and let $q(z)$ be univalent in $\mathbb{U}$; $q(0)=1$. Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,q_{\sigma }]$ for some $\sigma \in (0,1)$, where $q_{\sigma }(z)=q(\sigma z)$. If $f(z)\in \mathcal{H}$ satisfy the following conditions:

$$Re\left(\frac{\zeta q_{\sigma }^{″}(\zeta )}{q_{\sigma }^{\prime }(\zeta )}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right]}{q_{\sigma }^{\prime }(\zeta )}\right|\le |\frac{\alpha }{c}|\ell ,$$

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right)\prec h(z),$$

(25)

then

$${\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\prec q(z),$$

where $z\in \mathbb{U}$ and $\zeta \in \partial \mathbb{U}\backslash E(q_{\sigma }).$

Theorem 7.

Let $h(z)$ be univalent in $\mathbb{U}$. Let $\varphi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$. Suppose that the differential equation

$$\begin{array}{cc}\hfill \varphi & (q(z),(\frac{k}{b})\left(z{q}^{\prime }(z)+\frac{b}{k}q(z)\right),\hfill \\ \hfill \frac{k^2}{b(b+1)}& \left(z^2{q}^{″}(z)+(\frac{2b+1}{k}+1)z{q}^{\prime }(z)+\frac{b(b+1)}{k^2}q(z)\right),\hfill \\ \hfill \frac{k^3}{b(b+1)(b+2)}& (z^3{q}^{‴}(z)+3(\frac{b+1}{k}+1)z^2{q}^{″}(z)+(\frac{3b^2+6b+2}{k^2}+\frac{3(b+1)}{k}+1)z{q}^{\prime }(z)\hfill \\ \hfill & +\frac{b(b+1)(b+2)}{k^3}q(z));z)=h(z),\hfill \end{array}$$

(26)

has a solution $q(z)$ with $q(0)=1$, which satisfies (7). If $f(z)\in \mathcal{H}$ satisfies (24) and

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right),$$

is analytic in $\mathbb{U}$, then

$${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\prec q(z),$$

(27)

and $q(z)$ is the best dominant of (27).

Proof. 

By using Theorem 3 that $q(z)$ is a dominant of (24). Since $q(z)$ satisfies (26), it is also a solution of (24) and therefore $q(z)$ will be dominated by all dominants. Hence, $q(z)$ is the best dominant. □

Moreover, in a similar way, using Theorem 4, we have

Theorem 8.

Let $h(z)$ be univalent in $\mathbb{U}$. Let $\varphi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$. Suppose that the differential equation

$$\begin{array}{cc}\hfill \varphi & (q(z),(\frac{\alpha }{c})\left(z{q}^{\prime }(z)+\frac{c}{\alpha }q(z)\right),\hfill \\ \hfill \frac{{\alpha }^2}{c(c-1)}& \left(z^2{q}^{″}(z)+(\frac{2c-1}{\alpha }+1)z{q}^{\prime }(z)+\frac{c(c-1)}{{\alpha }^2}q(z)\right),\hfill \\ \hfill \frac{{\alpha }^3}{c(c-1)(c-2)}& (z^3{q}^{‴}(z)+3(\frac{c-1}{\alpha }+1)z^2{q}^{″}(z)+(\frac{3c^2-6c+2}{{\alpha }^2}+\frac{3(c-1)}{\alpha }+1)z{q}^{\prime }(z)\hfill \\ \hfill & +\frac{c(c-1)(c-2)}{{\alpha }^3}q(z));z)=h(z),\hfill \end{array}$$

(28)

has a solution $q(z)$ with $q(0)=1$, which satisfies (15). If $f(z)\in \mathcal{H}$ satisfies (25) and

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right),$$

is analytic in $\mathbb{U}$, then

$${\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\prec q(z).$$

(29)

and $q(z)$ is the best dominant of (29).

In the case $q(z)=1+Mz,(M>0)$ and in view of the Definition 7, the class of admissible functions ${\Psi }_{\Gamma }[\Omega ,q]$ denoted by ${\Psi }_{\Gamma }[\Omega ,M]$ is defined below.

Definition 9.

Let $\Omega \subseteq \mathcal{C}$ and $M>0$. The class of admissible functions ${\Psi }_{\Gamma }[\Omega ,M]$ consists of those functions $\varphi :{\mathbb{C}}^4\times \mathcal{U}⟶\mathbb{C}$ that satisfy the admissibility condition

$$\begin{array}{cc}\hfill \varphi & (1+M{e}^{i\theta },1+(\frac{k}{b})\left(\frac{b}{k}+\ell \right)M{e}^{i\theta },\hfill \\ \hfill 1+\frac{k^2}{b(b+1)}& \left(L+\left(\frac{b(b+1)}{k^2}+\ell (\frac{2b+1}{k}+1)\right)M{e}^{i\theta }\right),\hfill \\ \hfill 1+\frac{k^3}{b(b+1)(b+2)}& (N+3L(\frac{b+1}{k}+1)+(\frac{b(b+1)(b+2)}{k^3}\hfill \\ \hfill & +\ell (\frac{3b^2+6b+2}{k^2}+\frac{3(b+1)}{k}+1))M{e}^{i\theta });z)\notin \Omega ,\hfill \end{array}$$

where $z\in \mathbb{U}$, $Re(L{e}^{-i\theta })\ge \ell (\ell -1)M$ and $Re(N{e}^{-i\theta })\ge 0$ for all real θ and $\ell \in \mathbb{N}\backslash \left\{1\right\}$.

Corollary 5.

Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. If $f(z)\in \mathcal{H}$ satisfy the following conditions:

$$\left|\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right]\right|\le |\frac{k}{b}|\ell M,$$

(30)

and

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right)\in \Omega ,$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Furthermore, with Definition 8, we can define the following:

Definition 10.

Let $\Omega \subseteq \mathcal{C}$ and $M>0$. The class of admissible functions ${\Psi }_{\Gamma }[\Omega ,M]$ consists of those functions $\varphi :{\mathbb{C}}^4\times \mathbb{U}⟶\mathbb{C}$ that satisfy the admissibility condition

$$\begin{array}{cc}\hfill \varphi & (1+M{e}^{i\theta },1+(\frac{\alpha }{c})\left(\frac{c}{\alpha }+\ell \right)M{e}^{i\theta },\hfill \\ \hfill 1+\frac{{\alpha }^2}{c(c-1)}& \left(L+\left(\frac{c(c-1)}{{\alpha }^2}+\ell (\frac{2c-1}{\alpha }+1)\right)M{e}^{i\theta }\right),\hfill \\ \hfill 1+\frac{{\alpha }^3}{c(c-1)(c-2)}& (N+3L(\frac{c-1}{\alpha }+1)+(\frac{c(c-1)(c-2)}{{\alpha }^3}\hfill \\ \hfill & +\ell (\frac{3c^2-6c+2}{{\alpha }^2}+\frac{3(c-1)}{\alpha }+1))M{e}^{i\theta });z)\notin \Omega ,\hfill \end{array}$$

where $z\in \mathbb{U}$, $Re(L{e}^{-i\theta })\ge \ell (\ell -1)M$ and $Re(N{e}^{-i\theta })\ge 0$ for all real θ and $\ell \in \mathbb{N}\backslash \left\{1\right\}$.

Corollary 6.

Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. If $f(z)\in \mathcal{H}$ satisfy the following conditions:

$$\left|\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right]\right|\le |\frac{\alpha }{c}|\ell M,$$

(31)

and

$$\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right)\in \Omega ,$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}-1\right|<M.$$

In the case $\Omega =q(\mathbb{U})=\left\{\omega :|\omega -1|<M,(M>0)\right\}$, we use notation ${\mathsf{\Phi }}_{\Gamma }\left[M\right]$ to the class ${\mathsf{\Phi }}_{\Gamma }[\Omega ,M]$.

Corollary 7.

Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. If $f(z)\in \mathcal{H}$ satisfy the conditions (30) and

$$\left|\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right)-1\right|<M,$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Putting $\varphi (a_1,a_2,a_3,a_4;z)=a_2=1+(\frac{k}{b})\left(\frac{b}{k}+\ell \right)M{e}^{i\theta }$ in Corollary 7, we have the following corollary:

Corollary 8.

Let $M>0$ and $b\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}$ with $Re(b)<\frac{-\ell }{2}$, $\ell \in \mathbb{N}\backslash \left\{1\right\}$. If $f(z)\in \mathcal{H}$ satisfy the condition (30), and also, if:

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z}-1\right|<M,$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Corollary 9.

Let $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. If $f(z)\in \mathcal{H}$ satisfy the conditions (31) and

$$\left|\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right)-1\right|<M,$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Putting $\varphi (a_1,a_2,a_3,a_4;z)=a_2=1+(\frac{\alpha }{c})\left(\frac{c}{\alpha }+\ell \right)M{e}^{i\theta }$ in Corollary 9, we have the following corollary:

Corollary 10.

Let $M>0$ and $c\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}$ with $Re(c)<\frac{-\ell }{2}$, $\ell \in \mathbb{N}\backslash \left\{1\right\}$. If $f(z)\in \mathcal{H}$ satisfy the conditions (31) and

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}-1\right|<M,$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Corollary 11.

Let $\ell \in \mathbb{N}\backslash 1$, $M>0$ and $b\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}$. If $f(z)\in \mathcal{H}$ satisfies the condition (30) and

$$\left|\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right]\right|\le |\frac{k}{b}|\ell M,$$

(32)

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Proof. 

Let $\varphi (a_1,a_2,a_3,a_4;z)=a_2-a_1.$ Using Corollary 5 with $\Omega =h(\mathbb{U})$ and

$$h(z)=|\frac{k}{b}|\ell M(z\in \mathbb{U}).$$

Now we show that $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. Since the condition (30) is satisfied from the condition (32) and

$$\begin{array}{cc}\hfill |\varphi & (1+M{e}^{i\theta },1+(\frac{k}{b})\left(\frac{b}{k}+\ell \right)M{e}^{i\theta },\hfill \\ \hfill 1+\frac{k^2}{b(b+1)}& \left(L+\left(\frac{b(b+1)}{k^2}+\ell (\frac{2b+1}{k}+1)\right)M{e}^{i\theta }\right),\hfill \\ \hfill 1+\frac{k^3}{b(b+1)(b+2)}& (N+3L(\frac{b+1}{k}+1)+(\frac{b(b+1)(b+2)}{k^3}\hfill \\ \hfill +\ell (\frac{3b^2+6b+2}{k^2}& +\frac{3(b+1)}{k}+1))M{e}^{i\theta });z)|=\left|\frac{k\ell M{e}^{i\theta }}{b}\right|=|\frac{k}{b}|\ell M,\hfill \end{array}$$

then we have the Corollary 11. □

Corollary 12.

Let $\ell \in \mathbb{N}\backslash \{1\}$, $M>0$ and $c\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}$. If $f(z)\in \mathcal{H}$ satisfies the condition (31) and

$$\left|\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right]\right|\le |\frac{\alpha }{c}|\ell M,$$

(33)

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Proof. 

Let $\varphi (a_1,a_2,a_3,a_4;z)=a_2-a_1.$ Using Corollary 6 with $\Omega =h(\mathbb{U})$ and

$$h(z)=|\frac{\alpha }{c}|\ell M(z\in \mathbb{U}).$$

Now we show that $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. Since the condition (31) is satisfied from the condition (33) and

$$\begin{array}{cc}\hfill |\varphi & (1+M{e}^{i\theta },1+(\frac{\alpha }{c})\left(\frac{c}{\alpha }+\ell \right)M{e}^{i\theta },\hfill \\ \hfill & 1+\frac{{\alpha }^2}{c(c-1)}\left(L+\left(\frac{c(c-1)}{{\alpha }^2}+\ell (\frac{2c-1}{\alpha }+1)\right)M{e}^{i\theta }\right),\hfill \\ \hfill & 1+\frac{{\alpha }^3}{c(c-1)(c-2)}(N+3L(\frac{c-1}{\alpha }+1)+(\frac{c(c-1)(c-2)}{{\alpha }^3}\hfill \\ \hfill & +\ell (\frac{3c^2-6c+2}{{\alpha }^2}+\frac{3(c-1)}{\alpha }+1))M{e}^{i\theta });z)|=\left|\frac{\alpha \ell M{e}^{i\theta }}{c}\right|=|\frac{\alpha }{c}|\ell M,\hfill \end{array}$$

then the corollary is completed. □

Corollary 13.

Let $\ell \in \mathbb{N}\backslash \left\{1\right\}$, $M>0$ and $b\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}$. If $f(z)\in \mathcal{H}$ satisfy the condition (30) and

$$\begin{array}{cc}\hfill \left|\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))\right]\right|& \le 2\left|\frac{k^3}{b(b+1)(b+2)}\right|\hfill \\ \hfill & \left(\left|\frac{2b+1}{k}+3\right|+\left|\frac{b(b+1)}{k^2}+\frac{2b+1}{k}+1\right|\right)M,\hfill \end{array}$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Proof. 

Let $\varphi (a_1,a_2,a_3,a_4;z)=a_4-a_3$. We use Corollary 5 with $\Omega =h(\mathbb{U})$

$$h(z)=2\left|\frac{k^3}{b(b+1)(b+2)}\right|\left(\left|\frac{2b+1}{k}+3\right|+\left|\frac{b(b+1)}{k^2}+\frac{2b+1}{k}+1\right|\right)Mz,(z\in \mathbb{U}).$$

Now we show that $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. Since

$$\begin{array}{cc}\hfill |& \varphi (1+M{e}^{i\theta },1+(\frac{k}{b})\left(\frac{b}{k}+\ell \right)M{e}^{i\theta },\hfill \\ \hfill & 1+\frac{k^2}{b(b+1)}(L+(\frac{b(b+1)}{k^2}+\ell (\frac{2b+1}{k}+1)\left)M{e}^{i\theta }\right),\hfill \\ \hfill & 1+\frac{k^3}{b(b+1)(b+2)}(N+3L(\frac{b+1}{k}+1)+(\frac{b(b+1)(b+2)}{k^3}\hfill \\ \hfill & +\ell (\frac{3b^2+6b+2}{k^2}+\frac{3(b+1)}{k}+1)\left)M{e}^{i\theta }\right);z)|\hfill \\ \hfill & =\left|\frac{k^3}{b(b+1)(b+2)}\left(N+(\frac{2b+1}{k}+3)L+\ell \left(\frac{b(b+1)}{k^2}+\frac{2b+1}{k}+1\right)M{e}^{i\theta }\right)\right|\hfill \\ \hfill & =\left|\frac{k^3{e}^{i\theta }}{b(b+1)(b+2)}\left(N{e}^{-i\theta }+(\frac{2b+1}{k}+3)L{e}^{-i\theta }+\ell \left(\frac{b(b+1)}{k^2}+\frac{2b+1}{k}+1\right)M\right)\right|\hfill \\ \hfill & \ge \left|\frac{k^3}{b(b+1)(b+2)}\right|(Re(N{e}^{-i\theta })+\left|\frac{2b+1}{k}+3\right|Re(L{e}^{-i\theta })\hfill \\ \hfill & +\ell \left|\frac{b(b+1)}{k^2}+\frac{2b+1}{k}+1\right|M)\hfill \\ \hfill & \ge \left|\frac{k^3}{b(b+1)(b+2)}\right|\left(\ell (\ell -1)M\left|\frac{2b+1}{k}+3\right|+\ell \left|\frac{b(b+1)}{k^2}+\frac{2b+1}{k}+1\right|M\right)\hfill \\ \hfill & \ge 2\left|\frac{k^3}{b(b+1)(b+2)}\right|\left(\left|\frac{2b+1}{k}+3\right|+\left|\frac{b(b+1)}{k^2}+\frac{2b+1}{k}+1\right|\right)M,\hfill \end{array}$$

we complete the proof of Corollary 13. □

Corollary 14.

Let $\ell \in \mathbb{N}\backslash \left\{1\right\}$, $M>0$ and $c\in \mathbb{C}\backslash {\mathbb{Z}}_0^{-}$. If $f(z)\in \mathcal{H}$ satisfy the condition (31) and

$$\begin{array}{cc}\hfill \left|\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))\right]\right|& \le 2\left|\frac{{\alpha }^3}{c(c-1)(c-2)}\right|\hfill \\ \hfill & \left(\left|\frac{2c-1}{\alpha }+3\right|+\left|\frac{c(c-1)}{{\alpha }^2}+\frac{2c-1}{\alpha }+1\right|\right)M,\hfill \end{array}$$

then

$$\left|\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}-1\right|<M.$$

Proof. 

Let $\varphi (a_1,a_2,a_3,a_4;z)=a_4-a_3.$ We use Corollary 6 with $\Omega =h(\mathbb{U})$

$$h(z)=2\left|\frac{{\alpha }^3}{c(c-1)(c-2)}\right|\left(\left|\frac{2c-1}{\alpha }+3\right|+\left|\frac{c(c-1)}{{\alpha }^2}+\frac{2c-1}{\alpha }+1\right|\right)Mz,(z\in \mathbb{U}).$$

Now we show that $\varphi \in {\Psi }_{\Gamma }[\Omega ,M]$. Since

$$\begin{array}{cc}\hfill |& \varphi (1+M{e}^{i\theta },1+(\frac{\alpha }{c})\left(\frac{c}{\alpha }+\ell \right)M{e}^{i\theta },\hfill \\ \hfill & 1+\frac{{\alpha }^2}{c(c-1)}(L+(\frac{c(c-1)}{{\alpha }^2}+\ell (\frac{2c-1}{\alpha }+1)\left)M{e}^{i\theta }\right),\hfill \\ \hfill & 1+\frac{{\alpha }^3}{c(c-1)(c-2)}(N+3L(\frac{c-1}{\alpha }+1)+(\frac{c(c-1)(c-2)}{{\alpha }^3}\hfill \\ \hfill & +\ell (\frac{3c^2-6c+2}{{\alpha }^2}+\frac{3(c-1)}{\alpha }+1))M{e}^{i\theta });z)|\hfill \\ \hfill & =|\frac{{\alpha }^3}{c(c-1)(c-2)}(N+(\frac{2c-1}{\alpha }+3)L+\ell \left(\frac{c(c-1)}{{\alpha }^2}+\frac{2c-1}{\alpha }+1\left)M{e}^{i\theta }\right)\right|\hfill \\ \hfill & =\left|\frac{{\alpha }^3{e}^{-i\theta }}{c(c-1)(c-2)}\left(N{e}^{-i\theta }+(\frac{2c-1}{\alpha }+3)L{e}^{-i\theta }+\ell (\frac{c(c-1)}{{\alpha }^2}+\frac{2c-1}{\alpha }+1)M\right)\right|\hfill \\ \hfill & \ge \left|\frac{{\alpha }^3}{c(c-1)(c-2)}\right|(Re(N{e}^{-i\theta })+\left|\frac{2c-1}{\alpha }+3\right|Re(L{e}^{-i\theta })\hfill \\ \hfill & +\ell \left|\frac{c(c-1)}{{\alpha }^2}+\frac{2c-1}{\alpha }+1\right|M)\hfill \\ \hfill & \ge \left|\frac{{\alpha }^3}{c(c-1)(c-2)}\right|\left(\ell (\ell -1)M\left|\frac{2c-1}{k}+3\right|+\ell \left|\frac{c(c-1)}{{\alpha }^2}+\frac{2c-1}{\alpha }+1\right|M\right)\hfill \\ \hfill & \ge 2\left|\frac{{\alpha }^3}{c(c-1)(c-2)}\right|\left(\left|\frac{2c-1}{\alpha }+3\right|+\left|\frac{c(c-1)}{{\alpha }^2}+\frac{2c-1}{\alpha }+1\right|\right)M,\hfill \end{array}$$

we complete the proof of Corollary 14. □

3. Third Order Differential Supordination with ${\mathcal{H}}_{\mathbf{\alpha },\mathbf{\beta }}^{\mathbf{\eta },\mathbf{k}}(\mathit{f}(\mathit{z}))$

Definition 11.

Let $\Omega \subseteq \mathbb{C}$ and $q(z)\in \mathbb{Q}$. The class of admissible functions ${\Psi }_{\Gamma }[\Omega ,q(z)]$ consists of those functions $\psi :{\mathbb{C}}^4\times \overline{\mathbb{U}}⟶\mathbb{C}$ that satisfy the admissibility condition

$$\psi (a_1,a_2,a_3,a_4;z)\notin \Omega ,$$

whenever

$$\begin{array}{cc}\hfill & a_1=q(\zeta ),a_2=\frac{k\zeta q^{\prime }(\zeta )+bq(\zeta )}{bm},\hfill \\ \hfill Re& \left(\frac{(b+1)(a_3-a_1)}{k(a_2-a_1)}-\frac{2b+1}{k}\right)\le \frac{1}{m}Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}+1\right),\hfill \\ \hfill Re& (\frac{(b+1)(b+2)(a_4-a_1)-3(b+1)(b+k+1)(a_3-a_1)}{k^2(a_2-a_1)}+\frac{3b(b+1)+1}{k^2}\hfill \\ \hfill & +\frac{6b+3}{k}+2)\le \frac{1}{m^2}Re\left(\frac{{\zeta }^2{q}^{‴}(\zeta )}{q^{\prime }(\zeta )}\right),\hfill \end{array}$$

where $z\in \mathbb{U}$; $\zeta \in \partial \mathbb{U}$, $m\in \mathbb{N}\backslash \left\{1\right\}$ and $b=\eta +k$.

Theorem 9.

Let $\varphi \in {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$. If $f(z)\in \mathcal{H}$ and $\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}\in {\mathbb{Q}}_1$ satisfy the following conditions:

$$\begin{array}{cc}\hfill & Re\left(\frac{z{q}^{″}(z)}{q^{\prime }(z)}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))\right]}{q^{\prime }(z)}\right|\le |\frac{k}{b}|m,\hfill \\ \hfill & \left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},\hfill \end{array}$$

(34)

are univalent, and

$$\Omega \subset \{\varphi (\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\left):z\in \mathbb{U}\right\},$$

(35)

then

$$q(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}.$$

Proof. 

Let the functions $p(z)$ and $\psi$ are defined by (9) and (13). Since $\varphi \in {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$, therefore (14) and (35) give

$$\Omega \subset \psi (p(z),z{p}^{\prime }(z),z^2{p}^{″}(z),z^3{p}^{‴}(z);z).$$

The admissible condition for $\varphi \in {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$ is equivalent to the admissible condition for $\psi$ in Definition 6 with $n=2$. Therefore, $\psi \in {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$ and by using (34) and Theorem 2, we have $q(z)\prec p(z)$ which yields $q(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}.$ Therefore, we complete the proof of Theorem 9. □

Moreover, in a similar way, we can define the following:

Definition 12.

Let $\Omega \subseteq \mathbb{C}$ and $q(z)\in \mathbb{Q}$. The class of admissible functions ${\Psi }_{\Gamma }[\Omega ,q(z)]$ consists of those functions $\psi :{\mathbb{C}}^4\times \overline{\mathbb{U}}⟶\mathbb{C}$ that satisfy the admissibility condition:

$$\psi (a_1,a_2,a_3,a_4;z)\notin \Omega ,$$

whenever

$$\begin{array}{cc}\hfill & a_1=q(\zeta ),a_2=\frac{\alpha \zeta q^{\prime }(\zeta )+cq(\zeta )}{cm},\hfill \\ \hfill Re& \left(\frac{(c-1)(a_3-a_1)}{\alpha (a_2-a_1)}-\frac{2c-1}{\alpha }\right)\le \frac{1}{m}Re\left(\frac{\zeta q^{″}(\zeta )}{q^{\prime }(\zeta )}+1\right),\hfill \\ \hfill Re& (\frac{(c-1)(c-2)(a_4-a_1)-3(c-1)(c+\alpha -1)(a_3-a_1)}{{\alpha }^2(a_2-a_1)}+\frac{3c(c-1)+1}{{\alpha }^2}\hfill \\ \hfill & +\frac{6c-3}{\alpha }+2)\le \frac{1}{m^2}Re\left(\frac{{\zeta }^2{q}^{‴}(\zeta )}{q^{\prime }(\zeta )}\right),\hfill \end{array}$$

where $z\in \mathbb{U}$; $\zeta \in \partial \mathbb{U}$, $m\in \mathbb{N}\backslash \left\{1\right\}$ and $c=\alpha +\beta$.

With the assist of Definition 12 and Theorem 4, we have the following theorem

Theorem 10.

Let $\varphi \in {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$. If $f(z)\in \mathcal{H}$ and $\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}\in {\mathbb{Q}}_1$ satisfy the following conditions:

$$\begin{array}{cc}\hfill & Re\left(\frac{z{q}^{″}(z)}{q^{\prime }(z)}\right)\ge 0,\left|\frac{\frac{1}{z}\left[{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))-{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))\right]}{q^{\prime }(z)}\right|\le |\frac{\alpha }{c}|m,\hfill \\ \hfill & \left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},\hfill \end{array}$$

(36)

are univalent, and

$$\Omega \subset \left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},$$

then

$$q(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}.$$

If $\Omega \ne \mathbb{C}$ is a simply connected domain, then $\Omega =h(\mathbb{U})$ for some conformal mapping $h(z)$ of $\mathbb{U}$ onto $\Omega .$ In this case, the class ${\Psi }_{\Gamma }^{\prime }[h(u),q(z)]$ is written as ${\Psi }_{\Gamma }^{\prime }[h,q].$ The following theorem is a direct consequence of Theorems 3 and 4.

Theorem 11.

Let $\varphi \in {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$ and $h(z)$ be analytic function in $\mathbb{U}$, and $f(z)\in \mathcal{H}$ and $\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}\in {\mathbb{Q}}_1$ satisfy the condition (34). If

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},$$

is univalent function in $\mathbb{U}$, and

$$h(z)\prec \varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right),$$

then

$$q(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}.$$

Theorem 12.

Let $\varphi \in {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$ and $h(z)$ be analytic function in $\mathbb{U}$. If $f(z)\in \mathcal{H}$ and $\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}\in {\mathbb{Q}}_1$ satisfy the condition (36). If

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},$$

is univalent function in $\mathbb{U}$ and

$$h(z)\prec \varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right),$$

then

$$q(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}.$$

Theorem 13.

Let $h(z)$ be analytic function in $\mathbb{U}$ and let $\psi :{\mathbb{C}}^4\times \stackrel{\_}{\mathbb{U}}⟶\mathbb{C}$ and ψ is given by (13). Suppose that the differential (26) has a solution $q(z)\in {\mathbb{Q}}_1$, and $f(z)\in \mathcal{H}$ satisfy the condition (34). If

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},$$

is univalent function in $\mathbb{U}$, and

$$h(z)\prec \varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right),$$

(37)

then

$$q(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}.$$

and $q(z)$ is the best subordinant of relation (36).

Proof. 

The proof is similar to Theorem 7 and it is being omitted here. □

Combining both Theorems 5 and 11, we have the following sandwich result:

Corollary 15.

Let $h_1(z)$ and $q_1(z)$ be analytic functions in $\mathbb{U}$, also, let $h_2(z)$ be univalent in $\mathbb{U}$, $q_2(z)\in {\mathbb{Q}}_1$ with $q_1(0)=q_2(0)=1$ and $\varphi \in {\Psi }_{\Gamma }[\Omega ,q(z)]\cap {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)].$ If $f(z)\in \mathcal{H}$ and $\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}\in {\mathbb{Q}}_1\cap \mathcal{H}$,

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},$$

is univalent function in $\mathbb{U}$, and the conditions (22) and (34) are satisfied, also let

$$h_1(z)\prec \varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +1,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +2,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta +3,k}(f(z))}{z};z\right)\prec h_2(z),$$

then

$$q_1(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z}\prec q_2(z).$$

The proof of the following theorem is similar to Theorem 8; therefore, we omitted it.

Theorem 14.

Let $h(z)$ be analytic function in $\mathbb{U}$ and let $\psi :{\mathbb{C}}^4\times \stackrel{\_}{\mathbb{U}}⟶\mathbb{C}$ and ψ is given by (21). Suppose that the differential (28) has a solution $q(z)\in {\mathbb{Q}}_1$.If $f(z)\in \mathcal{H}$ satisfy the condition (36). If

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},$$

is univalent function in $\mathbb{U}$, and

$$h(z)\prec \varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right),$$

(38)

then

$$q(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}.$$

and $q(z)$ is the best subordinant of (38).

By combining Theorems 8 and 12, we obtain the following sandwich type result.

Corollary 16.

Let $h_1(z)$ and $q_1(z)$ be analytic functions in $\mathbb{U}$ and let $h_2(z)$ be univalent in $\mathbb{U}$, $q_2(z)\in {\mathbb{Q}}_1$ with $q_1(0)=q_2(0)=1$ and $\varphi \in {\Psi }_{\Gamma }[\Omega ,q(z)]\cap {\Psi }_{\Gamma }^{\prime }[\Omega ,q(z)]$. If $f(z)\in \mathcal{H}$ and $\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}\in {\mathbb{Q}}_1\cap \mathcal{H}$,

$$\left\{\varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right):z\in \mathbb{U}\right\},$$

is univalent function in $\mathbb{U}$, and the conditions (23) and (36) are satisfied; also let

$$h_1(z)\prec \varphi \left(\frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -1}^{\eta ,k}(f(z))}{z},\frac{{\mathcal{H}}_{\alpha ,\beta -2}^{\eta ,k}(f(z))}{z};z\right)\prec h_2(z),$$

then

$$q_1(z)\prec \frac{{\mathcal{H}}_{\alpha ,\beta +1}^{\eta ,k}(f(z))}{z}\prec q_2(z).$$

4. Conclusions

By using the method of third-order differential subordination and superordination, we obtained many interesting results concerning the subordination and superordination properties of analytic functions associated with the operator ${\mathcal{H}}_{\alpha ,\beta }^{\eta ,k}(f)$.