Majorization Problems for Subclasses of Meromorphic Functions Defined by the Generalized $\mathfrak{q}$-Sălăgean Operator ^†

Abstract

Using the generalized $\mathfrak{q}$-Sălăgean operator, we introduce a new class of meromorphic functions in a punctured unit disk ${\mathcal{U}}^{\ast }$ and investigate a majorization problem associated with this class. The principal tool employed in this analysis is the recently established $\mathfrak{q}$-Schwarz–Pick lemma. We investigate a majorization problem for meromorphic functions when the functions of the right hand side of the majorization belongs to this class. The main tool for this investigation is the generalization of Nehari’s lemma for the Jackson’s $\mathfrak{q}$-difference operator ${\partial }_{\mathfrak{q}}$ given recently by Adegani et al.

1. Introduction and Definitions

Let $\Sigma$ denote the class of meromorphic functions of the form

$$f\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{a}_k{z}^k,z\in {\mathcal{U}}^{\ast },$$

(1)

where ${\mathcal{U}}^{\ast }$ is the punctured unit disc defined by ${\mathcal{U}}^{\ast }:=\{z\in \mathbb{C}:0<|z|<1\}=\mathcal{U}\setminus \left\{0\right\}$. If $f\in \Sigma$ has the form (1) and g has the Laurent power series expansion

$$g\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{c}_k{z}^k,z\in {\mathcal{U}}^{\ast },$$

then the well-known Hadamard (or convolution) product of f and g is defined by (see, for example [1], p. 246)

$$(f\ast g)\left(z\right):={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{a}_k{c}_k{z}^k,z\in {\mathcal{U}}^{\ast }.$$

The class ${\Sigma }_s\left(\delta \right)$ of the starlike functions of order δ, $0\le \delta <1$, is the subclass of $\Sigma$, consisting of the functions $f\in \Sigma$ that satisfy

$$-Re{\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}>\delta ,z\in {\mathcal{U}}^{\ast },$$

and ${\Sigma }_s:={\Sigma }_s\left(0\right)$ is the well-known class of starlike meromorphic functions (see [2]).

Let f and g be two analytic functions in the open unit disk $\mathcal{U}$ or in the punctured one ${\mathcal{U}}^{\ast }$. We say that f is majorized by g (see [3]), written as follows:

$$f\left(z\right)\ll g\left(z\right),$$

(2)

if there exists a function $\mathrm{\Omega }$ analytic in $\mathcal{U}$ (or in ${\mathcal{U}}^{\ast }$) and satisfying $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in \mathcal{U}$ (or in $z\in {\mathcal{U}}^{\ast }$), such that

$$f\left(z\right)=\mathrm{\Omega }\left(z\right)g\left(z\right),z\in \mathcal{U}(\mathrm{or}\mathrm{in}z\in {\mathcal{U}}^{\ast }).$$

The point $z_0=0$ is an isolated singular point for the function $\mathrm{\Omega }$, while the $\mathrm{\Omega }$ analytic in ${\mathcal{U}}^{\ast }$ is represented with $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in {\mathcal{U}}^{\ast }$. Therefore, according to the Cauchy–Riemann removability criterion for singularities (see, for example, [4], Theorem 4.8.3, p. 128), the point $z_0=0$ will be a removable singularity of $\mathrm{\Omega }$; thus, without loss of generality, we can assume that $\mathrm{\Omega }$ is the analytic in $\mathcal{U}$, such that $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in \mathcal{U}$.

Also, according to [5], we say that f is subordinate to g, denoted as follows:

$$f\left(z\right)\prec g\left(z\right),$$

(3)

if there exists a function $\tilde{\omega }$ analytic in $\mathcal{U}$, with $\tilde{\omega }\left(0\right)=0$ and $\left|\tilde{\omega }\left(z\right)\right|<1$ for all $z\in \mathcal{U}$, such that $f=g\circ \tilde{\omega }$. Lemma 2.1. p. 36 of [6] shows that if the function g is univalent in $\mathcal{U}$, then $f\left(z\right)\prec g\left(z\right)$ if and only if

$$f\left(0\right)=g\left(0\right),f\left(\mathcal{U}\right)\subset g\left(\mathcal{U}\right).$$

Like in [7], the function f is said to be quasi-subordinate to g if there exists a function $\omega$ analytic in $\mathcal{U}$, with $\left|\omega \right(z\left)\right|\le 1$, $z\in \mathcal{U}$, such that the quotient function ${\displaystyle \frac{f}{\omega }}$ is analytic in $\mathcal{U}$ and

$${\displaystyle \frac{f\left(z\right)}{\omega \left(z\right)}}\prec g\left(z\right),$$

(4)

and we denote this quasi-subordination by

$$f\left(z\right){\prec }_{q}g\left(z\right).$$

(5)

Therefore, by the definition of subordination, the quasi-subordination (4) is equivalent to the fact that there exists a function $\tilde{\omega }$ analytic in $\mathcal{U}$ with $|\tilde{\omega }\left(z\right)|\le |z|$ in $\mathcal{U}$, such that

$$f\left(z\right)=\omega \left(z\right)g\left(\tilde{\omega }\left(z\right)\right),z\in \mathcal{U}.$$

(6)

Note that if we set $\omega \equiv 1$ in (6), then the quasi-subordination (5) reduces to the subordination (3), while for the case $\tilde{\omega }\left(z\right)\equiv z$ in (6), the quasi-subordination (5) becomes the majorization (2).

In the 1990’s, the notion of a $\mathfrak{q}$-starlike function was defined in Geometric Function Theory using $\mathfrak{q}$-calculus methods [8]. Following that, this line of inquiry developed by introducing a number of $\mathfrak{q}$-calculus operators used for various investigations, such as defining new classes of analytic functions and obtaining a variety of characteristics for them. Also, many studies deal with different geometric properties and coefficients estimates for the functions of this class.

The $\mathfrak{q}$-analogue of the derivative and integral operator was described by Jackson [9,10] who also suggested some of their applications. Kanas and Răducanu defined the $\mathfrak{q}$-analogue of the Ruscheweyh differential operator using the idea of convolution; hence, they started the line of inquiry where the classical operators were adapted to the $\mathfrak{q}$-calculus aspects embedded in Geometric Function Theory [11]. The study of such types of properties was first suggested in [11], and researchers quickly adopted the concept, such as Mahmood and Sokół [12], Aldweby and Darus [13], and many other research scholars that studied, over time, various classes of analytic functions defined using the Ruscheweyh differential operator’s $\mathfrak{q}$-analogue.

We will now introduce the fundamental idea of the $\mathfrak{q}$-calculus that was developed by Jackson [10], which is useful for our future research. The Jackson’s $\mathfrak{q}$-difference operator ${\partial }_{\mathfrak{q}}$, $0<\mathfrak{q}<1$, for a function f, is defined by

$${\partial }_{\mathfrak{q}}f\left(z\right):=\left\{\begin{array}{ccc}{\displaystyle \frac{f\left(z\right)-f\left(\mathfrak{q}z\right)}{(1-\mathfrak{q})z}},\hfill & \mathrm{if}\hfill & z\ne 0,\hfill \\ f^{\prime }\left(0\right),\hfill & \mathrm{if}\hfill & z=0,\hfill \end{array}\right.$$

assuming that the function f is differentiable at $z_0=0$. It follows easily that

$${\partial }_{\mathfrak{q}}\left(\sum _{k=1}^{\infty }{a}_k{z}^k\right)=\sum _{k=1}^{\infty }{\left[k\right]}_{\mathfrak{q}}{a}_k{z}^{k-1},$$

where the $\mathfrak{q}$-integer ${\left[k\right]}_{\mathfrak{q}}$ is defined by (see, for details, [9,10,14])

$${\left[k\right]}_{\mathfrak{q}}:={\displaystyle \frac{1-{\mathfrak{q}}^k}{1-\mathfrak{q}}}=1+\sum _{n=1}^{k-1}{\mathfrak{q}}^n,{\left[0\right]}_{\mathfrak{q}}:=0.$$

Also, the following fundamental laws hold for the $\mathfrak{q}$-difference operator:

$$\begin{array}{cc}\hfill & {\partial }_{\mathfrak{q}}\left(\sigma f\left(z\right)\pm \tau g\left(z\right)\right)=\sigma {\partial }_{\mathfrak{q}}f\left(z\right)\pm \tau {\partial }_{\mathfrak{q}}g\left(z\right),\sigma \tau \in \mathbb{C}\hfill \\ \hfill & {\partial }_{\mathfrak{q}}\left(f\left(z\right)g\left(z\right)\right)=f\left(\mathfrak{q}z\right){\partial }_{\mathfrak{q}}\left(g\left(z\right)\right)+g\left(z\right){\partial }_{\mathfrak{q}}\left(f\left(z\right)\right),\hfill \\ \hfill & {\partial }_{\mathfrak{q}}{\displaystyle \frac{f\left(z\right)}{g\left(z\right)}}={\displaystyle \frac{{\partial }_{\mathfrak{q}}\left(f\left(z\right)\right)g\left(z\right)-f\left(z\right){\partial }_{\mathfrak{q}}\left(g\left(z\right)\right)}{g\left(\mathfrak{q}z\right)g\left(z\right)}},g\left(\mathfrak{q}z\right)g\left(z\right)\ne 0.\hfill \end{array}$$

In [15], Aouf et al. generalized the $\mathfrak{q}$-Sălăgean operator, introducing the operator ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^s:\Sigma \to \Sigma$, $s\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$, $\varsigma \ge 0$, $0<\mathfrak{q}<1$, defined as follows:

$$\begin{array}{cc}\hfill & {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{0}f\left(z\right):=f\left(z\right)\hfill \\ \hfill & {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{1}f\left(z\right)=\left(1-\varsigma \right)f\left(z\right)+{\displaystyle \frac{\varsigma }{z}}{\partial }_{\mathfrak{q}}\left(z^{2}f\left(z\right)\right),\hfill \\ \hfill & .......................................\hfill \\ \hfill & {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)=\left(1-\varsigma \right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s-1}f\left(z\right)+{\displaystyle \frac{\varsigma }{z}}{\partial }_{\mathfrak{q}}\left(z^2{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s-1}f\left(z\right)\right),s\in \mathbb{N}.\hfill \end{array}$$

Then,

$${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)={\displaystyle \frac{1}{z}}+\sum _{k=1}^{\infty }{\left[1+\varsigma \left({[k+2]}_{\mathfrak{q}}-1\right)\right]}^s{a}_k{z}^k,$$

(7)

and from (7), we get

$$\varsigma {\mathfrak{q}}^{2}z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\right)={\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)-\left(1+\varsigma \mathfrak{q}\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right),\varsigma >0.$$

(8)

By using the operator ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^s$, we define a new subclass of function $f\in \Sigma$ as follows:

Definition 1.

Let $-1\le B<A\le 1$, $\eta \in {\mathbb{C}}^{\ast }:=\mathbb{C}\setminus \left\{0\right\}$, $\alpha \in \mathbb{R}\setminus \{-1,1\}$ and ${\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\le 1$. A function $f\in \Sigma$ is said to be in the class ${\wp }_{\varsigma ,\mathfrak{q}}^s(\alpha ,\eta ;A,B)$ of meromorphic functions of complex order $\eta \ne 0$ in ${\mathcal{U}}^{\ast }$ if and only if

$$1-{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{\mathfrak{q}z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)}}+1\right)-\alpha \left|-{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{\mathfrak{q}z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)}}+1\right)\right|\prec {\displaystyle \frac{1+Az}{1+Bz}}.$$

(9)

The assumptions of this definition were made because of the following reasons: The condition $\alpha \in \mathbb{R}\setminus \{-1,1\}$ is necessary for obtaining the inequality (19), while the assumption ${\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\le 1$ was made to assure that the denominator of the fraction from the right-hand side of (18) doesn’t vanishes in $\mathcal{U}$, and consequently the function given by this fraction is analytic in $\mathcal{U}$.

In particular, for $A=1$, $B=-1$ and $\alpha =0$, we denote the class

$$\begin{array}{cc}\hfill {\wp }_{\varsigma ,\mathfrak{q}}^s\left(\eta \right)& :={\wp }_{\varsigma ,\mathfrak{q}}^s(0,\eta ;1,-1)\hfill \\ \hfill & =\left\{f\in \Sigma :Re\left[1-{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)}}+1\right)\right]>0,z\in \mathcal{U}\right\},\hfill \end{array}$$

(10)

and note the following:

(i)

for $\eta =(1-\delta )cos\theta {\mathrm{e}}^{-i\theta }$$\left(\left|\theta \right|\le {\displaystyle \frac{\pi }{2}},0\le \delta <1\right)$, the class ${\wp }_{\varsigma ,\mathfrak{q}}^s\left(\eta \right)={\wp }_{\varsigma ,\mathfrak{q}}^s\left((1-\delta )cos\theta {\mathrm{e}}^{-i\theta }\right)$$=:{\wp }_{\varsigma ,\mathfrak{q}}^s(\delta ,\theta )$ called the generalized class of meromorphic $\theta$-spiral-like functions of order $\delta$$\left(0\le \delta <1\right)$ if

$$Re\left({\mathrm{e}}^{i\theta }{\displaystyle \frac{z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)}}\right)<-\delta cos\theta ,z\in \mathcal{U};$$

(ii)

taking $s=0$ and using the fact that $\underset{\mathfrak{q}\to 1^{-}}{lim}{\partial }_{\mathfrak{q}}f\left(z\right)=f^{\prime }\left(z\right)$, $z\in {\mathcal{U}}^{\ast }$, for $\mathfrak{q}\to 1^{-}$, the class ${\wp }_{\varsigma ,\mathfrak{q}}^0(0,\eta ;1,-1)$ reduces to the class

$$\Sigma \left(\eta \right):=\left\{f\in \Sigma :Re\left[1-{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}+1\right)\right]>0,z\in \mathcal{U}\right\},$$

where $\eta \in {\mathbb{C}}^{\ast }$;

(iii)

for $s=0$ and $\eta =1-\xi$, the class ${\Sigma }^{\ast }\left(\xi \right):={\wp }_{\varsigma ,\mathfrak{q}}^0(0,1-\xi ;1,-1)$$\left(0\le \xi <1\right)$ represents the meromorphic starlike univalent function of order $\xi$ in ${\mathcal{U}}^{\ast }$ (see [2,16] for $p=1$).

Majorization problems for univalent functions involve finding conditions under which one univalent function majorizes another and has applications in various areas like the Geometric Function Theory and General Theory of Conformal Mappings.

A general problem connected with the majorization for multivalent functions is to determine simple sufficient conditions that imply the majorization between the images of two functions by different operators. This is a challenging problem since the multivalent functions have more complex behavior compared to the univalent functions, while understanding majorization in the context of multivalent functions is crucial for studying the properties of functions with multiple analytic branches. By establishing majorization criteria for multivalent functions, researchers can gain better insights into the behavior and structure of these functions.

Numerous scholars have written extensively about majorization issues for both univalent and multivalent functions. Both MacGregor [3] and Altintas et al. [17] (see also [18]) have examined a majorization problem for the normalized classes of starlike functions. See, for instance, refs. [19,20,21,22,23,24] for recent expository works on majorization problems for meromorphic univalent and p-valent functions; these articles deal with majorization problems for different classical subclasses of univalent functions.

Inspired by the previously stated research, the author of this paper uses the generalized $\mathfrak{q}$-Sălăgean operator to investigate the majorization problem for the new class ${\wp }_{\varsigma ,\mathfrak{q}}^s(\alpha ,\eta ;A,B)$ of meromorphic functions of complex order.

2. Main Results

Throughout the sequel, we will presume, unless otherwise noted, that

$$-1\le B<A\le 1,\eta \in {\mathbb{C}}^{\ast },s\in {\mathbb{N}}_0,\varsigma \ge 0,0<\mathfrak{q}<1.$$

Theorem 1.

Let the functions $f\in \Sigma$ and $g\in {\wp }_{\varsigma ,\mathfrak{q}}^s(\alpha ,\eta ;A,B)$. Suppose that ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\ll {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)$, and let Ω be the function that realizes this majorization. Assume that

$$1-\mu -(1+\varsigma \mathfrak{q})(\tau +\mu )>0,$$

(11)

where

$$\begin{array}{cc}\hfill \mu & :=\mu (r,\mathfrak{q})=\underset{\left|z\right|\le r}{sup}\left|\mathrm{\Omega }\left(\mathfrak{q}z\right)\right)|=\underset{\left|\zeta \right|=\mathfrak{q}r}{max}\left|\mathrm{\Omega }\left(\zeta \right)\right|\in [0,1],0<\mathfrak{q}<1,\hfill \\ \hfill \tau & :=\tau \left(r\right)=\underset{\left|z\right|\le r}{sup}\left|\mathrm{\Omega }\left(z\right)\right|=\underset{\left|\zeta \right|=r}{max}\left|\mathrm{\Omega }\left(\zeta \right)\right|\in [0,1],\hfill \end{array}$$

If

$$r_0:=sup\left\{\rho \in (0,1):\phi \left(r\right)\ge 0,r\in (0,\rho )\right\},$$

where

$$\begin{array}{cc}\hfill \phi \left(r\right):=& (1-\mu )\left(1-r^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\right)r\right]\hfill \\ \hfill & -\left[(1+\varsigma \mathfrak{q})(\tau +\mu )+\varsigma {\mathfrak{q}}^{2}r(1+\mu \tau )\right](1+|B\left|r\right),\hfill \end{array}$$

(12)

then

$$\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)\right|\le \left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|,0<\left|z\right|<r_0.$$

Proof. 

Since $g\in {\wp }_{\varsigma ,\mathfrak{q}}^s(\alpha ,\eta ;A,B)$, it follows from (9), that

$$\begin{array}{c}\hfill 1-{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{\mathfrak{q}z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)}}+1\right)-\alpha \left|-{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{\mathfrak{q}z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)}}+1\right)\right|\\ \hfill ={\displaystyle \frac{1+A\tilde{\omega }\left(z\right)}{1+B\tilde{\omega }\left(z\right)}},z\in \mathcal{U},\end{array}$$

(13)

with $\tilde{\omega }\left(z\right)=c_{1}z+c_2{z}^2+\dots \in \mathfrak{B}$, where $\mathfrak{B}$ denotes the class of analytic functions in $\mathcal{U}$ that satisfies the conditions $\tilde{\omega }\left(0\right)=0$ and $|\tilde{\omega }\left(z\right)|\le 1$, $z\in \mathcal{U}$.

Denoting

$$\widehat{\omega }\left(z\right):=1-{\displaystyle \frac{1}{\eta }}\left({\displaystyle \frac{\mathfrak{q}z{\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)}}+1\right),z\in \mathcal{U},$$

(14)

Equation (13) is equivalent to

$$\widehat{\omega }\left(z\right)-\alpha \left|\widehat{\omega }\left(z\right)-1\right|={\displaystyle \frac{1+A\tilde{\omega }\left(z\right)}{1+B\tilde{\omega }\left(z\right)}},z\in \mathcal{U},$$

or

$$\widehat{\omega }\left(z\right)={\displaystyle \frac{1+\frac{A-B\alpha {\mathrm{e}}^{-i{\theta }_0}}{1-\alpha {\mathrm{e}}^{-i{\theta }_0}}·\tilde{\omega }\left(z\right)}{1+B\tilde{\omega }\left(z\right)}},z\in \mathcal{U},$$

(15)

where ${\theta }_0:=\theta \left(z\right)=arg\left(\widehat{\omega }\left(z\right)-1\right)\in [0,2\pi )$. Using the relation (15), it follows from (14) that

$${\displaystyle \frac{\mathfrak{q}z{\partial }_{\mathfrak{q}}{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)}}=-{\displaystyle \frac{1+\left[\frac{(A-B)\eta }{1-\alpha {\mathrm{e}}^{-i{\theta }_0}}+B\right]\tilde{\omega }\left(z\right)}{1+B\tilde{\omega }\left(z\right)}},z\in \mathcal{U},$$

(16)

and, using (8) in (16), we obtain

$${\displaystyle \frac{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)}{{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)}}={\displaystyle \frac{1+\left[-\frac{(A-B)\varsigma \mathfrak{q}\eta }{1-\alpha {\mathrm{e}}^{-i{\theta }_0}}+B\right]\tilde{\omega }\left(z\right)}{1+B\tilde{\omega }\left(z\right)}},z\in \mathcal{U},$$

or equivalently

$${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)={\displaystyle \frac{1+B\tilde{\omega }\left(z\right)}{1+\left[-\frac{(A-B)\varsigma \mathfrak{q}\eta }{1-\alpha {\mathrm{e}}^{-i{\theta }_0}}+B\right]\tilde{\omega }\left(z\right)}}·{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right),z\in \mathcal{U}.$$

(17)

Since $\tilde{\omega }\in \mathfrak{B}$, from the well-known Schwarz lemma, we have $|\tilde{\omega }\left(z\right)|\le |z|$ for all $z\in \mathcal{U}$. According to the triangle inequality, together with the assumption ${\displaystyle \frac{(A-B)\mathfrak{q}\left|\varsigma \eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\le 1$, from the Definition 1, the relation (17) leads to

$$\begin{array}{cc}\hfill {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)& \le {\displaystyle \frac{1+\left|B\right|\left|z\right|}{1-\left|-\frac{(A-B)\varsigma \mathfrak{q}\eta }{1-\alpha {\mathrm{e}}^{-i{\theta }_0}}+\mathcal{B}\right|\left|z\right|}}\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1+\left|B\right|\left|z\right|}{1-\left[\frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}+\left|B\right|\right]\left|z\right|}}\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|,z\in \mathcal{U},\hfill \end{array}$$

(18)

where we used $\alpha \in \mathbb{R}\setminus \{-1,1\}$ and

$$\left|1-\alpha {\mathrm{e}}^{i{\theta }_0}\right|\ge min\left\{\left|1-\alpha {\mathrm{e}}^{i\theta }\right|:\theta \in [0,2\pi )\right\}=|1-\left|\alpha \right||>0.$$

(19)

Further, since ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f$ is majorized by ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g$, and these two functions belong to the class $\Sigma$, we have

$${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)=\mathrm{\Omega }\left(z\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right),z\in {\mathcal{U}}^{\ast },$$

(20)

where $\mathrm{\Omega }$ is an analytic function in $\mathcal{U}$, such that $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in \mathcal{U}$.

Applying the $\mathfrak{q}$-differentiation rule in the both sides of (20) and multiplying by z, we obtain the following:

$$z{\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\right)=z{\partial }_{\mathfrak{q}}\left(\mathrm{\Omega }\left(z\right)\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)+z\mathrm{\Omega }\left(\mathfrak{q}z\right){\partial }_{\mathfrak{q}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\right),z\in {\mathcal{U}}^{\ast }.$$

(21)

Using (8) and (20) in (21), it follows that the left-hand side of (21) will be

$${\displaystyle \frac{1}{\varsigma {\mathfrak{q}}^2}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)-\left(1+\varsigma \mathfrak{q}\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\right),$$

(22)

while, for a similar reason, the right-hand side of (21) becomes

$$z{\partial }_{\mathfrak{q}}\left(\mathrm{\Omega }\left(z\right)\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)+{\displaystyle \frac{\mathrm{\Omega }\left(\mathfrak{q}z\right)}{\varsigma {\mathfrak{q}}^2}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)-\left(1+\varsigma \mathfrak{q}\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\right).$$

(23)

From (21), both of the right-hand sides of (22) and (23) will be equal, that is

$$\begin{array}{c}{\displaystyle \frac{1}{\varsigma {\mathfrak{q}}^2}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)-\left(1+\varsigma \mathfrak{q}\right)\mathrm{\Omega }\left(z\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\right)\hfill \\ \hfill =z{\partial }_{\mathfrak{q}}\left(\mathrm{\Omega }\left(z\right)\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)+{\displaystyle \frac{\mathrm{\Omega }\left(\mathfrak{q}z\right)}{\varsigma {\mathfrak{q}}^2}}\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)-\left(1+\varsigma \mathfrak{q}\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\right)\iff \end{array}$$

$$\begin{array}{c}{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)-\left(1+\varsigma \mathfrak{q}\right)\mathrm{\Omega }\left(z\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\hfill \\ \hfill =\varsigma {\mathfrak{q}}^{2}z{\partial }_{\mathfrak{q}}\left(\mathrm{\Omega }\left(z\right)\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)+\mathrm{\Omega }\left(\mathfrak{q}z\right)\left({\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)-\left(1+\varsigma \mathfrak{q}\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\right),\end{array}$$

or

$$\begin{array}{c}\hfill {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)=\left[(1+\varsigma \mathfrak{q})\left(\mathrm{\Omega }\left(z\right)-\mathrm{\Omega }\left(\mathfrak{q}z\right)\right)+\varsigma {\mathfrak{q}}^{2}z{\partial }_{\mathfrak{q}}\left(\mathrm{\Omega }\left(z\right)\right)\right]{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)\\ \hfill +\mathrm{\Omega }\left(\mathfrak{q}z\right){\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right),z\in {\mathcal{U}}^{\ast }.\end{array}$$

(24)

Since $\mathrm{\Omega }$ is an analytic in $\mathcal{U}$ and satisfies the inequality $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in \mathcal{U}$, it follows Lemma 2.1 of [25] that

$$\left|{\partial }_q\left(\mathrm{\Omega }\left(z\right)\right)\right|\le {\displaystyle \frac{\left|1-\overline{\mathrm{\Omega }\left(\mathfrak{q}z\right)}\mathrm{\Omega }\left(z\right)\right|}{1-{\left|z\right|}^2\mathfrak{q}}},z\in \mathcal{U}.$$

(25)

According to the inequalities (18) and (25), from (24) and the triangle inequality, we obtain

$$\begin{array}{c}\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)\right|\le \left[(1+\varsigma \mathfrak{q})\left|\mathrm{\Omega }\left(z\right)-\mathrm{\Omega }\left(\mathfrak{q}z\right)\right|+\varsigma {\mathfrak{q}}^2\left|z\right|{\displaystyle \frac{\left|1-\overline{\mathrm{\Omega }\left(\mathfrak{q}z\right)}\mathrm{\Omega }\left(z\right)\right|}{1-{\left|z\right|}^2\mathfrak{q}}}\right]·\hfill \\ \hfill {\displaystyle \frac{1+\left|B\right|\left|z\right|}{1-\left[\frac{(A-\mathcal{B})\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}+\left|B\right|\right]\left|z\right|}}\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|+\left|\mathrm{\Omega }\left(\mathfrak{q}z\right)\right|\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|,z\in {\mathcal{U}}^{\ast },\end{array}$$

or

$$\begin{array}{cc}\hfill \left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)\right|\le & \left\{\left[(1+\varsigma \mathfrak{q})\left|\mathrm{\Omega }\left(z\right)-\mathrm{\Omega }\left(\mathfrak{q}z\right)\right|+\varsigma {\mathfrak{q}}^2\left|z\right|{\displaystyle \frac{\left|1-\overline{\mathrm{\Omega }\left(\mathfrak{q}z\right)}\mathrm{\Omega }\left(z\right)\right|}{1-{\left|z\right|}^2\mathfrak{q}}}\right]·\right.\hfill \\ \hfill & \left.{\displaystyle \frac{1+\left|B\right|\left|z\right|}{1-\left[\frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}+\left|B\right|\right]\left|z\right|}}+\left|\mathrm{\Omega }\left(\mathfrak{q}z\right)\right|\right\}\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|,z\in {\mathcal{U}}^{\ast }.\hfill \end{array}$$

(26)

The inequality (26) could be written in the form

$$\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)\right|\le {\displaystyle \frac{\psi \left(\mathrm{\Omega }\right(z),\mathrm{\Omega }(\mathfrak{q}z\left)\right)}{\left(1-{\left|z\right|}^2\mathfrak{q}\right)\left[1-\left(\frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}+\left|B\right|\right)\left|z\right|\right]}}\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|,z\in {\mathcal{U}}^{\ast },$$

where

$$\begin{array}{c}\psi \left(\mathrm{\Omega }\left(z\right),\mathrm{\Omega }\left(\mathfrak{q}z\right)\right):=\left|\mathrm{\Omega }\left(\mathfrak{q}z\right)\right|\left(1-{\left|z\right|}^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\right)\left|z\right|\right]+\hfill \\ \hfill \left[(1+\varsigma \mathfrak{q})\left|\mathrm{\Omega }\left(z\right)-\mathrm{\Omega }\left(\mathfrak{q}z\right)\right|\left(1-{\left|z\right|}^2\mathfrak{q}\right)+\varsigma {\mathfrak{q}}^2\left|z\right|\left|1-\overline{\mathrm{\Omega }\left(\mathfrak{q}z\right)}\mathrm{\Omega }\left(z\right)\right|\right](1+|B\left|\right|z\left|\right),z\in \mathcal{U}.\end{array}$$

If we denote for $0<r<1$ the values

$$\begin{array}{cc}\hfill \mu & :=\mu (r,\mathfrak{q})=\underset{\left|z\right|\le r}{sup}\left|\mathrm{\Omega }\left(\mathfrak{q}z\right)\right)|=\underset{\left|\zeta \right|=\mathfrak{q}r}{max}\left|\mathrm{\Omega }\left(\zeta \right)\right|\in [0,1],0<\mathfrak{q}<1,\hfill \\ \hfill \tau & :=\tau \left(r\right)=\underset{\left|z\right|\le r}{sup}\left|\mathrm{\Omega }\left(z\right)\right|=\underset{\left|\zeta \right|=r}{max}\left|\mathrm{\Omega }\left(\zeta \right)\right|\in [0,1],\hfill \end{array}$$

then

$$\begin{array}{c}\psi \left(\mathrm{\Omega }\left(z\right),\mathrm{\Omega }\left(\mathfrak{q}z\right)\right)\le \psi (\mu ,\tau ;z):=\mu \left(1-{\left|z\right|}^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\right)\left|z\right|\right]+\hfill \\ \hfill \left[(1+\varsigma \mathfrak{q})(\tau +\mu )\left(1-{\left|z\right|}^2\mathfrak{q}\right)+\varsigma {\mathfrak{q}}^2\left|z\right|(1+\mu \tau )\right](1+|B\left|\right|z\left|\right),\left|z\right|\le r<1,\end{array}$$

and consequently

$$\begin{array}{cc}\hfill {\displaystyle \frac{\psi \left(\mathrm{\Omega }\right(z),\mathrm{\Omega }(\mathfrak{q}z\left)\right)}{\left(1-{\left|z\right|}^2\mathfrak{q}\right)\left[1-\left(\frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}+\left|B\right|\right)\left|z\right|\right]}}\le & {\displaystyle \frac{\psi (\mu ,\tau ;z)}{\left(1-{\left|z\right|}^2\mathfrak{q}\right)\left[1-\left(\frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}+\left|B\right|\right)\left|z\right|\right]}}\hfill \\ \hfill & =:\mathrm{\Psi }(\mu ,\tau ,z),\left|z\right|\le r<1.\hfill \end{array}$$

Now, we will determine sufficient conditions on the parameters, such that $\mathrm{\Psi }(\mu ,\tau ,z)\le 1$ whenever $\left|z\right|\le r<1$, for some $r\in (0,1)$. Noting that under our assumptions the denominator of $\mathrm{\Psi }(\mu ,\tau ,z)$ is positive, the inequality $\mathrm{\Psi }(\mu ,\tau ,z)\le 1$, $\left|z\right|\le r<1$, is equivalent to

$$\begin{array}{cc}\hfill & \chi (\mu ,\tau ,z):=(1-\mu )\left(1-{\left|z\right|}^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\right)\left|z\right|\right]\hfill \\ \hfill & -\left[(1+\varsigma \mathfrak{q})(\tau +\mu )(1-{\left|z\right|}^2\mathfrak{q})+\varsigma {\mathfrak{q}}^2\left|z\right|(1+\mu \tau )\right](1+|B\left|\right|z\left|\right)\ge 0,\left|z\right|\le r<1.\hfill \end{array}$$

(27)

The next problem is to find the value $r\in (0,1)$, such that

$$m\left(r\right):=inf\left\{\chi (\mu ,\tau ,z):\left|z\right|\le r<1\right\}\ge 0.$$

(i)

For the first term of $\chi (\mu ,\tau ,z)$, given by (27), we easily obtain the following:

$$\begin{array}{c}\hfill inf\left\{(1-\mu )\left(1-{\left|z\right|}^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\right)\left|z\right|\right]:\left|z\right|\le r<1\right\}\\ \hfill =(1-\mu )\left(1-r^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{(\mathcal{A}-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\right)r\right];\end{array}$$

(28)

• hence, the infimum of this term is attained for $\left|z\right|=r$.

(ii)

For the second term of $\chi (\mu ,\tau ,z)$, we have

$$\begin{array}{c}inf\left\{-\left[(1+\varsigma \mathfrak{q})(\tau +\mu )\left(1-{\left|z\right|}^2\mathfrak{q}\right)+\varsigma {\mathfrak{q}}^2\left|z\right|(1+\mu \tau )\right](1+|B\left|\right|z\left|\right):\left|z\right|\le r<1\right\}\hfill \\ \hfill =-sup\left\{\left[(1+\varsigma \mathfrak{q})(\tau +\mu )\left(1-{\left|z\right|}^2\mathfrak{q}\right)+\varsigma {\mathfrak{q}}^2\left|z\right|(1+\mu \tau )\right](1+|B\left|\right|z\left|\right):\left|z\right|\le r<1\right\};\end{array}$$

• hence, it is not obvious where the above supremum is attained on $\left|z\right|\le r<1$.

That is because the supremum for the first term from the bracket of the above sum is attained at $z=0$, while for the second term, it is attained from the bracket, and for the final factor, it is attained at $\left|z\right|=r$.

Therefore, we could only to find a lower bound for this infimum, that is

$$\begin{array}{cc}\hfill & inf\left\{-\left[(1+\varsigma \mathfrak{q})(\tau +\mu )\left(1-{\left|z\right|}^2\mathfrak{q}\right)+\varsigma {\mathfrak{q}}^2\left|z\right|(1+\mu \tau )\right](1+|B\left|\right|z\left|\right):\left|z\right|\le r<1\right\}\hfill \\ \hfill & =-sup\left\{\left[(1+\varsigma \mathfrak{q})(\tau +\mu )\left(1-{\left|z\right|}^2\mathfrak{q}\right)+\varsigma {\mathfrak{q}}^2\left|z\right|(1+\mu \tau )\right](1+|B\left|\right|z\left|\right):\left|z\right|\le r<1\right\}\hfill \\ \hfill & >-\left[(1+\varsigma \mathfrak{q})(\tau +\mu )+\varsigma {\mathfrak{q}}^{2}r(1+\mu \tau )\right](1+|B\left|r\right).\hfill \end{array}$$

(29)

Consequently, using (28) and (29), we achieve the following:

$$\begin{array}{cc}\hfill \chi (\mu ,\tau ,z)>& (1-\mu )\left(1-r^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{(A-B)\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+\left|B\right|\right)r\right]\hfill \\ \hfill & -[(1+\varsigma \mathfrak{q})(\tau +\mu )+\varsigma {\mathfrak{q}}^{2}r(1+\mu \tau )](1+|B\left|r\right)=\phi \left(r\right),\left|z\right|\le r<1,\hfill \end{array}$$

(30)

Thus, a sufficient condition for having the inequality (27) is that the right-hand side of (30) should be non-negative; that is, the function $\phi$ given by (12) satisfies $\phi \left(r\right)\ge 0$ for some $r_0$ that will be determined below.

If we assume that $\phi \left(0\right)>0$ is equivalent to the inequality (11), it follows that $\phi \left(r\right)\ge 0$ if $0\le r\le r_0$ where

$$r_0:=sup\left\{\rho \in (0,1):\phi \left(r\right)\ge 0,r\in (0,\rho )\right\},$$

and the proof of our result is complete. □

3. Consequences and Special Cases

In this section, we will give some particular forms and examples obtained from Theorem 1 for different choices of the parameters. Thus, taking $A=1$ and $B=-1$ in Theorem 1, we obtain the next result.

Corollary 1.

Let $f\in \Sigma$ and $g\in {\wp }_{\varsigma ,\mathfrak{q}}^s(\alpha ,\eta ;1,-1)$. Suppose that ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\ll {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)$, and let Ω be the function that realizes this majorization. Assume that the inequality (11) holds, where μ and τ are defined in Theorem 1.

If

$$r_0:=sup\left\{\rho \in (0,1):\psi \left(r\right)\ge 0,r\in (0,\rho )\right\},$$

where

$$\psi \left(r\right):=(1-\mu )\left(1-r^2\mathfrak{q}\right)\left[1-\left({\displaystyle \frac{2\varsigma \mathfrak{q}\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+1\right)r\right]-\left[(1+\varsigma \mathfrak{q})(\tau +\mu )+\varsigma {\mathfrak{q}}^{2}r(1+\mu \tau )\right](1+r),$$

then

$$\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)\right|\le \left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|,0<\left|z\right|<r_0.$$

For $\alpha =0$, the Corollary 1 leads to the following example:

Example 1.

Let the functions $f\in \Sigma$ and $g\in {\wp }_{\varsigma ,\mathfrak{q}}^s\left(\eta \right)$, where ${\wp }_{\varsigma ,\mathfrak{q}}^s\left(\eta \right)$ is defined by (10). Suppose that ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\ll {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)$, and let Ω be the function that realizes this majorization. Assume that the inequality (11) holds, where μ and τ are defined in Theorem 1.

If

$$\widehat{r}:=sup\left\{\rho \in (0,1):\widehat{\psi }\left(r\right)\ge 0,r\in (0,\rho )\right\},$$

where

$$\widehat{\psi }\left(r\right):=(1-\mu )\left(1-r^2\mathfrak{q}\right)\left[1-\left(2\varsigma \mathfrak{q}\left|\eta \right|+1\right)r\right]-[(1+\varsigma \mathfrak{q})(\tau +\mu )+\varsigma {\mathfrak{q}}^{2}r(1+\mu \tau )](1+r),$$

then

$$\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)\right|\le \left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|,0<\left|z\right|<\widehat{r}.$$

The next special case is obtained from the Corollary 1 by taking $\mathfrak{q}\to 1^{-}$, $\varsigma =1$ and $s=0$; hence, we should assume that $g\in {\wp }_{1,1}^0(\alpha ,\eta ;1,-1)$. It follows immediately that the assumption becomes $f\left(z\right)\ll g\left(z\right)$, while the inequality of the conclusion will be

$$\left|{\mathfrak{D}}_{1,1}^{1}f\left(z\right)\right|\le \left|{\mathfrak{D}}_{1,1}^{1}g\left(z\right)\right|,$$

that is equivalent, according to the definition of this operator, to

$$\left|2f\left(z\right)+z{f}^{\prime }\left(z\right)\right|\le \left|2g\left(z\right)+z{g}^{\prime }\left(z\right)\right|.$$

Corollary 2.

Let $f\in \Sigma$ and $g\in {\wp }_{1,1}^0(\alpha ,\eta ;1,-1)$. Suppose that $f\left(z\right)\ll g\left(z\right)$, and let Ω be the function that realizes this majorization. Assume that

$$1-\mu -2(\tau +\mu )>0,$$

(31)

where μ and τ are defined in Theorem 1.

If

$$\tilde{r}:=sup\left\{\rho \in (0,1):\tilde{\psi }\left(r\right)\ge 0,r\in (0,\rho )\right\},$$

where

$$\tilde{\psi }\left(r\right):=(1-\mu )\left(1-r^2\right)\left[1-\left({\displaystyle \frac{2\left|\eta \right|}{\left|1-\left|\alpha \right|\right|}}+1\right)r\right]-\left[2(\tau +\mu )+r(1+\mu \tau )\right](1+r),$$

then the inequality

$$\left|2f\left(z\right)+z{f}^{\prime }\left(z\right)\right|\le \left|2g\left(z\right)+z{g}^{\prime }\left(z\right)\right|$$

(32)

holds for $0<\left|z\right|<\tilde{r}$.

By setting $\eta =1-\delta$$(0\le \delta <1)$ in the Corollary 2, we obtain the following example, mentioning that ${\wp }_{1,1}^0(\alpha ,1-\delta ;1,-1)$ extends the class $\Sigma \left(\eta \right)$ obtained for $\alpha =0$:

Example 2.

Let the functions $f\in \Sigma$ and $g\in {\wp }_{1,1}^0(\alpha ,1-\delta ;1,-1)$, $0\le \delta <1$. Suppose that $f\left(z\right)\ll g\left(z\right)$, and let Ω be the function that realizes this majorization. Assume that the inequality (31) holds, where μ and τ are defined in Theorem 1.

If

$$r^{\ast }:=sup\left\{\rho \in (0,1):{\psi }^{\ast }\left(r\right)\ge 0,r\in (0,\rho )\right\},$$

where

$${\psi }^{\ast }\left(r\right):=(1-\mu )\left(1-r^2\right)[1-\left({\displaystyle \frac{2\left(1-\delta \right)}{\left|1-\left|\alpha \right|\right|}}+1\right)r]-\left[2(\tau +\mu )+r(1+\mu \tau )\right](1+r),$$

then the inequality (32) holds for $0<\left|z\right|<r^{\ast }$.

Taking $\eta =(1-\delta )cos\theta {\mathrm{e}}^{-i\theta }$$\left(\left|\theta \right|\le {\displaystyle \frac{\pi }{2}},0\le \delta <1\right)$ in Corollary 2, we obtain:

Example 3.

Let $f\in \Sigma$ and $g\in {\wp }_{1,1}^0\left(\alpha ,(1-\delta )cos\theta {\mathrm{e}}^{-i\theta };1,-1\right)$, $0\le \delta <1$. Suppose that $f\left(z\right)\ll g\left(z\right)$, and let Ω be the function that realizes this majorization. Assume that the inequality (31) holds, where μ and τ are defined in Theorem 1.

If

$$r^{\circ }:=sup\left\{\rho \in (0,1):{\psi }^{\circ }\left(r\right)\ge 0,r\in (0,\rho )\right\},$$

where

$${\psi }^{\circ }\left(r\right):=(1-\mu )\left(1-r^2\right)\left[1-\left({\displaystyle \frac{2\left(1-\delta \right)cos\theta }{\left|1-\left|\alpha \right|\right|}}+1\right)r\right]-\left[2(\tau +\mu )+r(1+\mu \tau )\right](1+r),$$

then the inequality (32) holds for $0<\left|z\right|<r^{\circ }$.

Remark 1.

The main theorem and all the results of the above corollaries and examples depend on the function Ω being an analytic in $\mathcal{U}$, such that $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in \mathcal{U}$, which realizes the majorizations ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\ll {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)$.

1.

Like a special case, more exactly as a circular transform that maps the open unit disk onto itself, we could consider the function:

$$\mathrm{\Omega }\left(z\right)={\mathrm{e}}^{i\mathrm{\Theta }}{\displaystyle \frac{z-a}{1-\overline{a}z}},\Theta \in \mathbb{R},\left|a\right|<1,$$

that represents the group of bilinear transforms that maps $\mathcal{U}$ onto itself.

2.

Another function Ω that realizes the majorizations ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\ll {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)$ could be

$$\mathrm{\Omega }\left(z\right)={\mathrm{e}}^{\lambda z}-1,0<\lambda \le ln2,$$

which is analytic in $\mathcal{U}$ with $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in \mathcal{U}$.

In both of these cases, more exactly in the first one for $a\in (-1,1)$, we could see that the values of μ and τ given in Theorem 1 are attained in the real axe and could be easily determined, which proves the existence of many examples and special cases where our results could be used.

4. Concluding Remarks

Finally, we conclude that new majorization results could be obtained for some subclasses of $\Sigma$ defined by using the generalized $\mathfrak{q}$-Sălăgean operator ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^s$, defined in [15] by using the main tool given by Lemma 2.1 of [25], that is, the Nehari’s inequality for the Jackson’s $\mathfrak{q}$-difference operator.

In the main result, we obtained simple sufficient conditions, such as the subordination such that for $f\in \Sigma$ and $g\in {\wp }_{\varsigma ,\mathfrak{q}}^s(\alpha ,\eta ;A,B)$, the majorization ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\ll {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)$ implies $\left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}f\left(z\right)\right|\le \left|{\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s+1}g\left(z\right)\right|$ in the disk $0<\left|z\right|<r_0$, where $r_0$ is determined with the aid of the function $\mathrm{\Omega }$ that realizes this majorization.

For a given function $\mathrm{\Omega }$ that realizes the majorization and for different choices of the parameters, we could determine the values of $\mu$ and $\tau$, defined in Theorem 1, like we showed in Section 3. Since another generalization of Nehari’s inequality given by Lemma 1 in [24] was recently obtained, it is interesting to find a result corresponding to our main one, which was obtained by using this lemma instead of Lemma 2.1 of [25]. Moreover, it remains an open question to determine which of the above-mentioned lemmas gives better results under some additional assumptions.

The main result could be used by choosing the functions $g\in {\wp }_{\varsigma ,\mathfrak{q}}^s(\alpha ,\eta ;A,B)$ and the function $\mathrm{\Omega }$ analytic in $\mathcal{U}$, such that $\left|\mathrm{\Omega }\right(z\left)\right|\le 1$, $z\in \mathcal{U}$, which realizes the majorizations ${\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}f\left(z\right)\ll {\mathfrak{D}}_{\varsigma ,\mathfrak{q}}^{s}g\left(z\right)$, while the above-mentioned radius $r_0$ could be determined such that the modules inequality of our result holds in this disk.

We believe that these results, where new generalized Nehari’s inequalities are used for the $\mathfrak{q}$-difference operator, could be helpful for further studies involving majorization problems for different new subclasses of meromorphic functions defined by using this operator.