Janowski-Type q-Classes Involving Higher-Order q-Derivatives and Fractional Integral Operators

Abstract

In this paper, we address the lack of general Janowski-type subclasses for analytic functions involving higher-order q-derivatives, unifying cases with both positive and negative coefficients. Using a combination of higher-order q-derivative techniques and Janowski subordination, we introduce two new q-analytic classes and derive sharp coefficient inequalities that fully characterize them. Our main theorems provide explicit coefficient bounds, distortion and neighborhood inclusion results, extending the classical Goodman–Ruscheweyh theory to the q-calculus setting. Applications are given to fractional q-integral operators, in particular to the q-Jung–Kim–Srivastava operator, and the results reduce to several known cases as $q\to 1^{-}$.

1. Introduction and Preliminaries

Let $\mathcal{H}\left(\mathbb{D}\right)$ denote the collection of complex-valued functions that are analytic on the open unit disk,

$$\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}.$$

We define ${\mathcal{A}}_n$ as the subset of $\mathcal{H}\left(\mathbb{D}\right)$ consisting of functions that satisfy the normalization condition

$${\mathcal{A}}_n=\{f\in \mathcal{H}\left(\mathbb{D}\right):f\left(z\right)=z^n+a_{n+1}{z}^{n+1}+\dots ,z\in \mathbb{D}\},$$

(1)

where $n\in \mathbb{N}=\left\{1,2,3\dots \right\}$. When $n=1$, this family coincides with the classical class of normalized analytic functions $\mathcal{A}={\mathcal{A}}_1$.

Definition 1

([1]). Given two functions f and g analytic in $\mathbb{D}$, the function f is subordinate to g in $\mathbb{D}$, denoted by

$$f\left(z\right)\prec g\left(z\right),z\in \mathbb{D},$$

if there exists a Schwarz function $w\left(z\right)$, analytic in $\mathbb{D}$, with $w\left(0\right)=0$ and $\left|w\right(z\left)\right|<1$, such that

$$f\left(z\right)=g\left(w\right(z\left)\right),z\in \mathbb{D}.$$

The study of analytic and multivalent function classes has a long and rich history, beginning with classical subclasses such as starlike and convex functions and evolving through the introduction of various operator-based frameworks. A key milestone was Janowski’s subordination approach, which provided a flexible way to define families of analytic functions via transformations associated with conic domains. Subsequent works by Aouf (see [2]), Hayami–Owa (see [3]), and others refined these ideas for multivalent functions, while Srivastava and coauthors (see [4]) developed generalized differential and integral operators that allowed a systematic treatment of broader subclasses. Over the years, the focus has increasingly shifted toward constructing function classes that are stable under differential or integral operators, admit sharp coefficient estimates, and display inclusion, distortion, and neighborhood properties analogous to their classical counterparts. Motivated by these developments, the present paper incorporates higher-order differential operators in the framework of q-calculus, aiming to extend and unify several existing families of analytic functions through Janowski-type subordination.

In recent years, q-calculus has emerged as a focal area of research owing to its broad applications across many branches of mathematics and key areas of physics. The q-derivative has proved to be an essential tool for analyzing numerous subclasses of analytic functions, while the q-difference operator plays a central role in the theory of basic hypergeometric series and is frequently used in quantum physics (see [5,6]). The notion of q-starlike functions dates to 1990 with foundational work by Ismail and coauthors (see [7]). A systematic framework for embedding q-calculus into Geometric Function Theory was later developed by Srivastava, notably via generalized basic hypergeometric functions (see [4]). Since then, many contributions have enriched the subject. For example, Arif and collaborators [8] introduced the Noor integral operator in the q-calculus setting via convolution methods, thereby generating new subclasses of analytic functions. Parallel investigations [9] examined generalized differential operators within q-calculus and their action on newly introduced families of analytic functions. Wongsai Jai and Sukantamala [10] carried out an in-depth study of subclasses defined by q-starlikeness, highlighting several analytic properties. Srivastava and coauthors (see [11]) broadened the scope by incorporating q-starlike functions associated with conic regions, and subsequent works explored interactions with Janowski functions (see [12,13]). More recently, Srivastava’s comprehensive survey (see [14]) synthesized these developments, discussed extended q-analogues in fractional contexts, emphasized their connections to q-differential operators, and outlined their role in geometric function theory. This survey now serves as a central reference for researchers active in the field.

Several recent works have highlighted the role of q-calculus in geometric function theory, particularly through the use of higher-order q-derivatives, fractional operators, and Janowski-type functions (see for example [15,16,17,18]).

We now recall certain notions and symbols from the q-calculus that will be essential in the subsequent analysis.

The idea underlying q-analogues, also referred to as q-extensions of classical results, is encapsulated in the identity,

$$\underset{q\to 1^{-}}{\lim }{\displaystyle \frac{1-q^j}{1-q}}=j,j\in \mathbb{N}.$$

(2)

The expression ${\displaystyle \frac{1-q^j}{1-q}}$ is commonly called the basic number and denoted by ${\left[j\right]}_q$.

The corresponding q-factorial is defined as

$${\left[j\right]}_q!=\left\{\begin{array}{c}{\left[j\right]}_q·{\left[j-1\right]}_q\cdots {\left[1\right]}_q,\mathrm{for}j=1,2,\dots ;\\ 1,\mathrm{for}j=0.\end{array}\right.$$

(3)

As $q\to 1^{-}$, q-analogues recover their classical forms; in particular, ${\left[j\right]}_q\to j$.

It is worth noting the basic-number relation

$${[j+n-r]}_q={\left[j\right]}_q+q^j{[n-r]}_q,j,n,r\in {\mathbb{N}}_0,$$

(4)

which follows directly from the definition ${\left[m\right]}_q={\displaystyle \frac{1-q^m}{1-q}},m\in {\mathbb{N}}_0$.

Jackson, in [19,20], introduced the q-derivative operator $D_q$ acting on functions $f\left(z\right)\in \mathcal{A}$ by

$$\left(D_{q}f\right)\left(z\right)={\displaystyle \frac{f\left(z\right)-f\left(qz\right)}{z\left(1-q\right)}},z\ne 0,0<q<1;\left(D_{q}f\right)=f^{\prime }\left(0\right).$$

(5)

In the limit $q\to 1^{-}$, $\left(D_{q}f\right)\left(z\right)$ reduces to the ordinary derivative $f^{\prime }\left(z\right)$.

For the monomial $f\left(z\right)=z^j$, one has

$$D_q\left(z^j\right)={\displaystyle \frac{1-q^j}{1-q}}·z^{j-1}={\left[j\right]}_q{z}^{j-1},$$

(6)

and consequently, as $q\to 1^{-}$,

$$D_q\left(z^j\right)\to j{z}^{j-1}=f^{\prime }\left(z\right).$$

Using Equation (5), we obtain

$$\left(D_{q}f\right)\left(z\right)=1+\sum _{j=2}^{\infty }{\displaystyle \frac{1-q^j}{1-q}}{a}_j{z}^{j-1},z\ne 0.$$

The operator $D_q$ satisfies the following rules for f, g $\in \mathcal{A}$ and $a,b\in \mathbb{C}$:

$$D_q\left(\left(af\left(z\right)\right)\pm bg\left(z\right)\right)=a{D}_{q}f\left(z\right)\pm b{D}_{q}g\left(z\right),$$

$$D_q\left(f\left(z\right)g\left(z\right)\right)=g\left(z\right)D_{q}f\left(z\right)+f\left(qz\right)D_{q}g\left(z\right),$$

$$D_q\left({\displaystyle \frac{f\left(z\right)}{g\left(z\right)}}\right)={\displaystyle \frac{g\left(z\right)D_{q}f\left(z\right)-f\left(z\right)D_{q}g\left(z\right)}{g\left(z\right)g\left(qz\right)}},g\left(z\right)g\left(qz\right)\ne 0.$$

The q-integral of f over $[0,z]$ is given by (see [19])

$${\int }_0^{z}f\left(t\right)d_{q}t=z\left(1-q\right)\sum _{j=0}^{\infty }{q}^{j}f\left(q^{j}z\right),$$

provided the series converges.

In the limit $q\to 1^{-}$, this reduces to the usual integral

$$\underset{q\to 1}{\lim }{\int }_0^{z}f\left(t\right)d_{q}t={\int }_0^{z}f\left(t\right)dt.$$

Higher-order q-derivatives are defined recursively by

$$D_q^{\left(0\right)}f\left(z\right)=f\left(z\right),D_q^{\left(r\right)}f\left(z\right)=D_q\left(D_q^{(r-1)}f\left(z\right)\right).$$

Explicitly, the q-derivative of the function $f\left(z\right)$ of order r is given by

$$D_q^{\left(r\right)}f\left(z\right)={\displaystyle \frac{{\left[n\right]}_q!}{{[n-r]}_q!}}{z}^{n-r}+\sum _{j=1}^{\infty }{\displaystyle \frac{{[j+n]}_q!}{{[j+n-r]}_q!}}{a}_{j+n}{z}^{j+n-r},$$

(7)

for $0\le r\le n-1$ and $r\in {\mathbb{N}}_0=\left\{0,1,2,\dots \right\}$.

The definition of the q-Gamma function ${\Gamma }_q\left(x\right)$ is given by

$${\Gamma }_q\left(x\right)={\left(1-q\right)}^{1-x}\prod _{j=0}^{\infty }{\displaystyle \frac{1-q^{j+1}}{1-q^{j+x}}},x>0,$$

which satisfies the following fundamental properties:

$${\Gamma }_q\left(x+1\right)={\left[x\right]}_q{\Gamma }_q\left(x\right),$$

and

$${\Gamma }_q\left(j+1\right)={\left[j\right]}_q!,$$

where $j\in \mathbb{N}$ and the q-factorial ${\left[j\right]}_q!$ is defined in Equation (3).

We next revisit several key definitions related to fractional q-calculus operators for complex-valued functions $f\left(z\right)$.

Definition 2

([21]). The fractional q-integral of order $\alpha >0$ for the function f is defined by

$$\left({\mathcal{I}}_{q,z}^{\alpha }f\right)\left(z\right)=D_{q,z}^{-\alpha }f\left(z\right)={\displaystyle \frac{1}{{\Gamma }_q\left(\alpha \right)}}{\int }_0^z{(z-qt)}_{\alpha -1}f\left(t\right)d_{q}t,$$

with $f\left(z\right)$ analytic in a simply connected region of the complex plane that contains the origin. Here ${(z-qt)}_{\alpha -1}$ denotes the standard fractional q-shifted kernel used in fractional q-calculus, taken on the branch for which $\log \left(z-qt\right)$ is real whenever $z-qt>0$. Throughout, we assume the principal branch and that the resulting integrals/series are absolutely convergent for $z\in \mathbb{D}$.

Definition 3

([21]). The fractional q-derivative of order α for the function f is defined by

$$\left({\mathcal{D}}_{q,z}^{\alpha }f\right)\left(z\right)={\mathcal{D}}_{q,z}{\mathcal{I}}_{q,z}^{1-\alpha }f\left(z\right)={\displaystyle \frac{1}{{\Gamma }_q(1-\alpha )}}{D}_{q,z}{\int }_0^z{(z-qt)}_{-\alpha }f\left(t\right)d_{q}t,0\le \alpha <1.$$

The theory of univalent and multivalent functions has seen major developments through the introduction of various subclasses defined by analytic, geometric, or operator-based properties. A fruitful approach involves the use of differential operators and subordination principles to describe starlike or convex function classes. When extended to the framework of q-calculus, these ideas give rise to discrete analogues such as q-starlike or q-convex functions.

For $-1\le N<M\le 1$ and $0\le \beta <n$, Aouf [22] defined the class $P(M,N;n;\beta )$, which represents a subclass of the family ${\mathcal{A}}_n$ and comprises functions represented by

$$p\left(z\right)=n+n_{1}z+n_2{z}^2+n_3{z}^3+\cdots ,$$

with the property that

$$p\left(z\right)\prec {\displaystyle \frac{n+[n·N+(M-N\left)\right(n-\beta \left)\right]·z}{1+N·z}}.$$

Aouf (see [22]) introduced the subclass ${\mathcal{S}}^{\ast }(M,N;n;\beta )$ of multivalent quasi-starlike functions defined by a subordination involving the derivative $z{f}^{\prime }\left(z\right)/f\left(z\right)$ and the fractional transformation of Janowski-type. Later, authors like Srivastava et al. (see [23,24,25]) developed further classes using higher-order q-derivatives and generalized subordination functions.

To make the presentation self-contained, we summarize below the main symbols, operators, and parameters used throughout the paper (Table 1).

Table 1. Notation and Parameters.

Symbol	Meaning/Definition	Range/Domain
${\left[j\right]}_q$	Basic q-number, ${\left[j\right]}_q={\displaystyle \frac{1-q^j}{1-q}}$	$j\in \mathbb{N},0<q<1$
${\left[j\right]}_q!$	q-factorial, ${\left[j\right]}_q!={\left[1\right]}_q{\left[2\right]}_q\cdots {\left[j\right]}_q$	$j\in \mathbb{N}$
$D_{q}f\left(x\right)$	Jackson q-derivative, $D_{q}f\left(z\right)={\displaystyle \frac{f\left(z\right)-f\left(qz\right)}{(1-q)z}}$	$q\in (0,1)$
	for $z\ne 0;D_{q}f\left(0\right)=f^{\prime }\left(0\right)$
$D_q^{\left(r\right)}f\left(x\right)$	Higher-order q-derivative	$r\in {\mathbb{N}}_0$
	$D_q^{\left(0\right)}f=f,D_q^{\left(r\right)}f=D_q\left(D_q^{(r-1)}f\right)$
${\displaystyle {\int }_0^{z}f\left(t\right)d_{q}t}$	q-integral: ${\displaystyle z(1-q)\sum _{j=0}^{\infty }{q}^{j}f\left(q^{j}z\right)}$	$q\in (0,1)$
$\mathcal{H}\left(\mathbb{D}\right)$	All complex-valued functions analytic in D	—
${\mathcal{A}}_n$	$\{f\in \mathcal{H}\left(\mathbb{D}\right):f\left(z\right)=z^n+a_{n+1}{z}^{n+1}+\dots \}$	$n\in \mathbb{N}$
${\mathcal{T}}_n$	Negative coefficient subclass of ${\mathcal{A}}_n$	$j,n\in \mathbb{N}$
$M,N$	Real parameters	$-1\le N<M\le 1$
$\beta$	Real parameter	$0\le \beta <{\left[p\right]}_q$
$\xi$	Real parameter	$\xi \in [0,1]$
n	Valency (normalization order in ${\mathcal{A}}_n$)	$n\in \mathbb{N}$
r	Order of q-derivative	$r\in {\mathbb{N}}_0$, $r\le n-1$
q	Deformation parameter	$q\in (0,1)$

Note: All undefined symbols have their standard meaning in q-calculus.

In this work, we address the absence of a unified framework for Janowski-type subclasses of analytic functions involving higher-order q-derivatives, encompassing both positive and negative coefficient cases. By employing higher-order q-derivative operators and Janowski subordination, we define two new normalized q-classes within the standard family ${\mathcal{A}}_n$, which encompass subclasses with both positive and negative coefficients, thereby extending and unifying several classical families in geometric function theory. Sharp coefficient inequalities are obtained, together with structural properties such as invariance under arithmetic and weighted means, parameter monotonicity, and distortion bounds. We also establish Goodman–Ruscheweyh–type neighborhood inclusions and apply the results to fractional q-calculus, particularly to the q-Jung–Kim–Srivastava operator, with all main theorems reducing to known results as $q\to 1^{-}$.

Let us define the generalized Janowski-type function as follows:

$${\varphi }_{M,N;\beta }^{{\left[p\right]}_q}\left(z\right)={\displaystyle \frac{{\left[p\right]}_q+(N{\left[p\right]}_q+\left(M-N\right)\left({\left[p\right]}_q-\beta \right))z}{1+Nz}},$$

(8)

where $-1\le N<M\le 1$ and $0\le \beta <{\left[p\right]}_q,p\in \mathbb{N}$, $p\le n$.

Definition 4.

Let $n\in \mathbb{N}$, $r\in {\mathbb{N}}_0=\left\{0,1,2,\dots \right\}$ with $0\le r\le n-1$, $q\in (0,1)$, $\xi \in [0,1]$, $0\le \beta <{[n-r]}_q$ and $M,N\in \mathbb{R}$ with $-1\le N<M\le 1$. A function $f\in {\mathcal{A}}_n$ is said to belong to the class ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ if

$${\displaystyle \frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}}\prec {\varphi }_{M,N;\beta }^{{[n-r]}_q}\left(z\right),z\in \mathbb{D},$$

(9)

that is, there exists a Schwarz function $w\left(z\right)$, analytic in $\mathbb{D}$ with $w\left(0\right)=0$ and $\left|w\right(z\left)\right|<1$, such that

$${\displaystyle \frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}}={\varphi }_{M,N;\beta }^{{[n-r]}_q}\left(w\left(z\right)\right),$$

with ${\varphi }_{M,N;\beta }^{{[n-r]}_q}$ given by Equation (8).

Remark 1.

Because the series representations of $D_q^{\left(r\right)}f$ and $D_q^{(r+1)}f$ are absolutely and uniformly convergent on compact subsets of $\mathbb{D}$, all termwise operations are justified.

Remark 2.

Throughout, we assume, without repeated mention, that the following conditions hold: $z\in \mathbb{D}$, $n\in \mathbb{N}$, $r\in {\mathbb{N}}_0=\left\{0,1,2,\dots \right\}$ with $0\le r\le n-1$, $q\in (0,1)$, $\xi \in [0,1]$, $0\le \beta <{[n-r]}_q$, and $M,N\in \mathbb{R}$ with $-1\le N<M\le 1$.

Remark 3.

The class ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to several known subclasses of multivalent functions for particular choices of parameters:

• If $q\to 1^{-}$, $\xi =1$, and $r=0$, then ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to the class ${\mathcal{S}}^{\ast }(M,N;n;\beta )$ introduced by Aouf (see [22]) where

  $${\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\prec {\displaystyle \frac{n+[nN+(M-N\left)\right(n-\beta \left)\right]z}{1+Nz}}.$$
• If $N=-1$, $M=1$, $q\to 1^{-}$ and $\xi =1$, then ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ becomes the class of n-valent starlike functions of order β, ${\mathcal{S}}_n^{\ast }\left(\beta \right)$.
• If $r=0$, $\xi =1$, $\beta =0$ and $q\to 1^{-}$, then ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ specializes to the class ${\mathcal{S}}_n^{\ast }(M,N)$ investigated by Hayami and Owa [3].
• If $n=1$, $r=0$, $\xi =1$, $q\to 1^{-}$ then ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to the class ${\mathcal{S}}_n^{\ast }(M,N;\beta )$ proposed by Polatoglu et al. [26].
• If $n=1$, $r=0$, $\xi =1$, $q\to 1^{-}$ and $\beta =0$, then ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to the subclass ${\mathcal{S}}^{\ast }(M,N)$ considered by Janowski [27] and was further studied by Goel and Mehrok [28].
• If $N=-1$, $M=1$, $q\to 1^{-},n=1$, $r=0$, $\xi =1$, then ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to ${\mathcal{S}}^{\ast }\left(\beta \right)$.
• If $N=-1$, $M=1$, $q\to 1^{-},n=1$, $r=0$, $\xi =1$ and $\beta =0$, then ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to ${\mathcal{S}}^{\ast }$.

We denote by ${\mathcal{T}}_n$ the subclass of ${\mathcal{A}}_n$ consisting of all functions having negative coefficients, namely

$$f\left(z\right)=z^n-\sum _{j=1}^{\infty }\left|a_{j+n}\right|z^{j+n}.$$

(10)

We furthermore introduce ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )={\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )\bigcap {\mathcal{T}}_n$.

Remark 4.

By assigning specific values to the parameter values, we recover several well-known subclasses of analytic functions in $\mathbb{D}$ with negative coefficients:

• If $q\to 1^{-}$, $\xi =1$ and $r=0$, then ${\mathcal{TM}}_n^{\left(0\right)}(M,N;1,\beta )$ becomes the class introduced and studied by Aouf (see [2]).
• If $N=-1$, $M=1$, $q\to 1^{-}$, $\xi =1$ and $r=0$, then ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to the class ${\mathcal{P}}_n^{\ast }\left(\beta \right)$ proposed by Sekine and Owa (see [29]).
• If $N=-\alpha$, $M=\alpha$, $q\to 1^{-}$, $\xi =1$, $r=0$, $\beta =1$, $n=1$, then ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ reduces to the class ${\mathcal{D}}^{\ast }\left(\alpha \right)$ investigated by Kim and Lee (see [30]).

For $\xi =0$, we denote the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;0,\beta )$ by ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$.

Lemma 1

([31]). Let $-1\le N_2\le N_1<M_1\le M_2\le 1$, then

$${\displaystyle \frac{1+M_{1}z}{1+N_{1}z}}\prec {\displaystyle \frac{1+M_{2}z}{1+N_{2}z}}.$$

2. Coefficient Characterizations and Fundamental Properties of the Class ${\mathcal{TM}}_{\mathit{n},\mathit{q}}^{\left(\mathit{r}\right)}(\mathit{M},\mathit{N};\mathit{\xi },\mathit{\beta })$

In this section, we derive a coefficient inequality that characterizes functions belonging to the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$. Building on this characterization, we establish sharp coefficient bounds and investigate inclusion properties of various subclasses. Furthermore, we examine the behavior of these classes under weighted and arithmetic mean operations, prove several distortion theorems, and derive related corollaries. These results provide a unified framework for analyzing geometric properties of q-multivalent functions with negative coefficients.

Theorem 1.

Suppose $f\left(z\right)=z^n-{\sum }_{j=1}^{\infty }\left|a_{j+n}\right|z^{j+n}$ is analytic in the unit disk $\mathbb{D}$. The function f belongs to the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ if and only if

$$\sum _{j=1}^{\infty }{\displaystyle \frac{{[n+j]}_q!}{{[j+n-r]}_q!}}\left\{\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|+\left|{\chi }_{j,q}\right|\right\}\left|a_{j+n}\right|\le C,$$

(11)

where

$$\begin{array}{ccc}\hfill {\chi }_{j,q}& =& \left(-\xi M+N{q}^j\right){[n-r]}_q+\xi \beta \left(M-N\right)+N{\left[j\right]}_q,\hfill \end{array}$$

(12)

and

$$\begin{array}{ccc}\hfill C& =& \left(M-N\right)\left({[n-r]}_q-\beta \right){\displaystyle \frac{{\left[n\right]}_q!}{{[n-r]}_q!}}.\hfill \end{array}$$

(13)

Proof. 

The subordination condition defining the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ can be written equivalently as

$$\left|{\displaystyle \frac{\frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}-{[n-r]}_q}{N{[n-r]}_q+\left(M-N\right)\left({[n-r]}_q-\beta \right)-N\frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}}}\right|<1.$$

(14)

Using Equation (4) and substituting the series expansion for $D_q^{\left(r\right)}f$ and $D_q^{(r+1)}f$, we obtain

$$\left|{\displaystyle \frac{{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}\left\{{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right\}{a}_{j+n}{z}^{j+n-r}}{C{z}^{n-r}+{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}{\chi }_{j,q}{a}_{j+n}{z}^{j+n-r}}}\right|<1.$$

Taking $\left|z\right|=1$ and applying the triangle inequality to numerator and denominator, we get

$${\displaystyle \frac{{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|\left|a_{j+n}\right|}{C-{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}\left|{\chi }_{j,q}\right|\left|a_{j+n}\right|}}<1,$$

which leads directly to Equation (11), where ${\chi }_{j,q}$ and C are given by Equations (12) and (13).

In the reverse direction, let $f\left(z\right)=z^n-{\sum }_{j=1}^{\infty }{a}_{j+n}{z}^{j+n}$ belong to class ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$. So, we have

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{\frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}-{[n-r]}_q}{N{[n-r]}_q+\left(M-N\right)\left({[n-r]}_q-\beta \right)-N\frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}}}\right|\hfill \\ & =& \left|{\displaystyle \frac{-{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}\left\{{\left[j\right]}_q+{[n-r]}_q\left(q^j-\xi \right)\right\}{a}_{j+n}{z}^{j+n-r}}{C{z}^{n-r}-{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}\left\{\left(\xi M-N{q}^j\right){[n-r]}_q-\xi \beta \left(M-N\right)-N{\left[j\right]}_q\right\}{a}_{j+n}{z}^{j+n-r}}}\right|<1.\hfill \end{array}$$

Fix $\theta \in \mathbb{R}$ and write $z=\rho e^{i\theta }$ with $0<\rho <1$. To pass from the complex inequality to the real-part bound, we choose z such that the transformed ratio is real, and use the inequality $\mathrm{Re}w\le \left|w\right|$. We have

$$\mathrm{Re}\left\{{\displaystyle \frac{{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|\left|a_{j+n}\right|z^{j+n-r}}{C{z}^{n-r}-{\sum }_{j=1}^{\infty }\frac{{[n+j]}_q!}{{[j+n-r]}_q!}\left|{\chi }_{j,q}\right|\left|a_{j+n}\right|z^{j+n-r}}}\right\}<1.$$

The limit as $\rho \to 1^{-}$ is then justified by dominated convergence, owing to the absolute and uniform convergence of the underlying series. So, we obtain

$$\sum _{j=1}^{\infty }{\displaystyle \frac{{[n+j]}_q!}{{[j+n-r]}_q!}}\left\{\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|+\left|{\chi }_{j,q}\right|\right\}\left|a_{j+n}\right|\le C,$$

and the proof is now complete. □

Remark 5.

Since the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ is a subset of ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ it suffices for the function $f\left(z\right)=z^n+{\sum }_{j=1}^{\infty }{a}_{j+n}{z}^{j+n}$ to satisfy Equation (11) of the previous theorem in order to be a member of ${\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$.

In order to illustrate the definition of the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$, we provide an explicit example for specific parameter choices.

Example 1.

Let $n=2$, $r=0$, $q=\frac{1}{2}$, $\xi =\frac{1}{2}$, $M=1$, $N=0$, $\beta =\frac{1}{2}$. We have

$${\left[1\right]}_q=1,{\left[2\right]}_q=\frac{3}{2},{\left[3\right]}_q=\frac{7}{4},{\left[2\right]}_q!=\frac{3}{2}.$$

Hence $C=1·\left(\frac{3}{2}-\frac{1}{2}\right)·1=1$, where C is given by Equation (13). From Theorem 1, the extremal function corresponding to $j=1$ is

$$f\left(z\right)=z^2-{\displaystyle \frac{C}{K}}{z}^3,$$

where

$$\begin{array}{ccc}\hfill K& =& {\displaystyle \frac{{\left[3\right]}_q!}{{\left[3\right]}_q!}}\left({|1+\left[2\right]}_q{(q-\xi )|+|(-\xi M+Nq)\left[2\right]}_q+\xi \beta (M-N)+N|\right)\hfill \\ & =& \frac{7}{4}\left(|1+\frac{3}{2}\left(0\right)|+|-\frac{1}{2}+\frac{1}{4}|\right)=\frac{35}{16}.\hfill \end{array}$$

Therefore,

$$f\left(z\right)=z^2-{\displaystyle \frac{16}{35}}{z}^3$$

belongs to the class ${\mathcal{TM}}_{2,1/2}^{\left(0\right)}(1,0;\frac{1}{2},\frac{1}{2})$ and satisfies Equation (11) exactly.

Corollary 1.

Let $f\left(z\right)$ be given by Equation (10). If $f\left(z\right)$$\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$, then for every $j\ge 1$ the coefficient bound

$$\left|a_{j+n}\right|\le {\displaystyle \frac{\left(M-N\right)\left({[n-r]}_q-\beta \right){\left[n\right]}_q!{[j+n-r]}_q!}{{[n+j]}_q!{[n-r]}_q!\left\{\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|+\left|{\chi }_{j,q}\right|\right\}}},$$

holds, where ${\chi }_{j,q}=\left(-\xi M+N{q}^j\right){[n-r]}_q+\xi \beta \left(M-N\right)+N{\left[j\right]}_q$. This estimate is sharp in each coordinate: for every fixed j, equality is attained by the extremal function

$$f\left(z\right)=z^n-\sum _{j=1}^{\infty }{\displaystyle \frac{\left(M-N\right)\left({[n-r]}_q-\beta \right){\left[n\right]}_q!{[j+n-r]}_q!}{{[n+j]}_q!{[n-r]}_q!\left\{\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|+\left|{\chi }_{j,q}\right|\right\}}}{z}^{j+n}.$$

For $q\to 1^{-}$, $\xi =1$ and $r=0$ in Theorem 1, we obtain the following result previously obtained by Aouf (see [2])

Corollary 2.

The function $f\left(z\right)=z^n-{\sum }_{j=1}^{\infty }\left|a_{j+n}\right|z^{j+n}$ belongs to the class ${\mathcal{TM}}_n^{\left(0\right)}(M,N;1,\beta )$ if and only if

$$\sum _{j=1}^{\infty }\left(j+\left|\left(-M+N\right)n+\beta \left(M-N\right)+Nj\right|\right)\left|a_{j+n}\right|\le \left(M-N\right)\left(n-\beta \right).$$

For $\xi =0$ in Theorem 1, we obtain the following result:

Corollary 3.

Let $f\left(z\right)=z^n-{\sum }_{j=1}^{\infty }\left|a_{j+n}\right|z^{j+n}$ analytic in the unit disk $\mathbb{D}$. The function f belongs to the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$ if and only if

$$\sum _{j=1}^{\infty }{\displaystyle \frac{{[n+j]}_q!}{{[j+n-r-1]}_q!}}\left|a_{j+n}\right|\le {\displaystyle \frac{C}{1+N}},$$

(15)

where C is given by Equation (13) and $-1<N<M\le 1.$

Theorem 2.

Let the function $f\left(z\right)=z^n-{\sum }_{j=1}^{\infty }\left|a_{j+n}\right|z^{j+n}$ belong to the class ${\mathcal{TM}}_n^{\left(r\right)}(M,N;\beta )$, where $-1<N<M\le 1$. Then

$$\sum _{j=1}^{\infty }\left|a_{j+n}\right|\le {\displaystyle \frac{\left(M-N\right)\left({[n-r]}_q-\beta \right)}{\left(N+1\right){[n+1]}_q}}.$$

(16)

Proof. 

Applying Corollary 3, Equation (15) yields

$$\begin{array}{ccc}& & \left(N+1\right){\displaystyle \frac{{[n+1]}_q!}{{[n-r]}_q!}}\sum _{j=1}^{\infty }\left|a_{j+n}\right|\hfill \\ & \le & \sum _{j=1}^{\infty }{\displaystyle \frac{{[n+j]}_q!}{{[j+n-r-1]}_q!}}\left(N+1\right)\left|a_{j+n}\right|\hfill \\ & \le & \left(M-N\right)\left({[n-r]}_q-\beta \right){\displaystyle \frac{{\left[n\right]}_q!}{{[n-r]}_q!}},\hfill \end{array}$$

hence Equation (16) follows immediately. □

Utilizing Theorem 1, we can establish that the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ remains invariant under both weighted and arithmetic means.

Theorem 3.

Let

$$f_1\left(z\right)=z^n-\sum _{j=1}^{\infty }\left|a_{j+n}\right|z^{j+n},f_2\left(z\right)=z^n-\sum _{j=1}^{\infty }\left|b_{j+n}\right|z^{j+n},$$

be two functions in the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$. For any $k\in [-1,1]$, define

$$F_k\left(z\right)={\displaystyle \frac{1-k}{2}}{f}_1\left(z\right)+{\displaystyle \frac{1+k}{2}}{f}_2\left(z\right).$$

Then $F_k\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$.

Proof. 

For ease of notation, set:

$$W_j={\displaystyle \frac{{[n+j]}_q!}{{[j+n-r]}_q!}}\left\{|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q|+\left|{\chi }_{j,q}\right|\right\}\ge 0,$$

(17)

where ${\chi }_{j,q}$ is given by Equation (12) and C is given by Equation (13). By the coefficient characterization Theorem 1, the functions $f_1,f_2\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ if and only if

$$\sum _{j=1}^{\infty }{W}_j\left|a_{j+n}\right|\le C,\sum _{j=1}^{\infty }{W}_j\left|b_{j+n}\right|\le C.$$

(18)

In terms of its series representation, $F_k$ takes the form:

$$F_k\left(z\right)=z^n-\sum _{j=1}^{\infty }{c}_{j+n}{z}^{j+n},\mathrm{where}{c}_{j+n}={\displaystyle \frac{1-k}{2}}\left|a_{j+n}\right|+{\displaystyle \frac{1+k}{2}}\left|b_{j+n}\right|.$$

Since $k\in [-1,1]$, we have $\frac{1-k}{2}\ge 0$ and $\frac{1+k}{2}\ge 0$, hence $c_{j+n}\ge 0$ and the required sign structure of the coefficients is preserved.

Using linearity and Equation (18),

$$\begin{array}{cc}\hfill \sum _{j=1}^{\infty }{W}_j{c}_{j+n}& ={\displaystyle \frac{1-k}{2}}\sum _{j=1}^{\infty }{W}_j\left|a_{j+n}\right|+{\displaystyle \frac{1+k}{2}}\sum _{j=1}^{\infty }{W}_j\left|b_{j+n}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1-k}{2}}C+{\displaystyle \frac{1+k}{2}}C=C.\hfill \end{array}$$

Therefore the characterization Equation (11) holds for the coefficients of $F_k$, and thus $F_k\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$. This proves that the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ is closed under the weighted mean. □

Theorem 4.

Let

$$f_l\left(z\right)=z^n-\sum _{j=1}^{\infty }\left|a_{j+n,l}\right|z^{j+n},l=1,2,\dots ,m,$$

be functions in the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$. Then their arithmetic mean

$$G\left(z\right)={\displaystyle \frac{1}{m}}\sum _{l=1}^m{f}_l\left(z\right),$$

also belongs to ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$.

Proof. 

The function G can be expressed in the same negative-coefficients series representation:

$$G\left(z\right)=z^n-\sum _{j=1}^{\infty }{c}_{j+n}{z}^{j+n},c_{j+n}≔{\displaystyle \frac{1}{m}}\sum _{l=1}^m\left|a_{j+n,l}\right|.$$

Clearly $c_{j+n}\ge 0$, so the coefficient pattern required by the subclass is preserved.

In view of Equations (17) and (13) and by Theorem 1, a function $f\left(z\right)=z^n-{\sum }_{j=1}^{\infty }\left|c_{j+n}\right|z^{j+n}$ belongs to ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ if and only if

$$\sum _{j=1}^{\infty }{W}_j\left|c_{j+n}\right|\le C.$$

(19)

Since each $f_l\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$, we have for every $l=1,\dots ,m$,

$$\sum _{j=1}^{\infty }{W}_j\left|a_{j+n,l}\right|\le C.$$

Using linearity and the definition of $c_{j+n}$,

$$\begin{array}{cc}\hfill \sum _{j=1}^{\infty }{W}_j{c}_{j+n}& =\sum _{j=1}^{\infty }{W}_j{\displaystyle \frac{1}{m}}\sum _{l=1}^m\left|a_{j+n,l}\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{m}}\sum _{l=1}^m\sum _{j=1}^{\infty }{W}_j\left|a_{j+n,l}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{m}}\sum _{l=1}^{m}C=C.\hfill \end{array}$$

Thus Equation (19) holds for G, and by Theorem 1 we conclude $G\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$. This proves closure under arithmetic means. □

The next theorem establishes a monotonicity property with respect to the parameters.

Theorem 5.

Let $0\le {\beta }_2\le {\beta }_1<{[n-r]}_q$ and $-1\le N<M_1\le M_2\le 1$. Then

$${\mathcal{TM}}_{n,q}^{\left(r\right)}(M_1,N;\xi ,{\beta }_1)\subset {\mathcal{TM}}_{n,q}^{\left(r\right)}(M_2,N;\xi ,{\beta }_2).$$

Proof. 

Let $f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M_1,N;\xi ,{\beta }_1)$. From Equation (9), we have

$${\displaystyle \frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}}\prec {\displaystyle \frac{{[n-r]}_q+(N{[n-r]}_q+\left(M_1-N\right)\left({[n-r]}_q-{\beta }_1\right))z}{1+Nz}}.$$

From $0\le {\beta }_2\le {\beta }_1<{[n-r]}_q$ and $-1\le N<M_1\le M_2\le 1$, we obtain

$$-1\le N+{\displaystyle \frac{\left(M_1-N\right)\left({[n-r]}_q-{\beta }_1\right)}{{[n-r]}_q}}\le N+{\displaystyle \frac{\left(M_2-N\right)\left({[n-r]}_q-{\beta }_2\right)}{{[n-r]}_q}}\le 1.$$

The middle inequality in the chain follows because the expression $N+{\displaystyle \frac{\left(M-N\right)\left({[n-r]}_q-\beta \right)}{{[n-r]}_q}}$ increases with M and decreases with $\beta$.

In view of Lemma 1, we get

$${\displaystyle \frac{z{D}_q^{(r+1)}f\left(z\right)}{(1-\xi )\frac{{\left[n\right]}_q!}{{[n-r]}_q!}{z}^{n-r}+\xi D_q^{\left(r\right)}f\left(z\right)}}\prec {\displaystyle \frac{{[n-r]}_q+(N{[n-r]}_q+\left(M_2-N\right)\left({[n-r]}_q-{\beta }_2\right))z}{1+Nz}}.$$

Hence, $f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M_2,N;\xi ,{\beta }_2)$. □

In the sequel, we establish the following distortion theorems.

Theorem 6.

If $f\in \mathcal{T}{\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$, then for $\left|z\right|=\rho$, $0<\rho <1$,

$${\rho }^n-{\displaystyle \frac{C}{W_1}}{\rho }^{n+1}\le \left|f\left(z\right)\right|\le {\rho }^n+{\displaystyle \frac{C}{W_1}}{\rho }^{n+1},$$

(20)

where C is given in Equation (13) and

$$W_1={\displaystyle \frac{{[n+1]}_q!}{{[n+1-r]}_q!}}\left\{|1+{[n-r]}_q(q-\xi )|+\left|\left(-\xi M+Nq\right){[n-r]}_q+\xi \beta \left(M-N\right)+N\right|\right\}.$$

(21)

The bounds in Equation (20) are sharp and are achieved by the function $f\left(z\right)$ defined by

$$f\left(z\right)=z^n-{\displaystyle \frac{C}{W_1}}{z}^{n+1},$$

(22)

at $z=\rho$ and $z=\rho exp\left(\left(2k+1\right)\pi i\right)$.

Proof. 

By Theorem 1, one readily gets

$$W_1\sum _{j=1}^{\infty }\left|a_{j+n}\right|\le \sum _{j=1}^{\infty }{W}_j\left|a_{j+n}\right|\le C,$$

and thus

$$\left|f\left(z\right)\right|\le {\rho }^n+\sum _{j=1}^{\infty }\left|a_{j+n}\right|{\rho }^{n+j}\le {\rho }^n+{\rho }^{n+1}\sum _{j=1}^{\infty }\left|a_{j+n}\right|\le {\rho }^n+{\displaystyle \frac{C}{W_1}}{\rho }^{n+1}.$$

A similar estimate yields the lower bound.

$$\left|f\left(z\right)\right|\ge {\rho }^n-\sum _{j=1}^{\infty }\left|a_{j+n}\right|{\rho }^{n+j}\ge {\rho }^n-{\rho }^{n+1}\sum _{j=1}^{\infty }\left|a_{j+n}\right|\ge {\rho }^n-{\displaystyle \frac{C}{W_1}}{\rho }^{n+1}.$$

The extremal function $f\left(z\right)=z^n-{\displaystyle \frac{C}{W_1}}{z}^{n+1}$ attains equality in (20) at $z=\rho$ (for the upper bound) and $z=\rho {\exp }^{(2k+1)\pi i}$ (for the lower bound), since in these cases $\left|f\left(\rho \right)\right|={\rho }^n-{\displaystyle \frac{C}{W_1}}{\rho }^{n+1}$. This confirms the sharpness of both bounds.

Hence, the proof is finished. □

For $q\to 1^{-}$, $\xi =1$ and $r=0$, we obtain the following result previously obtained by Aouf (see [2]).

Corollary 4.

If $f\in \mathcal{T}{\mathcal{M}}_n^{\left(0\right)}(M,N;1,\beta )$, then for $\left|z\right|=\rho$, $0<\rho <1$,

$${\rho }^n-{\displaystyle \frac{\left(M-N\right)\left(n-\beta \right)}{1+\left|N+\left(N-M\right)\left(n-\beta \right)\right|}}{\rho }^{n+1}\le \left|f\left(z\right)\right|\le {\rho }^n+{\displaystyle \frac{\left(M-N\right)\left(n-\beta \right)}{1+\left|N+\left(N-M\right)\left(n-\beta \right)\right|}}{\rho }^{n+1}.$$

For $\xi =0$, we get:

Corollary 5.

If $f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$ and $N\ge 0$, then for $\left|z\right|=\rho$, $0<\rho <1$,

$$\left|f\left(z\right)\right|\ge {\rho }^n-{\displaystyle \frac{C}{w_1}}{\rho }^{n+1},\left|f\left(z\right)\right|\le {\rho }^n+{\displaystyle \frac{C}{w_1}}{\rho }^{n+1},$$

(23)

where C is given in Equation (13) and

$$w_1={\displaystyle \frac{{\left[n+1\right]}_q!}{{\left[n+1-r\right]}_q!}}\left\{1+{\left[n-r\right]}_{q}q+Nq{\left[n-r\right]}_q+N\right\}.$$

(24)

The bounds in Equation (23) are sharp and are achieved by

$$f\left(z\right)=z^n-{\displaystyle \frac{C}{w_1}}{z}^{n+1}$$

(25)

at $z=\rho$ and $z=\rho e^{(2k+1)\pi i}$.

The proof of the next result is analogous to that of Theorem 6, and is therefore omitted.

Theorem 7.

If $f\in \mathcal{T}{\mathcal{M}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$, then for $\left|z\right|=\rho$, $0<\rho <1$,

$$\left|D_{q}f\left(z\right)\right|\ge {\left[n\right]}_q{\rho }^{n-1}-{[n+1]}_q{\displaystyle \frac{C}{W_1}}{\rho }^n,$$

(26)

and

$$\left|D_{q}f\left(z\right)\right|\le {\left[n\right]}_q{\rho }^{n-1}+{[n+1]}_q{\displaystyle \frac{C}{W_1}}{\rho }^n,$$

(27)

where C and $W_1$ are given by Equations (13) and (21). The bounds in Equations (26) and (27) are sharp and are achieved by the function specified by Equation (20).

For $q\to 1^{-}$, $\xi =1$ and $r=0$, we obtain the following result previously obtained by Aouf (see [2]).

Corollary 6.

If $f\in \mathcal{T}{\mathcal{M}}_n^{\left(0\right)}(M,N;1,\beta )$, then for $\left|z\right|=\rho$, $0<\rho <1$,

$$n{\rho }^{n-1}-{\displaystyle \frac{\left(n+1\right)\left(M-N\right)\left(n-\beta \right)}{1+\left|N+\left(N-M\right)\left(n-\beta \right)\right|}}{\rho }^n\le \left|f^{\prime }\left(z\right)\right|\le n{\rho }^{n-1}+{\displaystyle \frac{\left(n+1\right)\left(M-N\right)\left(n-\beta \right)}{1+\left|N+\left(N-M\right)\left(n-\beta \right)\right|}}{\rho }^n.$$

For $\xi =0$, we get:

Corollary 7.

If $f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$ and $N\ge 0$, then for $\left|z\right|=\rho$, $0<\rho <1$,

$$\begin{array}{ccc}\hfill \left|D_{q}f\left(z\right)\right|& \ge & {\left[n\right]}_q{\rho }^{n-1}-{\left[n+1\right]}_q{\displaystyle \frac{C}{w_1}}{\rho }^n,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \left|D_{q}f\left(z\right)\right|& \le & {\left[n\right]}_q{\rho }^{n-1}+{\left[n+1\right]}_q{\displaystyle \frac{C}{w_1}}{\rho }^n,\hfill \end{array}$$

(29)

where C is given in Equation (13) and $w_1$ is given in Equation (24). The bounds are sharp and are achieved by Equation (25).

Example 2.

Let’s take

$$n=1,r=0,q=0.7,\xi =\frac{1}{2},M=0.8,N=-0.2,\beta =0.3.$$

Then ${[n-r]}_q={\left[1\right]}_q=1$, ${\left[n\right]}_q!={\left[1\right]}_q!=1$, hence

$$C=(0.8-(-0.2\left)\right)(1-0.3)=0.7,$$

where C is given by (13). We also record the basic q-numbers (with $q=0.7$):

$${\left[1\right]}_q={\displaystyle \frac{1-0.7}{1-0.7}}=1,{\left[2\right]}_q={\displaystyle \frac{1-0.7^2}{1-0.7}}={\displaystyle \frac{1-0.49}{0.3}}=1.7.$$

We give now coefficient bounds from Theorem 1 (for $j=1,2$): For $n=1,r=0$, Theorem 1 gives

$$\left|a_{j+1}\right|\le {\displaystyle \frac{(M-N)\left({\left[1\right]}_q-\beta \right){\left[1\right]}_q!{[j+1]}_q!}{{[1+j]}_q!{\left[1\right]}_q!\left({|\left[1\right]}_q(\xi -q^j)-{\left[j\right]}_q|+|{\chi }_{j,q}|\right)}}={\displaystyle \frac{C}{|(\xi -q^j)-{\left[j\right]}_q|+|{\chi }_{j,q}|}},$$

with ${\chi }_{j,q}$ given by Equation (12). For $j=1$: ${\left[j\right]}_q=1,q^j=0.7.$

$$|(\xi -q^1)-{\left[1\right]}_q|=|(0.5-0.7)-1|=1.2,$$

$${\chi }_{1,q}=(-\frac{1}{2}·0.8+(-0.2)·0.7)·1+\frac{1}{2}·0.3·\left(1.0\right)+(-0.2)·1=-0.59,\left|{\chi }_{1,q}\right|=0.59.$$

Hence,

$$\left|a_2\right|\le {\displaystyle \frac{0.7}{1.2+0.59}}\approx 0.391.$$

For $j=2$: ${\left[j\right]}_q={\left[2\right]}_q=1.7,q^j=0.49.$

$$|(\xi -q^2)-{\left[2\right]}_q|=|(0.5-0.49)-1.7|=1.69,$$

$${\chi }_{2,q}=(-0.5·0.8+(-0.2)·0.49)·1+0.5·0.3·\left(1.0\right)+(-0.2)·1.7=-0.688,\left|{\chi }_{2,q}\right|=0.688.$$

Hence,

$$\left|a_3\right|\le {\displaystyle \frac{0.7}{1.69+0.688}}\approx 0.294.$$

Distortion estimate at $\rho =0.8$. For $n=1,r=0$,

$${\displaystyle \frac{{\left[2\right]}_q!}{{\left[2\right]}_q!}}{=1,|1+\left[1\right]}_q(q-\xi )|=|1+(0.7-0.5)|=1.2,$$

Thus,

$$W_1=1.2+0.59=1.79,{\displaystyle \frac{C}{W_1}}={\displaystyle \frac{0.7}{1.79}}\approx 0.391,$$

where $W_1$ and C are given by Equations (12) and (13).

Therefore, for $\left|z\right|=\rho =0.8$ and $n=1$,

$$\left|f\left(z\right)\right|\in \left[\rho -\frac{C}{W_1}{\rho }^2,\rho +\frac{C}{W_1}{\rho }^2\right]=\left[0.8-0.391·0.64,0.8+0.391·0.64\right]\approx [0.550,1.050].$$

Note. The distortion theorem requires $\xi \le q$ (satisfied since $0.5\le 0.7$) and $(-\xi M+Nq){[n-r]}_q+\xi \beta (M-N)+N\ge 0$. For this parameter choice, the last quantity equals $-0.59<0$; if one chooses $N\ge 0$, the hypothesis is strictly satisfied.

3. Neighborhood Inclusions

In this section, we investigate neighborhood inclusion properties for the function class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$. Following the classical approach initiated by Goodman and Ruscheweyh, we introduce a suitable notion of $\mu$-neighborhood for functions in ${\mathcal{T}}_n$ and establish sufficient conditions under which such neighborhoods are contained in the class ${\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$. These results provide a natural extension of earlier neighborhood theorems to the setting of q-calculus and multivalent analytic functions with negative coefficients.

In line with the classical approach of Goodman [32] and Ruscheweyh [33] (cf. [34]), we introduce the $\mu$-neighborhood of a function $f\left(z\right)\in {\mathcal{T}}_n$ by

$${\mathcal{N}}_{\mu }(f;h)=\left\{h\left(z\right)=z^n-\sum _{j=1}^{\infty }\left|d_{j+n}\right|z^{j+n}\in {\mathcal{T}}_n:\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-1\right]}_q!}}\left|\left|a_{j+n}\right|-\left|d_{j+n}\right|\right|\le \mu \right\},$$

(30)

with $\mu >0$.

Theorem 8.

Let $f\in {\mathcal{TM}}_{n,q}^{(r+1)}(M,N;\beta )$. Then ${\mathcal{N}}_{\mu }(f;h)\subset {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$, where

$$\mu ={\displaystyle \frac{(M-N)\left({\left[n-r\right]}_q-\beta \right)\left({\left[n-r\right]}_q-1\right){\left[n\right]}_q!}{(1+N){\left[n-r\right]}_q{\left[n-r\right]}_q!}},$$

(31)

with $-1<N<M\le 1$.

Proof. 

If $f\in {\mathcal{TM}}_{n,q}^{(r+1)}(M,N;\beta )$, Corollary 3 implies

$$\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-2\right]}_q!}}\left|a_{j+n}\right|\le {\displaystyle \frac{C}{1+N}},$$

where C is given by Equation (13).

Hence

$$\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!{\left[j+n-r-1\right]}_q}{{\left[j+n-r-1\right]}_q!}}\left|a_{j+n}\right|\le {\displaystyle \frac{C}{1+N}},$$

so that

$${\left[n-r\right]}_q\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-1\right]}_q!}}\left|a_{j+n}\right|\le \sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-2\right]}_q!}}\left|a_{j+n}\right|\le {\displaystyle \frac{C}{1+N}},$$

which is equivalent to

$$\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-1\right]}_q!}}\left|a_{j+n}\right|\le {\displaystyle \frac{C}{(1+N){\left[n-r\right]}_q}}.$$

(32)

Now take $h\left(z\right)=z^n-{\sum }_{j=1}^{\infty }\left|d_{j+n}\right|z^{j+n}\in {\mathcal{N}}_{\mu }(f;h)$, where $\mu$ is given in Equation (31). From Equation (30) we have

$$\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-1\right]}_q!}}\left|\left|a_{j+n}\right|-\left|d_{j+n}\right|\right|\le \mu ,\mu >0.$$

(33)

From Equations (32) and (33) it follows that

$$\begin{array}{cc}\hfill \sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-1\right]}_q!}}\left|d_{j+n}\right|& \le \sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-1\right]}_q!}}\left|a_{j+n}\right|\hfill \\ \hfill & +\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[n+j\right]}_q!}{{\left[j+n-r-1\right]}_q!}}\left|\left|a_{j+n}\right|-\left|d_{j+n}\right|\right|\hfill \\ \hfill & \le {\displaystyle \frac{C}{(1+N){\left[n-r\right]}_q}}+\mu ={\displaystyle \frac{(M-N)({\left[n-r\right]}_q-\beta ){\left[n\right]}_q!}{(1+N){\left[n-r\right]}_q{\left[n-r\right]}_q!}}\hfill \\ \hfill & +{\displaystyle \frac{(M-N)({\left[n-r\right]}_q-\beta )({\left[n-r\right]}_q-1){\left[n\right]}_q!}{(1+N){\left[n-r\right]}_q{\left[n-r\right]}_q!}}\hfill \\ \hfill & ={\displaystyle \frac{(M-N)({\left[n-r\right]}_q-\beta ){\left[n\right]}_q!}{(1+N){\left[n-r\right]}_q!}}={\displaystyle \frac{C}{1+N}}.\hfill \end{array}$$

Therefore, applying Corollary 3 again, we conclude that $h\left(z\right)\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$. □

For $\beta =0$, we obtain

Corollary 8.

Let $f\in {\mathcal{TM}}_{n,q}^{(r+1)}(M,N;0)$. Then ${\mathcal{N}}_{\mu }(f;h)\subset {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;0)$, where

$$\mu ={\displaystyle \frac{(M-N)-({\left[n-r\right]}_q-1){\left[n\right]}_q!}{(1+N){\left[n-r\right]}_q!}},$$

with $-1<N<M\le 1$.

4. Applications to Fractional q-Integral Operators

In this section, we investigate how the newly defined function classes behave under the action of fractional q-integral operators, with a particular focus on the q-Jung–Kim–Srivastava operator.

By employing the fractional q-integral operator for $f\in {\mathcal{T}}_n$, the q-Jung–Kim–Srivastava integral operator is defined as follows (see [35]):

$$\begin{array}{ccc}\hfill \left({\mathcal{JKS}}_{q,n}^{-\alpha }f\right)\left(z\right)& =& {\displaystyle \frac{{\Gamma }_q(c+\alpha +n)}{{\Gamma }_q(c+n)}}{z}^{1-c-\alpha }{D}_{q,z}^{-\alpha }\left(z^{c-1}f\left(z\right)\right)\hfill \\ & =& z^n-\sum _{j=1}^{\infty }{\displaystyle \frac{{\Gamma }_q(c+\alpha +n){\Gamma }_q(c+j+n)}{{\Gamma }_q(c+n){\Gamma }_q(c+j+n+\alpha )}}\left|a_{j+n}\right|z^{j+n},\hfill \end{array}$$

(34)

where $c>-n$, $\alpha >0$.

Remark 6.

For $n=1$ and $q\to 1^{-}$ in Equation (34), we obtain the Jung–Kim–Srivastava integral operator defined in [36] as

$$\left({\mathcal{JKS}}^{-\alpha }f\right)\left(z\right)=z-\sum _{j=1}^{\infty }{\displaystyle \frac{\Gamma \left(c+\alpha +1\right)\Gamma (c+j+1)}{\Gamma (c+1)\Gamma (c+j+1+\alpha )}}\left|a_{j+1}\right|z^{j+1}.$$

Theorem 9.

If $f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ and $c>-n$, $\alpha >0$, then ${\mathcal{JKS}}_{q,n}^{-\alpha }f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$.

Proof. 

Let $f\left(z\right)=z^n-{\sum }_{j=1}^{\infty }\left|a_{j+n}\right|z^{j+n}$ and

$$\left({\mathcal{JKS}}_{q,n}^{-\alpha }f\right)\left(z\right)=z^n-\sum _{j=1}^{\infty }{\displaystyle \frac{{\Gamma }_q(c+\alpha +n){\Gamma }_q(c+j+n)}{{\Gamma }_q(c+n){\Gamma }_q(c+j+n+\alpha )}}\left|a_{j+n}\right|z^{j+n}.$$

${\mathcal{JKS}}_{q,n}^{-\alpha }f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$ if

$$A=\sum _{j=1}^{\infty }{\displaystyle \frac{{[n+j]}_q!}{{[j+n-r]}_q!}}\left\{\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|+\left|{\chi }_{j,q}\right|\right\}{\displaystyle \frac{{\Gamma }_q(c+\alpha +n){\Gamma }_q(c+j+n)}{{\Gamma }_q(c+n){\Gamma }_q(c+j+n+\alpha )}}\left|a_{j+n}\right|\le C,$$

where ${\chi }_{j,q}$ and C are given by Equations (12) and (13).

Since $c>-n$ and $\alpha >0$, all arguments of ${\Gamma }_q$ are positives. Using that ${\Gamma }_q$ is log-convex on $(0,\infty )$ for $q\in (0,1)$, the function $x\mapsto {\displaystyle \frac{{\Gamma }_q(x+\alpha )}{{\Gamma }_q\left(x\right)}}$ is increasing; hence, for every $j\in \mathbb{N}$,

$${\displaystyle \frac{{\Gamma }_q(c+\alpha +n)}{{\Gamma }_q(c+n)}}\le {\displaystyle \frac{{\Gamma }_q(c+j+n+\alpha )}{{\Gamma }_q(c+j+n)}}$$

implies that

$${\displaystyle \frac{{\Gamma }_q(c+\alpha +n){\Gamma }_q(c+j+n)}{{\Gamma }_q(c+n){\Gamma }_q(c+j+n+\alpha )}}\le 1.$$

So, it is evident that

$$A\le \sum _{j=1}^{\infty }{\displaystyle \frac{{[n+j]}_q!}{{[j+n-r]}_q!}}\left\{\left|{[n-r]}_q\left(\xi -q^j\right)-{\left[j\right]}_q\right|+\left|{\chi }_{j,q}\right|\right\}\left|a_{j+n}\right|.$$

By Equation (11), we have

$$A\le C,$$

hence ${\mathcal{JKS}}_{q,n}^{-\alpha }f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\xi ,\beta )$. □

Corollary 9.

If $f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$ and $c>-n$, $0\le \alpha <1$, then ${\mathcal{JKS}}_{q,n}^{-\alpha }f\in {\mathcal{TM}}_{n,q}^{\left(r\right)}(M,N;\beta )$.

5. Concluding Remarks and Further Directions

In this work, we have introduced two new Janowski-type q-classes of normalized analytic functions defined through higher-order q-derivatives. We derived sharp coefficient inequalities, established inclusion and neighborhood results, investigated distortion properties, and analyzed the behavior of these classes under fractional q-integral operators, with a special focus on the q-Jung–Kim–Srivastava operator. These findings provide a unified framework that encompasses several classical subclasses as $q\to 1^{-}$ and underline the versatility of q-calculus in geometric function theory.

Future research may address extensions involving alternative fractional q-operators and convolution structures, as well as the study of extremal problems, Fekete–Szegő functionals, and subordination/superordination results. Applications to basic hypergeometric functions and quantum calculus represent additional promising directions.