Application of the Quasi-Hadamard Product to Subclasses of Analytic Functions Involving the q-Difference Operator

Abstract

In this study, the subclasses ${\mathcal{V}}_{q,\varrho }^{\ell }(c_{𝚥},\lambda ),$${\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )$ and ${\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )$ of analytic functions using the q-difference operator are defined and investigated. Thus, we obtained some results for the quasi-Hadamard product on these classes of analytic functions. Furthermore, connections between our results and some previously established results are outlined in this study.

1. Introduction

We consider the class $\mathcal{A}$ of analytic (holomorphic) functions with the following expansion:

$$f\left(z\right)=z+\sum _{𝚥=2}^{\infty }{a}_{𝚥}{z}^{𝚥}$$

(1)

in the open unit disc $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}.$ A function $f\left(z\right)$ is referred to as univalent in $\mathbb{D}$ if it is injective in $\mathbb{D}.$ We denote by $\mathcal{S}$ the subclass of $\mathcal{A}$ containing the univalent functions.

One of the most challenging subjects that have recently drawn the attention of mathematicians from the field of geometric theory of analytic univalent functions is the quantum (or q-) calculus theory. This has as its main focuses the generalization of integration and differentiation operators.

Even if its historical roots started with the work of Bernoulli and Euler, it has become increasingly more useful in modern mathematics, mechanics and physics, due to its extensive areas of application. Thus, it has a wide variety of applications in quantum mechanics and operator theory. Furthermore, such applications have been developed lately in the domain of geometric function theory.

Quantum calculus can be considered an approach to examining the calculus without using the limits. Jackson [1,2] discovered the most crucial step in q-calculus and established the useful formulas for q-integral and q-derivative operators. Its applications in fields such as regular fractional calculus, including combinatorics, orthogonal polynomials and fundamental hypergeometric functions, have generated many studies (see, for example, [3,4,5,6] and others).

In this study, we use q-calculus to define and investigate some classes of analytic functions. We first provide several definitions and notations of q-calculus.

Definition 1.

The q-derivative of a function $f\left(z\right)$ of the form (1) was defined by Jackson [1] as:

$$D_{q}f\left(z\right)=\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z}=1+\sum _{𝚥=2}^{\infty }{\left[𝚥\right]}_q{a}_{𝚥}{z}^{𝚥-1}(z\ne 0,0<q<1),$$

(2)

where

$${\left[𝚥\right]}_q=\left\{\begin{array}{ccc}\sum _{𝚤=0}^{𝚥-1}{q}^{𝚤}=1+q+q^2+\cdots +q^{𝚥-1}\hfill & & (𝚥\in \mathbb{N})\\ \hfill & & \\ 0\hfill & & (𝚥=0).\end{array}\right.$$

(3)

It can be noticed that:

$$\underset{q\to 1^{-}}{lim}{D}_{q}f\left(z\right)=f^{\prime }\left(z\right).$$

Jackson [1] also defined the q-integral of a function $f\left(z\right)$ of the form (1), as:

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

for convergent series and:

$$\begin{array}{ccc}\hfill \underset{q\to 1^{-}}{lim}{\int }_0^{z}f\left(t\right)d_{q}t& =& {\int }_0^{z}f\left(t\right)dt,\hfill \end{array}$$

where ${\int }_0^{z}f\left(t\right)dt$ is the ordinary integral.

Aouf et al. [7] defined the linear q-difference operator ${\widehat{\mathcal{D}}}_{q,\gamma }^m:\mathcal{A}\to \mathcal{A}$ as follows:

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{D}}}_{q,\gamma }^{0}f\left(z\right)& =& f\left(z\right)\hfill \\ \hfill {\widehat{\mathcal{D}}}_{q,\gamma }^{1}f\left(z\right)& =& \left(1-\gamma \right)f\left(z\right)+\gamma z{D}_{q}f\left(z\right)={\widehat{\mathcal{D}}}_{q,\gamma }f\left(z\right)\hfill \\ & =& z+\sum _{𝚥=2}^{\infty }\left(1+\gamma ({\left[𝚥\right]}_q-1)\right)a_{𝚥}{z}^{𝚥}\hfill \\ & ⋮& \hfill \\ \hfill {\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)& =& {\widehat{\mathcal{D}}}_{q,\gamma }\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m-1}f\left(z\right)\right),\hfill \end{array}$$

(4)

where $\gamma \ge 0,m\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ and $0<q<1.$ It follows from (4) that:

$${\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)=z+\sum _{𝚥=2}^{\infty }{\left(1+\gamma ({\left[𝚥\right]}_q-1)\right)}^m{a}_{𝚥}{z}^{𝚥}.$$

(5)

Remark 1.

It can be noticed that the operator ${\widehat{\mathcal{D}}}_{q,\gamma }^m$ generalizes a number of well-known, previously researched operators. We will now highlight some of the important special cases:

(i) 

For $\gamma =1$, we obtain ${\mathcal{S}}_q^m$, studied by Govindaraj and Sivasubramanian [8];

(ii) 

For $q\to 1^{-}$, we obtain $D_{\gamma }^m$, studied by Al-Oboudi [9];

(iii) 

For $q\to 1^{-}$ and $\gamma =1$, we obtain $D^m$, studied by Sălăgean [10].

Using the linear q-difference operator given by (5), Aouf et al. [7] introduced the subclass ${\mathcal{Y}}_q^m(\gamma ,\delta )$ of q-starlike functions of order $\delta$ in $\mathbb{D}$ and also the subclass ${\mathcal{K}}_q^m(\gamma ,\delta )$ of q-convex functions of order $\delta$ in $\mathbb{D}$ as follows:

$$\mathrm{Re}\left(\frac{z{D}_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)}{{\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)}\right)<\delta (0<q<1,\delta >1),$$

(6)

or, equivalently

$$\left|\frac{\frac{z{D}_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)}{{\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)}-1}{\frac{z{D}_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)}{{\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)}-(2\delta -1)}\right|<1$$

and

$$\mathrm{Re}\left(\frac{D_q\left(z{D}_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)\right)}{D_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)}\right)<\delta (0<q<1,\delta >1),$$

(7)

or, equivalently

$$\left|\frac{\frac{D_q\left(z{D}_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)\right)}{D_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)}-1}{\frac{D_q\left(z{D}_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)\right)}{D_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)}-(2\delta -1)}\right|<1,$$

respectively, where $\gamma \ge 0,m\in {\mathbb{N}}_0$ and $f\left(z\right)\in \mathcal{A}$. From (6) and (7), it follows that:

$${\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\in {\mathcal{K}}_q^m(\gamma ,\delta )⟺z{D}_q\left({\widehat{\mathcal{D}}}_{q,\gamma }^{m}f\left(z\right)\right)\in {\mathcal{Y}}_q^m(\gamma ,\delta ).$$

Remark 2.

It can be noticed that:

(i) 

If $\gamma =1$, then ${\mathcal{Y}}_q^m(1,\delta )={\mathcal{Y}}_q^m\left(\delta \right)$ and ${\mathcal{K}}_q^m(1,\delta )={\mathcal{K}}_q^m\left(\delta \right)$:

$$\mathrm{Re}\left(\frac{z{D}_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)}{{\widehat{\mathcal{D}}}_q^{m}f\left(z\right)}\right)<\delta (0<q<1,\delta >1,m\in {\mathbb{N}}_0),$$

or, equivalently

$$\left|\frac{\frac{z{D}_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)}{{\widehat{\mathcal{D}}}_q^{m}f\left(z\right)}-1}{\frac{z{D}_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)}{{\widehat{\mathcal{D}}}_q^{m}f\left(z\right)}-(2\delta -1)}\right|<1$$

and

$$\mathrm{Re}\left(\frac{D_q\left(z{D}_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)\right)}{D_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)}\right)<\delta (0<q<1,\delta >1,m\in {\mathbb{N}}_0),$$

or, equivalently

$$\left|\frac{\frac{D_q\left(z{D}_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)\right)}{D_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)}-1}{\frac{D_q\left(z{D}_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)\right)}{D_q\left({\widehat{\mathcal{D}}}_q^{m}f\left(z\right)\right)}-(2\delta -1)}\right|<1,$$

respectively.

(ii) 

For $m=0$, we obtain the subclasses, ${\mathcal{Y}}_q\left(\delta \right)$ of q-starlike functions of order δ in $\mathbb{D}$ and ${\mathcal{K}}_q\left(\delta \right)$ of q-convex functions of order δ in $\mathbb{D}$ as follows:

$$\mathrm{Re}\left(\frac{z{D}_{q}f\left(z\right)}{f\left(z\right)}\right)<\delta (0<q<1,\delta >1),$$

(8)

or, equivalently

$$\left|\frac{\frac{z{D}_{q}f\left(z\right)}{f\left(z\right)}-1}{\frac{z{D}_{q}f\left(z\right)}{f\left(z\right)}-(2\delta -1)}\right|<1$$

and

$$\mathrm{Re}\left(\frac{D_q\left(z{D}_{q}f\left(z\right)\right)}{D_{q}f\left(z\right)}\right)<\delta (0<q<1,\delta >1),$$

(9)

or, equivalently

$$\left|\frac{\frac{D_q\left(z{D}_{q}f\left(z\right)\right)}{D_{q}f\left(z\right)}-1}{\frac{D_q\left(z{D}_{q}f\left(z\right)\right)}{D_{q}f\left(z\right)}-(2\delta -1)}\right|<1,$$

respectively. From (8) and (9), it follows that:

$$f\left(z\right)\in {\mathcal{K}}_q\left(\delta \right)⟺z{D}_{q}f\left(z\right)\in {\mathcal{Y}}_q\left(\delta \right).$$

(iii) 

${lim}_{q\to 1^{-}}{\mathcal{Y}}_q^m(\gamma ,\delta )={\mathcal{Y}}^m(\gamma ,\delta )$ and ${lim}_{q\to 1^{-}}{\mathcal{K}}_q^m(\gamma ,\delta )={\mathcal{K}}^m(\gamma ,\delta )$

$$\mathrm{Re}\left(\frac{z{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{\prime }}{{\mathcal{D}}_{\gamma }^{m}f\left(z\right)}\right)<\delta (\delta >1,\gamma \ge 0,m\in {\mathbb{N}}_0),$$

or, equivalently

$$\left|\frac{\frac{z{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{\prime }}{{\mathcal{D}}_{\gamma }^{m}f\left(z\right)}-1}{\frac{z{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{\prime }}{{\mathcal{D}}_{\gamma }^{m}f\left(z\right)}-(2\delta -1)}\right|<1$$

and

$$\mathrm{Re}\left(1+\frac{z{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{\prime }}\right)<\delta (\delta >1,\gamma \ge 0,m\in {\mathbb{N}}_0),$$

or, equivalently

$$\left|\frac{\frac{z{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{\prime }}}{\frac{z{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}_{\gamma }^{m}f\left(z\right)\right)}^{\prime }}-2(\delta -1)}\right|<1,$$

respectively.

(iv) 

${lim}_{q\to 1^{-}}{\mathcal{Y}}_q^m(1,\delta )={\mathcal{Y}}^m\left(\delta \right)$ and ${lim}_{q\to 1^{-}}{\mathcal{K}}_q^m(1,\delta )={\mathcal{K}}^m\left(\delta \right)$:

$$\mathrm{Re}\left(\frac{z{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{\prime }}{{\mathcal{D}}^{m}f\left(z\right)}\right)<\delta (\delta >1,m\in {\mathbb{N}}_0),$$

or, equivalently

$$\left|\frac{\frac{z{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{\prime }}{{\mathcal{D}}^{m}f\left(z\right)}-1}{\frac{z{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{\prime }}{{\mathcal{D}}^{m}f\left(z\right)}-(2\delta -1)}\right|<1$$

and

$$\mathrm{Re}\left(1+\frac{z{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{\prime }}\right)<\delta (\delta >1,m\in {\mathbb{N}}_0),$$

or, equivalently

$$\left|\frac{\frac{z{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{\prime }}}{\frac{z{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{″}}{{\left({\mathcal{D}}^{m}f\left(z\right)\right)}^{\prime }}-2(\delta -1)}\right|<1,$$

respectively.

(v) 

${lim}_{q\to 1^{-}}{\mathcal{Y}}_q^0(\gamma ,\delta )=M\left(\delta \right)$ and ${lim}_{q\to 1^{-}}{\mathcal{K}}_q^0(\gamma ,\delta )=N\left(\delta \right),\left(\delta >1\right)$ (see Nishiwaki and Owa [11]);

(vi) 

${lim}_{q\to 1^{-}}{\mathcal{Y}}_q^0(\gamma ,\delta )=M\left(\delta \right)$ and ${lim}_{q\to 1^{-}}{\mathcal{K}}_q^0(\gamma ,\delta )=N\left(\delta \right),\left(1<\delta \le \frac{4}{3}\right)$ (see Uralegaddi et al. [12]).

We denote by $\mathcal{V}$ the subclass of analytic univalent functions having the form:

$$f\left(z\right)=a_{0}z+\sum _{𝚥=2}^{\infty }|a_{𝚥}|z^{𝚥}(a_0>0).$$

(10)

Furthermore, let:

$$f_{𝚤}\left(z\right)=a_{0,𝚤}z+\sum _{𝚥=2}^{\infty }|a_{𝚥,𝚤}|z^{𝚥}(a_{0,𝚤}>0),$$

(11)

and

$$g_{\kappa }\left(z\right)=b_{0,\kappa }z+\sum _{𝚥=2}^{\infty }|b_{𝚥,\kappa }|z^{𝚥}(b_{0,\kappa }>0),$$

(12)

the quasi-Hadamard product $(f_{𝚤}\ast g_{\kappa })\left(z\right)$ of the functions $f_{𝚤}\left(z\right)$ and $g_{\kappa }\left(z\right)$ is defined as:

$$(f_{𝚤}\ast g_{\kappa })\left(z\right)=a_{0,𝚤}{b}_{0,\kappa }z+\sum _{𝚥=2}^{\infty }|a_{𝚥,𝚤}||b_{𝚥,\kappa }|z^{𝚥}(𝚤,\kappa \in \mathbb{N}).$$

Following a similar manner, the quasi-Hadamard product of several functions can be defined. The definition of the quasi-Hadamard product for two or more functions was first given by Owa [13], which was then investigated by Kumar [14] for various function classes (see also [15,16]).

Let:

$${\mathcal{YV}}_q\left(\delta \right)={\mathcal{Y}}_q\left(\delta \right)\cap \mathcal{V}$$

and

$${\mathcal{KV}}_q\left(\delta \right)={\mathcal{K}}_q\left(\delta \right)\cap \mathcal{V}.$$

For $q\to 1^{-}$, we obtain the following particular classes:

(i)

$\mathcal{YV}\left(\delta \right)=\mathcal{Y}\left(\delta \right)\cap \mathcal{V};$

(ii)

$\mathcal{KV}\left(\delta \right)=\mathcal{K}\left(\delta \right)\cap \mathcal{V}.$

2. Preliminary Results

Following the technique of Aouf et al. [7], we formulate the following results that can be easily demonstrated:

Lemma 1.

A function $f\left(z\right)\in {\mathcal{YV}}_q\left(\delta \right)$ if and only if:

$$\sum _{𝚥=2}^{\infty }\left({\left[𝚥\right]}_q-\delta \right)|a_{𝚥}|\le (\delta -1)a_0(\delta >1,0<q<1).$$

(13)

Lemma 2.

A function $f\left(z\right)\in {\mathcal{KV}}_q\left(\delta \right)$ if and only if:

$$\begin{array}{c}\hfill \sum _{𝚥=2}^{\infty }{\left[𝚥\right]}_q\left({\left[𝚥\right]}_q-\delta \right)|a_{𝚥}|\le (\delta -1)a_0(\delta >1,0<q<1).\end{array}$$

(14)

We define a fixed function as follows:

$$\varrho \left(z\right)=z+\sum _{𝚥=2}^{\infty }{c}_{𝚥}{z}^{𝚥}(c_{𝚥}>0).$$

(15)

Now, we create the new classes mentioned below using the function defined by (15).

Definition 2.

Let $f\left(z\right)$ given by (10) belongs to ${\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )(c_{𝚥}\ge c_2>0;0<q<1)$ if and only if:

$$\sum _{𝚥=2}^{\infty }{c}_{𝚥}|a_{𝚥}|\le \lambda a_0(\lambda >0).$$

Definition 3.

Let $f\left(z\right)$ given by (10) belongs to ${\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )(c_{𝚥}\ge c_2>0)$ if and only if:

$$\sum _{𝚥=2}^{\infty }{\left[𝚥\right]}_q{c}_{𝚥}|a_{𝚥}|\le \lambda a_0(\lambda >0,0<q<1).$$

In addition, we provide the next class of analytical functions, which is important to the following discussion.

Definition 4.

Let ℓ be any fixed nonnegative real number. A function $f\left(z\right)\in {\mathcal{V}}_{q,\varrho }^{\ell }(c_{𝚥},\lambda )(c_{𝚥}\ge c_2>0)$ if and only if:

$$\sum _{𝚥=2}^{\infty }{\left[𝚥\right]}_q^{\ell }{c}_{𝚥}|a_{𝚥}|\le \lambda a_0(\lambda >0,0<q<1).$$

(16)

Remark  3.

For appropriate selections of $c_{𝚥},\lambda ,\ell ,q$ and $a_0=1$, we obtain:

(i) 

${\mathcal{V}}_{q,\varrho }^0(\left({\left[𝚥\right]}_q-\delta \right){(1+\gamma ({\left[𝚥\right]}_q-1))}^m,(\delta -1))={\mathcal{Y}}_q^m(\gamma ,\delta )(\gamma \ge 0,\delta >1,m\in {\mathbb{N}}_0,0<q<1)$ (see [7]);

(ii) 

${\mathcal{V}}_{q,\varrho }^1({\left[𝚥\right]}_q\left({\left[𝚥\right]}_q-\delta \right){(1+\gamma ({\left[𝚥\right]}_q-1))}^m,(\delta -1))={\mathcal{K}}_q^m(\gamma ,\delta )(\gamma \ge 0,\delta >1,m\in {\mathbb{N}}_0,0<q<1)$ (see [7]);

(iii) 

${\mathcal{V}}_{q,\varrho }^0(\left({\left[𝚥\right]}_q-\delta \right),(\delta -1))={\mathcal{Y}}_q\left(\delta \right)(0<q<1,\delta >1)$;

(iv) 

${\mathcal{V}}_{q,\varrho }^1({\left[𝚥\right]}_q\left({\left[𝚥\right]}_q-\delta \right),(\delta -1))={\mathcal{K}}_q\left(\delta \right)(0<q<1,\delta >1)$;

(v) 

${lim}_{q\to 1^{-}}{\mathcal{V}}_{q,\varrho }^0(\left({\left[𝚥\right]}_q-\delta \right),(\delta -1))=\mathcal{Y}\left(\delta \right)(1<\delta \le \frac{4}{3})$ (see [12]);

(vi) 

${lim}_{q\to 1^{-}}{\mathcal{V}}_{q,\varrho }^1({\left[𝚥\right]}_q\left({\left[𝚥\right]}_q-\delta \right),(\delta -1))=\mathcal{K}\left(\delta \right)(1<\delta \le \frac{4}{3})$ (see [12]);

(vii) 

${lim}_{q\to 1^{-}}{\mathcal{V}}_{q,\varrho }^0(\left({\left[𝚥\right]}_q-1\right){+|\left[𝚥\right]}_q-2\delta +1|,2(\delta -1))=M\left(\delta \right)(\delta >1,a_0=1)$ (see [11] and ([17] with $k=1$));

(viii) 

${lim}_{q\to 1^{-}}{\mathcal{V}}_{q,\varrho }^1{(\left[𝚥\right]}_q\left({\left[𝚥\right]}_q-1\right){+|\left[𝚥\right]}_q-2\delta +1|,2(\delta -1))=N\left(\delta \right)(\delta >1,a_0=1)$ (see [11] and ([17] with $k=1$)).

It is evident that ${\mathcal{V}}_{q,\varrho }^0(c_{𝚥},\lambda )={\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )$ and ${\mathcal{V}}_{q,\varrho }^1(c_{𝚥},\lambda )={\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )$. Furthermore, ${\mathcal{V}}_{q,\varrho }^{\zeta }(c_{𝚥},\lambda )\subset {\mathcal{V}}_{q,\varrho }^{\varsigma }(c_{𝚥},\lambda )$ if $\zeta >\varsigma \ge 0$, the containment being proper. Hence, for any positive integer ℓ, we have the inclusion relation:

$${\mathcal{V}}_{q,\varrho }^{\ell }(c_{𝚥},\lambda )\subset {\mathcal{V}}_{q,\varrho }^{\ell -1}(c_{𝚥},\lambda )\subset \cdots \subset {\mathcal{V}}_{q,\varrho }^2(c_{𝚥},\lambda )\subset {\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )\subset {\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda ).$$

Additionally, for nonnegative real integer ℓ, we observe that the class ${\mathcal{V}}_{q,\varrho }^{\ell }(c_{𝚥},\lambda )$ is nonempty as the function:

$$f\left(z\right)=a_{0}z+\sum _{𝚥=2}^{\infty }{\left[𝚥\right]}_q^{-\ell }\frac{\lambda a_0}{c_{𝚥}}{\chi }_{𝚥}{z}^{𝚥},$$

(17)

where ${\chi }_{𝚥}\ge 0,a_0>0$ and ${\sum }_{𝚥=2}^{\infty }{\chi }_{𝚥}\le 1,$ which satisfies the inequality (16).

This study’s goal is to analyze the quasi-Hadamard product of functions of classes ${\mathcal{V}}_{q,\varrho }^{\ell }(c_{𝚥},\lambda ),$ ${\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )$ and ${\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )$.

3. Main Results

In what follows we obtain some results regarding the behavior of the quasi-Hadamard product on our classes.

Theorem 1.

Let be $f_{𝚤}\left(z\right)$ of the form (11), from the class ${\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )$, for every $𝚤=1,2,\cdots ,s$ and $g_{\kappa }\left(z\right)$ of the form (12), from the class ${\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )$, for every $\kappa =1,2,\cdots ,r$. If $c_{𝚥}\ge {\left[𝚥\right]}_q\lambda (𝚥\in \mathbb{N})$, then the quasi-Hadamard product $f_1\ast f_2\ast \cdots \ast f_s\ast g_1\ast g_2\ast \cdots \ast g_r$ belongs to the class ${\mathcal{V}}_{q,\varrho }^{2s+r-1}(c_{𝚥},\lambda )$.

Proof. 

It suffices to demonstrate that:

$$\sum _{𝚥=2}^{\infty }\left[{\left[𝚥\right]}_q^{2s+r-1}{c}_{𝚥}\left(\prod _{𝚤=1}^s|a_{𝚥,𝚤}|.\prod _{\kappa =1}^r|b_{𝚥,\kappa }|\right)\right]\le \lambda \left(\prod _{𝚤=1}^s{a}_{0,𝚤}.\prod _{\kappa =1}^r{b}_{0,\kappa }\right).$$

Since $f_{𝚤}\in {\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )$, we have:

$$\sum _{𝚥=2}^{\infty }{\left[𝚥\right]}_q{c}_{𝚥}|a_{𝚥,𝚤}|\le \lambda a_{0,𝚤},(𝚤=1,2,\cdots ,s).$$

(18)

Therefore:

$$|a_{𝚥,𝚤}|\le \left(\frac{\lambda }{{\left[𝚥\right]}_q{c}_{𝚥}}\right)a_{0,𝚤},$$

and

$$|a_{𝚥,𝚤}|\le \frac{a_{0,𝚤}}{{\left[𝚥\right]}_q^2},$$

(19)

the inequalities (18) and (19) hold for every $𝚤=1,2,\cdots ,s$. Furthermore, since $g_{\kappa }\in {\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )$, we have:

$$\sum _{𝚥=2}^{\infty }{c}_{𝚥}|b_{𝚥,\kappa }|\le \lambda b_{0,\kappa },(\kappa =1,2,\cdots ,r).$$

(20)

Hence:

$$|b_{𝚥,\kappa }|\le \frac{b_{0,\kappa }}{{\left[𝚥\right]}_q},(\kappa =1,2,\cdots ,r).$$

(21)

Using (19) for every $𝚤=1,2,\cdots ,s$, (21) for every $\kappa =1,2,\cdots ,r-1$ and (20) for $\kappa =r$, we have:

$$\begin{array}{cc}\hfill \sum _{𝚥=2}^{\infty }& \left[{\left[𝚥\right]}_q^{2s+r-1}{c}_{𝚥}\left(\prod _{𝚤=1}^s|a_{𝚥,𝚤}|.\prod _{\kappa =1}^r|b_{𝚥,\kappa }|\right)\right]\hfill \\ & \le \sum _{𝚥=2}^{\infty }\left[{\left[𝚥\right]}_q^{2s+r-1}{c}_{𝚥}\left({\left[𝚥\right]}_q^{-2s}{\left[𝚥\right]}_q^{1-r}\prod _{𝚤=1}^s{a}_{0,𝚤}.\prod _{\kappa =1}^{r-1}{b}_{0,\kappa }\right)|b_{𝚥,r}|\right]\hfill \\ & =\left(\prod _{𝚤=1}^s{a}_{0,𝚤}.\prod _{\kappa =1}^{r-1}{b}_{0,\kappa }\right)\sum _{𝚥=2}^{\infty }{c}_{𝚥}|b_{𝚥,r}|\le \lambda \left(\prod _{𝚤=1}^s{a}_{0,𝚤}.\prod _{\kappa =1}^r{b}_{0,\kappa }\right).\hfill \end{array}$$

Hence, $f_1\ast f_2\ast \cdots \ast f_s\ast g_1\ast g_2\ast \cdots \ast g_r\in {\mathcal{V}}_{q,\varrho }^{2s+r-1}(c_{𝚥},\lambda )$.

We also can use (19) for $𝚤=1,2,\cdots ,s-1$, (21) for $\kappa =1,2,\cdots ,r$ and (18) for $𝚤=s.$ to obtain the required estimate. □

If in the proof of Theorem 1 we consider the quasi-Hadamard product functions $f_1\ast f_2\ast \cdots \ast f_s$ only, and if we apply (19) for $𝚤=1,2,\cdots ,s-1$, and (18) for $𝚤=s$, we obtain the following result:

Corollary 1.

Let $f_{𝚤}\left(z\right)$ be of the form (11), from the class ${\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )$, for every $𝚤=1,2,\cdots ,s$. If $c_{𝚥}\ge {\left[𝚥\right]}_q\lambda (𝚥\in \mathbb{N})$, then the quasi-Hadamard product $f_1\ast f_2\ast \cdots \ast f_s$ belongs to the class ${\mathcal{V}}_{q,\varrho }^{2s-1}(c_{𝚥},\lambda )$.

If in the proof of Theorem 1 we consider the quasi-Hadamard product functions $g_1\ast g_2\ast \cdots \ast g_r$ only, and if we apply (21) for $\kappa =1,2,\cdots ,r-1$, and (20) for $\kappa =r$, we obtain the following result:

Corollary 2.

Let $g_{\kappa }\left(z\right)$ be of the form (12), from the class ${\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )$, for every $\kappa =1,2,\cdots ,r$. If $c_{𝚥}\ge {\left[𝚥\right]}_q\lambda (𝚥\in \mathbb{N})$, then the quasi-Hadamard product $g_1\ast g_2\ast \cdots \ast g_r$ belongs to the class ${\mathcal{V}}_{q,\varrho }^{r-1}(c_{𝚥},\lambda )$.

Remark 4.

We notice that:

(i) 

Taking $c_{𝚥}=\left({\left[𝚥\right]}_q-\delta \right)(𝚥\ge 2,0<q<1)$ and $\lambda =(\delta -1)(\delta >1)$ in Corollary 2, we obtain similar results for the class ${\mathcal{YV}}_q\left(\delta \right);$

(ii) 

Taking $c_{𝚥}={\left[𝚥\right]}_q\left({\left[𝚥\right]}_q-\delta \right)(𝚥\ge 2,0<q<1)$ and $\lambda =(\delta -1)(\delta >1)$ in Corollary 1, we obtain similar results for the class ${\mathcal{KV}}_q\left(\delta \right);$

(iii) 

Taking $c_{𝚥}=\left({\left[𝚥\right]}_q-\delta \right)(𝚥\ge 2)$, $\lambda =(\delta -1)(1<\delta \le \frac{4}{3})$ and $q\to 1^{-}$ in Corollary 2, we obtain similar results for the class $\mathcal{Y}\left(\delta \right)$ (see [15]);

(iv) 

Taking $c_{𝚥}={\left[𝚥\right]}_q\left({\left[𝚥\right]}_q-\delta \right)(𝚥\ge 2)$, $\lambda =(\delta -1)(1<\delta \le \frac{4}{3})$ and $q\to 1^{-}$ in Corollary 1, we obtain similar results for the class $\mathcal{K}\left(\delta \right)$ (see [15]);

(v) 

Taking $c_{𝚥}=\left({\left[𝚥\right]}_q-1\right)+|{\left[𝚥\right]}_q-2\delta +1|(𝚥\ge 2)$, $\lambda =2(\delta -1)(\delta >1)$ and $q\to 1^{-}$ in Corollary 2, we obtain similar results for the class $M\left(\delta \right)$ (see [15]);

(vi) 

Taking $c_{𝚥}={\left[𝚥\right]}_q\left(\left({\left[𝚥\right]}_q-1\right)+|{\left[𝚥\right]}_q-2\delta +1|\right)(𝚥\ge 2)$, $\lambda =2(\delta -1)(\delta >1)$ and $q\to 1^{-}$ in Corollary 1, we obtain similar results for the class $N\left(\delta \right)$ (see [15]).

4. Conclusions

The q-difference operator was applied in this research paper to introduce the new subclasses ${\mathcal{V}}_{q,\varrho }^{\ell }(c_{𝚥},\lambda ),$ ${\mathcal{U}}_{q,\varrho }(c_{𝚥},\lambda )$ and ${\mathcal{V}}_{q,\varrho }(c_{𝚥},\lambda )$ of analytic functions. Thus, we obtained the main results regarding the quasi-Hadamard product of these subclasses of analytic functions. Moreover, we described some previously established results as particular cases derived from our classes.

Furthermore, as further investigations, it is interesting to outline that there are some aspects that connect the classical q-analysis, applied in this study with so-called $(p,q)$-analysis. Considering that, from the results obtained here on q-analogues, for $0<q<1$, one can easily derive some corresponding results on $(p;q)$-analogues, for $0<p;q<1$, by taking some specific parametric and also argument variations (see [5] (p. 340) and [18] (pp. 511–512) for further information on the fractional $(p;q)$-calculus).