Subclass of analytic functions defined by $q$-derivative operator associated with Pascal distribution series

1.   Introduction and definitions

Let $\mathcal{A}$ denote the class of the normalized functions of the form

which are analytic in the open unit disk $\mathbb{U} = \{z\in \mathbb{C} :\left \vert z\right \vert < 1\}.$ Further, let $\mathcal{T}$ be a subclass of $\mathcal{A}$ consisting of functions of the form,

A function $f\in \mathcal{A}$ is said to be in the class $\mathcal{R}^{\tau }(A, B)$, $\tau \in \mathbb{C}\backslash \{0\}$, $-1\leq B < A\leq 1,$ if it satisfies the inequality

This class was introduced by Dixit and Pal ^[13].

The theory of $q$-calculus operators are used in describing and solving various problems in applied science such as ordinary fractional calculus, optimal control, $q$-difference and $q$-integral equations, as well as geometric function theory of complex analysis. The application of $q$-calculus was initiated by Jackson ^[23]. Recently, many researchers studied $q$-calculus such as Srivastava et al. ^[52], Muhammad and Darus ^[31], Kanas and Răducanu ^[28], Aldweby and Darus ^[2,3,4] and Muhammad and Sokol ^[30]. For details on $q$-calculus one can refer ^{[1,5,6,7,9,20,23,25,38,39,43,44,46,48,49,50,51]} and also the reference cited therein.

For $0 < q < 1$ the Jackson's $\mathit{q}$-derivative of a function $f\in \mathcal{A}$ is, by definition, given as follows ^[23]

and

From (1.3), we have

where

is sometimes called the basic number $n$. If $q\rightarrow 1-, [n]_{q}\rightarrow n$.

For a function $h(z) = z^{n},$ we obtain

and

where $h^{\prime }$ is the ordinary derivative.

Using the above defined $q$-calculus, several subclasses belonging to the class $\mathcal{A}$ have already been investigated in geometric function theory. Ismail et al. ^[26] were the first who used the $q$-derivative operator $D_{q}$ to study the $q$-calculus analogous of the class $\mathcal{S }^{\ast }$ of starlike functions in $\mathbb{U}$ (see Definition 1.1 below). However, a firm footing of the $q$-calculus in the context of geometric function theory was presented mainly and basic (or $q$-) hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, (^[45], p.347 et seq.); see also ^[46]).

For $0 < q < 1$, we define the class $\mathcal{S}_{q}^{\ast }(\alpha)$ of $q$- starlike functions and the class $\mathcal{C}_{q}(\alpha)$ of $q$- convex functions of order $\alpha (0\leq \alpha < 1)$ (see, ^[26,40,41]), as below:

Definition 1.1. A function $f\in$ $\mathcal{A}$ is said to be in the class $\mathcal{S}_{q}^{\ast }(\alpha)$ if it satisfies

Definition 1.2. A function $f\in$ $\mathcal{A}$ is said to be in the class $\mathcal{C} _{q}(\alpha)$ if it satisfies

It is clear that $\lim \limits_{q\rightarrow 1-}\mathcal{S} _{q}^{\ast }(\alpha) = \mathcal{S}^{\ast }(\alpha)\mathcal{\ }$and $\lim \limits_{q\rightarrow 1-}\mathcal{C}_{q}(\alpha) = \mathcal{C}(\alpha),$ where $\mathcal{S}^{\ast }(\alpha)$ and $\mathcal{C}(\alpha)$ are, respectively, well-known starlike and convex functions of order $\alpha$ in $\mathbb{U}$.

We now introduce a new subclass of analytic functions defined by $q$ -derivative operator $D_{q}.$

Definition 1.3. A function $f\in$ $\mathcal{A}$ is said to be in the class $\mathcal{C} _{q}(\lambda, \alpha)$ if it satisfies

where $0\leq \alpha < 1, \mathit{\ }0\leq \lambda \leq 1.$

We write

A variable $X$ is said to be Pascal distribution if it takes the values $0, 1, 2, 3, \dots$ with probabilities $(1-s)^{m}$, $\dfrac{sm(1-s)^{m}}{1!}$, $\dfrac{s^{2}m(m+1)(1-s)^{m}}{2!}$, $\dfrac{s^{3}m(m+1)(m+2)(1-s)^{m}}{3!}, \dots,$ respectively, where $s$ and $m$ are called the parameters, and thus

Very recently, El-Deeb et al. ^[15] (see also, ^[10,34]) introduced a power series whose coefficients are probabilities of Pascal distribution, that is

where $m\geq 1$, $0\leq s\leq 1$, and we note that, by ratio test the radius of convergence of above series is infinity. We also define the series

Let consider the linear operator $\mathcal{I}_{s}^{m}:\mathcal{A}\rightarrow \mathcal{A}$ defined by the convolution or Hadamard product

where $m\geq 1$ and $0\leq s\leq 1$.

Motivated by several earlier results on connections between various subclasses of analytic and univalent functions, using hypergeometric functions (see for example, ^{[8,11,21,29,42,47]}), generalized Bessel functions (see for example, ^{[18,22,33,36]}), Struve functions (see for example, ^[12,24]), Poisson distribution series (see for example, ^{[14,16,19,32,35,37]}) and Pascal distribution series (see for example, ^{[10,15,17,34]}), in this paper we determine the necessary and sufficient condition for $\Phi _{s}^{m}$ to be in the class $\mathcal{TC}_{q}(\lambda, \alpha)$. Furthermore, we give sufficient condition for $\mathcal{I}_{s}^{m}\left(\mathcal{R}^{\tau }(A, B)\right) \subset \mathcal{TC}_{q}(\lambda, \alpha)$ and finally, we give necessary and sufficient condition for the function $f$ such that its image by the integral operator $\mathcal{G}_{s}^{m}f(z) = \int_{0}^{z}\frac{\Phi _{s}^{m}(t)}{t}dt$ belongs to the class $\mathcal{TC }_{q}(\lambda, \alpha)$.

To establish our main results, we need the following Lemmas.

Lemma 1.4. A function $f$ of the form (1.2) is in $\mathcal{TC} _{q}(\lambda, \alpha)$ if and only if it satisfies

where $0\leq \alpha < 1, \ 0\leq \lambda \leq 1$ and $z\in \mathbb{U}$.

Lemma 1.4 can be proved using the same technique as in ^[27].

Lemma 1.5. ^[13] If $f$ $\in$ $\mathcal{R}^{\tau }(A, B)$ is of the form (1.1), then

The result is sharp.

2.   Necessary and sufficient condition for $\Phi _{s}^{m}\in \mathcal{TC }_{q}( \lambda, \alpha)$

For convenience throughout in the sequel, we use the following identities that hold for $m\geq 1$ and $0\leq s < 1$:

By simple calculations we derive the following relations:

and

Unless otherwise mentioned, we shall assume in this paper that $0\leq \alpha < 1$ and$\mathit{\ }0\leq \lambda \leq 1,$ $0 < q < 1$ and $0\leq s < 1$.

Firstly, we obtain the necessary and sufficient conditions for $\Phi _{s}^{m}$ to be in the class $\mathcal{TC}_{q}(\lambda, \alpha).$

Theorem 2.1. Let $m\geq 1$ and $q\rightarrow 1-.$Then $\Phi _{s}^{m}\in \mathcal{TC}_{q}(\lambda, \alpha)$ if and only if

Proof. Since $\Phi _{s}^{m}$ is defined by (1.7), in view of Lemma 1.4 it is sufficient to show that

Since $[n]_{q}\rightarrow n,$ when $q\rightarrow 1-,$ we get

Writing

and using (2.2)–(2.5), we have

but this last expression is upper bounded by $1-\alpha$ if and only if (2.6) holds.

3.   Sufficient condition for $\mathcal{I}_{s}^{m}\left(\mathcal{R}^{ \tau }(A, B)\right) \subset \mathcal{TC}_{q}( \lambda, \alpha)$

Making use of Lemma 1.5, we will study the action of the Pascal distribution series on the class $\mathcal{TC}_{q}(\lambda, \alpha)$.

Theorem 3.1. Let $m\geq 1$ and $q\rightarrow 1-.$If $f\in \mathcal{R}^{\tau }(A, B)$ and the inequality

is satisfied then $\mathcal{I}_{s}^{m}f\in \mathcal{TC}_{q}(\lambda, \alpha).$

Proof. According to Lemma 1.4 it is sufficient to show that

Since $f\in \mathcal{R}^{\tau }(A, B)$, using Lemma 1.5 we have

therefore

Writing $n^{2}, n^{3}$ as given in (2.7) and (2.8)$,$ $n = n-1+1,$ and making use of (2.2)–(2.5), we get

but this last expression is upper bounded by $1-\alpha$ if and only if (3.1) holds.

4.   Integral operator

Theorem 4.1. Let $m\geq 1$ and $q\rightarrow 1-.$If the integral operator $\mathcal{G}_{s}^{m}$ is given by

then $\mathcal{G}_{s}^{m}\in \mathcal{TC}_{q}(\lambda, \alpha)$ if and only

Proof. According to (1.7) it follows that

Using Lemma 1.4, the function $\mathcal{G}_{q}^{m}(z)$ belongs to $\mathcal{TC}_{q}(\lambda, \alpha)$ if and only if

Now,

By a similar proof like those of Theorem 3.1 we get that $\mathcal{G} _{s}^{m}f\in \mathcal{TC}_{q}(\lambda, \alpha)$ if and only if (4.2) holds.

5.   Corollaries and consequences

Corollary 5.1. Let $m\geq 1$ and $q\rightarrow 1-.$Then $\Phi _{s}^{m}\in \mathcal{TC} _{q}(0, \alpha)$, if and only if

Corollary 5.2. Let $m\geq 1$ and $q\rightarrow 1-.\ $If $f\in \mathcal{R}^{\tau }(A, B)$ and the inequality

is satisfied then $\mathcal{I}_{s}^{m}f\in \mathcal{TC} _{q}(0, \alpha).$

Corollary 5.3. Let $m\geq 1$ and $q\rightarrow 1-.\ $If the integral operator $\mathcal{G} _{s}^{m}$ is given by (4.1), then $\mathcal{G}_{s}^{m}\in \mathcal{TC} _{q}(0, \alpha)$ if and only

6.   Conclusions

In this paper, we find the necessary and sufficient conditions and inclusion relations for Pascal distribution series to be in a subclass of analytic functions defined by $q$-derivative operator. Basic (or $q$-) series and basic (or $q$-) polynomials, especially the basic (or $q$-) hypergeometric functions and basic (or $q$-) hypergeometric polynomials, are applicable particularly in several diverse areas (see, for example, [^[45], pp.350–351] and [^[44], p.328]). Moreover, in this recently-published survey-cum-expository review article by Srivastava ^[44], the so-called ($p, q$)-calculus was exposed to be a rather trivial and inconsequential variation of the classical $q$-calculus, the additional parameter $p$ being redundant (see, for details, [^[44], p.340]). This observation by Srivastava ^[44] will indeed apply also to any attempt to produce the rather straightforward ($p, q$)-variations of the results which we have presented in this paper.

Acknowledgements

This research was funded by Universiti Kebangsaan Malaysia, grant number GUP-2019-032. The authors would like to thank the referees for their helpful comments and suggestions.

Conflicts of interest

The authors declare no conflict of interest.