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1.   Introduction and definition

Let us consider the following second-order linear homogeneous differential equation

The differential equation in (1.1) is famous Bessel's differential equation. Its solution is denoted by $ J_{u}(z) $ and known as Bessel function. The familiar representation of $ J_{u}(z) $ is given by (1.2) and is defined by particular solution of (1.1) as follows:

where $ \Gamma $ is the familiar Euler Gamma function. For a comprehensive study of Bessel function of first kind, see ^[9,30].

Let $ \mathcal{A} $ represents the class of all those functions which are analytic in the open unit disk

and having the series expansion of the form

Let $ \mathcal{S} $ be the subclass of $ \mathcal{A} $ consisting the functions that is univalent in $ E $ and satisfy the normalized conditions

Let two functions $ f $ and $ g $ are analytic in $ E $, then $ f $ is subordinate to $ g $, (written as $ f\prec g), $ if there exists a Schwarz function $ h\left(z\right) $, which is analytic in $ E $ with

such that

If $ g $ is univalent in $ E, $ then

For $ f\in \mathcal{A}, $ given by (1.3) and another function $ g, $ given by

then the Hadamard product (or convolution) of $ f(z) $ and $ g(z) $ is given by

In ^[34], Robertson introduced the class of starlike $ (\mathcal{S}^{\ast }) $ and class of convex $ (\mathcal{C}) $ functions and be defined as:

It can easily seen that

After that Srivastava and Owa investigated these subclasses in ^[43].

Let $ f\in \mathcal{A} $ and $ g\in \mathcal{S}^{\ast }, $ is said to be close to convex $ (\mathcal{K}) $ functions if and only if

Furthermore, Kanas and Wisniowska in ^[12] introduced subclasses of $ k $ -uniformly convex $ \left(k-\mathcal{UCV}\right) $ and $ \left(k-\mathcal{ST} \right) $ and be defined as:

and

Note that

For the further developments Kanas and Srivastava studied these subclasses $ \left(k-\mathcal{UCV}\right) $ and $ \left(k-\mathcal{ST}\right) $ of analytic functions in ^[11]. For particular value of $ k = 1 $, then $ k- \mathcal{UCV = } $ $ \mathcal{UCV} $ and $ k-\mathcal{ST = S}^{\ast } $

Kanas and Wisniowska ^[13,14] (see also ^[11] and ^[15]) defined these subclasses of analytic functions subject to the conic domain $ \Omega _{k}, $ where

For $ k = 0 $, the domain $ \Omega _{k} $ presents the right half plane, for $ 0 < k < 1 $, the domain $ \Omega _{k} $ presents hyperbola, for $ k = 1 $ its presents parabola and an ellipse for $ k > 1 $.

For this conic domain, the following functions play the role of extremal functions.

where

and

is chosen such that

Here $ R(t) $ is Legender's complete elliptic integral of first kind (see ^[13,14]).

Since the $ q $-calculus is being vastly used in different areas of mathematics and physics it is of great interest of researchers. In the study of Geometric Function Theory, the versatile applications of $ q $-derivative operator make it remarkably significant. Initially, in the year 1990, Ismail et al. ^[5] gave the idea of $ q $-starlike functions. Nevertheless, a firm foothold of the usage of the $ q $-calculus in the context of Geometric Function Theory was effectively established, and the use of the generalized basic (or $ q $-) hypergeometric functions in Geometric Function Theory was made by Srivastava (see for detail ^[37]). For the study of various families of analytic and univalent function, the quantum (or $ q $-) calculus has been used as a important tools. Jackson ^[7,8] first defined the $ q $-derivative and integral operator as well as provided some of their applications. The $ q $-Ruscheweyh differential operator was defined by Kanas and Raducanu in ^[10]. Recently, by using the concept of convolution Srivastava ^[40] introduced $ q $-Noor integral operator and studied some of its applications. Many $ q $-differential and $ q $-integral operators can be written in term of convolution, for detail we refer ^{[4,23,36,39,41]} see also ^[16,18]. Moreover, Srivastava et al. (see, for example, ^[35,44,45]) published a set of articles in which they concentrated upon the classes of $ q $-starlike functions related with the Janowski functions from several different aspects. Additionally specking, a recently-published survey-cum-expository review article by Srivastava ^[38] is potentially useful for researchers and scholars working on these topics. In this survey-cum-expository review article ^[38], the mathematical explanation and applications of the fractional $ q $-calculus and the fractional $ q $-derivative operators in Geometric Function Theory was systematically investigated. For other recent investigations involving the $ q $-calculus, one may refer to ^{[1,19,22,24,25,31,32,33]} and ^[17]. We remark in passing that, in the above-cited recently-published survey-cum-expository review article ^[38], 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 or superfluous (see, for details, [38, p. 340]). In order to have a better understanding of the present article we provide some notation and concepts of quantum (or $ q $-) calculus used in this article.

Definition 1. (^[10]). Let $ q\in (0, 1) $ and define the $ q $-number $ [\eta ]_{q} $ as:

Definition 2. Let $ q\in \left(0, 1\right), $ $ n\in \mathbb{N} $ and define the $ q $-factorial $ [n]_{q}! $

Definition 3. The $ q $-generalized Pochhammer symbol $ [a]_{n, q} $ be defined as:

and

Definition 4. The $ q $-Gamma function $ \Gamma _{q}(n) $ is defined by

The $ q $-Gamma function $ \Gamma _{q}(n) $ satisfies the following functional equation

Definition 5. (^[7]). For $ f\in \mathcal{A} $, and the $ q $-derivative operator or $ q $ -difference operator be defined as:

and

Definition 6. (^[5]). An analytic function $ f\in $ $ \mathcal{S}_{q}^{\ast } $ if

and

we can rewrite the conditions (1.7) as follows, (see ^[46]).

Here Serivastava et al. ^[39] (see also ^[42]) defined the following definition by making use of quantum (or $ q $-) calculus, principle of subordination and general conic domain $ \Omega _{k, q} $ as:

Definition 7. (^[39]). Let $ k\geq 0 $ and $ q\in \left(0, 1\right) $. A function $ p(z) $ is said to be in the class $ k-\mathcal{P}_{q} $ if and only if

and $ p_{k}(z) $ is given by (1.4).

Geometrically, the function $ p(z)\in k-\mathcal{P}_{q} $ takes all values from the domain $ \Omega _{k, q} $ which is defined as follows:

Remark 1. We see that

and

For $ q\rightarrow 1- $, then we have

where the class $ \mathcal{P}\left(\frac{k}{k+1}\right) $ introduced by Kanas and Wisniowska ^[13] and therefore,

Also for $ k = 0 $ and $ q\rightarrow 1-, $ we have

and

Remark 2. For $ q\rightarrow 1-, $ then $ \Omega _{k, q} = \Omega _{k}, $ where domain $ \Omega _{k} $ introduced by Kanas and Wisniowska in ^[13].

By Applying $ q $-derivative operator we introduce new subclasses of $ q $ -starlike functions, $ q $-convex functions, $ q $-close to convex functions and $ q $-quasi-convex functions as follows:

Definition 8. ^[42] For $ f\in \mathcal{A}, $ $ k\geq 0, $ then $ f\in k-\mathcal{ST}_{q} $ if and only if

Definition 9. ^[42] For $ f\in \mathcal{A}, $ $ k\geq 0, $ then $ f\in k-\mathcal{UCV}_{q} $ if and only if

It can easily seen that

Definition 10. ^[42] For $ f\in \mathcal{A}, $ $ k\geq 0, $ then $ f\in k-\mathcal{UCC}_{q} $ if and only if

Definition 11. ^[42] For $ f\in \mathcal{A}, $ $ k\geq 0, $ then $ f\in k-\mathcal{UQV}_{q} $ if and only if

Remark 3. For $ q\rightarrow 1-, $ then all theses newly defined subclasses reduces to the well-known subclasses of analytic functions introduced in ^[29].

The Jackson $ q $-Bessel functions and the Hahn-Exton $ q $-Bessel functions are, respectively, defined by

and

where $ z\in \mathbb{C}, $ $ u > -1, $ $ q\in \left(0, 1\right). $ The functions $ J_{u}^{1}(z, q) $ and $ J_{u}^{2}(z, q) $ are the $ q $-extensions of the classical Bessel functions of the first kind. For more study about $ q $-extensions of Bessel functions (see ^[6,20,21]). Since neither $ J_{u}^{1}(z, q) $ nor $ J_{u}^{2}(z, q) $ belongs to $ \mathcal{A} $, first we perform normalizations of $ J_{u}^{1}(z, q) $ and $ J_{u}^{2}(z, q) $ as:

Similarly

where

Now clearly, the functions $ f_{u}^{1}(z, q) $ and $ f_{u}^{2}(z, q)\in $ $ \mathcal{A}. $

Now, by using the above idea of convolution and normalized Jackson and Hahn-Exton $ q $-Bessel functions, we introduce a new operators $ B_{u}^{q} $ and $ B_{u, 1}^{q} $ as follows:

and

where

and

From the definition (1.11) and (1.12), it can easy to verify that

and

Finally Noor et al. introduced $ q $-Bernardi integral operator ^[28], which is defined by

Remark 4. For $ q\rightarrow 1- $, then $ L_{\lambda }^{q} = L $, introduced by Bernardi in ^[2].

2.   Preliminary results

Here we gave the generalization of two lemmas which was introduced in ^[3,27].

Lemma 1. Let $ h(z) $ be an analytic and convex univalent in $ E $ with

If $ p(z) $ is analytic in $ E $ and $ p(0) = 1, $ then

then

Proof. Suppose that $ h(z) $ is analytic and convex univalent in $ E $ and $ p(z) $ is analytic in $ E $. Letting $ q\rightarrow 1-, $ in (2.1), we have

then by Lemma in ^[26], we have

Lemma 2. Let an analytic functions $ p(z) $ and $ g(z) $ in open unit disk $ E $ with

suppose that $ h(z) $ be convex functions in $ E $ and let $ U\geq 0, $ then

then

Proof. Suppose that $ h(z) $ is convex in the open unit disk $ E $. Let $ p(z) $ and $ g(z) $ is analytic in $ E $ with $ \Re p(z) > 0 $ and $ h(0) = g(0) $. Letting $ q\rightarrow 1-, $ in (2.2), we have

then by Lemma in ^[27], we have

3.   Main results

Theorem 1. Let $ h(z) $ be convex univalent in $ E $ with $ \Re(h(z)) > 0 $ and $ h(0) = 1 $. If a function $ f\in \mathcal{A} $ satisfies the condition

then

Proof. Let

where $ p $ is an analytic function in $ E $ with $ p(0) = 1. $ By using (1.13) into (3.1), we have

Differentiating logarithmically with respect to $ z $, we have

By using Lemma $ 1, $ we get required result.

By taking $ q\rightarrow 1-, $ in Theorem 1, then we have the following result.

Corollary 1. Let $ h(z) $ be convex univalent in $ E $ with $ \Re(h(z) > 0 $ and $ h(0) = 1 $. If a function $ f\in \mathcal{A} $ satisfies the condition

then

Theorem 2. Let $ f\in \mathcal{A}. $ If $ B_{u}^{q}f(z)\in k-\mathcal{ST}_{q}, $ then $ B_{u+1}^{q}f(z)\in k-\mathcal{ST}_{q}. $

Proof. Let

From (1.13), we have

Differentiating logarithmically with respect to $ z $, we have

Since $ p_{k, q}\left(z\right) $ is convex univalent in $ E $ given by (1.8) and

The proof of the theorem 3.2 follows by Theorem 1 and condition (1.9).

For $ q\rightarrow 1-, $ in Theorem 2, then we have the following result.

Corollary 2. Let $ f\in \mathcal{A}. $ If $ B_{u}f(z)\in k-\mathcal{ST}, $ then $ B_{u+1}f(z)\in k-\mathcal{ST}. $

Theorem 3. Let $ f\in \mathcal{A}. $ If $ B_{u}^{q}f(z)\in k-\mathcal{UCV}_{q}, $ then $ B_{u+1}^{q}f(z)\in k-\mathcal{UCV}_{q}. $

Proof. By virtue of (1.10), and Theorem 2, we get

Hence Theorem 3 is complete.

For $ q\rightarrow 1-, $ in Theorem 3, then we have the following result.

Corollary 3. Let $ f\in \mathcal{A}. $ If $ B_{u}f(z)\in k-\mathcal{UCV}, $ then $ B_{u+1}f(z)\in k-\mathcal{UCV}. $

Theorem 4. Let $ f\in \mathcal{A}. $ If $ B_{u}^{q}f(z)\in k-\mathcal{UCC}_{q}, $ then $ B_{u+1}^{q}f(z)\in k-\mathcal{UCC}_{q}. $

Proof. Since

then

Letting

and

We see that $ h(z), H(z)\in \mathcal{A} $, in $ E $ with $ h(0) = H(0) = 1. $ By using Theorem 2, we have

and

Also note that

Differentiating both sides of (3.3), we obtain

By using the identity (1.13), we get

From (3.2), (3.4), and (3.5), we conclude that

On letting $ U = 0 $ and $ B(z) = \frac{1}{H(z)+\frac{\left[ u\right] _{q}}{q^{u}}}, $ we have

Apply Lemma 2, we have

where $ p_{k, q}\left(z\right) $ given by (1.8). Hence Theorem 4 is complete.

We can prove Theorem 5 by using a similar argument of Theorem 4

Theorem 5. Let $ f\in \mathcal{A}. $ If $ B_{u}^{q}f(z)\in k-\mathcal{UQC}_{q}, $ then $ B_{u+1}^{q}f(z)\in k-\mathcal{UQC}_{q}. $

Now in Theorem 6, we study the closure properties of the $ q $-Bernardi integral operator $ L_{\lambda }^{q}. $

Theorem 6. Let $ f\in \mathcal{A} $ and $ \lambda > -\left(\frac{2k}{2k+1+q} \right). $ If $ B_{u}^{q}f(z)\in k-\mathcal{ST}_{q}, $ then $ L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{ST}_{q}. $

Proof. From the definition of $ L_{\lambda }^{q}f(z) $ and the linearity of the operator $ B_{u}^{q}, $ we have

Substituting $ p(z) = \frac{zD_{q}\left(B_{u}^{q}L_{\lambda }^{q}f(z)\right) }{ B_{u}^{q}L_{\lambda }^{q}f(z)} $ in $ \left(3.6\right) $, we have

Differentiating (3.7) with respect to $ z $, we have

By Lemma 1, $ p(z)\prec $ $ p_{k, q}\left(z\right) $, since $ \Re \left(p_{k, q}\left(z\right) +\lambda \right) > 0. $ This completes the proof of Theorem 6.

By a similar argument we can prove Theorem 7 as below.

Theorem 7. Let $ f\in \mathcal{A} $ and $ \lambda > -\left(\frac{2k}{2k+1+q} \right). $ If $ B_{u}^{q}f(z)\in k-\mathcal{U}\mathcal{CV}_{q}, $ then $ L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{U}\mathcal{CV} _{q}. $

Theorem 8. Let $ f\in \mathcal{A} $ and $ \lambda > -\left(\frac{2k}{2k+1+q} \right). $ If $ B_{u}^{q}f(z)\in k-\mathcal{UCC}_{q}, $ then $ L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{UCC}_{q}. $

Proof. By definition, there exists a function

so that

Now from (3.6), we have

Since $ B_{u}^{q}g(z)\in k-\mathcal{ST}_{q}, $ by Theorem 6, we have $ L_{\lambda }^{q}\left(B_{u}^{q}g(z)\right) \in k-\mathcal{ST}_{q}. $ Taking

We see that $ H(z)\in \mathcal{A} $ in $ E $ with $ H(0) = 1, $ and

Now for

Thus we obtain

Differentiating both sides of (3.10), we obtain

Therefore from (3.9) and (3.11), we obtain

This in conjunction with (3.8) leads to

On letting $ U = 0 $ and $ B(z) = \frac{1}{H(z)+\lambda }, $ we have

Apply Lemma 2, we have

where $ p_{k, q}\left(z\right) $ given by (1.8). Hence Theorem 8 is complete.

We can prove Theorem 9 by using a similar argument of Theorem 8.

Theorem 9. Let $ f\in \mathcal{A} $ and $ \lambda > -\left(\frac{2k}{2k+1+q} \right). $ If $ B_{u}^{q}f(z)\in k-\mathcal{UQC}_{q}, $ then $ L_{\lambda }^{q}\left(B_{u}^{q}f(z)\right) \in k-\mathcal{UQC}_{q}. $

4.   Conclusions

Our present investigation is motivated by the well-established potential for the usages of the basic (or $ q $-) calculus and the fractional basic (or $ q $ -) calculus in Geometric Function Theory as described in a recently-published survey-cum-expository review article by Srivastava ^[38]. We have studied new family of analytic functions involving the Jackson and Hahn-Exton $ q $-Bessel functions and investigate their inclusion relationships and certain integral preserving properties bounded by generalized conic domain $ \Omega _{k, q} $. Also we discussed some applications of our main results by using the $ q $-Bernardi integral operator. The convolution operator $ B_{u, 1}^{q} $, which are defined by (1.12) will indeed apply to any attempt to produce the rather straightforward results which we have presented in this paper.

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, [38, p. 328]).

Moreover, in this recently-published survey-cum-expository review article by Srivastava ^[38], 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, [38, p. 340]). This observation by Srivastava ^[38] 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

The third author is partially supported by Universiti Kebangsaan Malaysia grant (GUP-2019-032).

Conflicts of interest

The authors declare that they have no competing interests.