$q$-Noor integral operator associated with starlike functions and $q$-conic domains

1.   Introduction

The quantum (or $q$-) calculus is an important area of study in the field of traditional mathematical analysis. Quantum calculus is a fascinating area of mathematical science with historical background, as well as a revived focus in the modern era. Quantum calculus is the modern name for the investigation of calculus without notation of limit. The quantum calculus or $q$-calculus began with Jackson in the early twentieth century, but this type of calculus had already been investigated by Euler and Jacobi. Recently $q$-calculus attract researchers for its wide applications in mathematics and related areas, such as number theory, combinatorics, orthogonal polynomials, basic hyper-geometric functions and other sciences. In recent years, the topic of $q$-calculus has attracted the attention of several researchers, and a variety of new results can be found in the papers ^[1,2,3,4] and the references cited therein.

A function $\hat{g}$ is analytic at a point $\xi _{0}$ if $\hat{g}^{^{\prime }}(\xi)$ exists at $\xi _{0}$ as well as in some neighborhood of $\xi _{0}.$ A function $\hat{g}(\xi)$ is analytic in $\mathbb{D}$ if $\hat{g}(\xi)$ is analytic at each point of $\mathbb{D}$. In most of the cases it is much harder to use a random domain, so Riemann mapping theorem allows us to replace $\text{it with open unit disk defined as:}$

An analytic function $\hat{g}$ is univalent in $\mathbb{U}$, if $\hat{g}(\xi _{1}) = \hat{g}(\xi _{2})$ then $\xi _{1} = \xi _{2}$. A function $\hat{g}(\xi)$ is said to be the class $\mathfrak{A}$ if it has a Taylor series of the form

A collection of functions of the form (1.1), which are analytic and univalent in $\mathbb{U}$ are placed in the class $\mathfrak{S.}$ An analytic function $p\left(\xi \right)$ having positive real part i.e., ${ \Re }\left \{ p\left(\xi \right) \right \} > 0$ and $p\left(0\right) = 1$ is placed in class $\mathfrak{P.}$ Or equivalently

The class of normalized convex functions is given by

Similarly, the class of normalized starlike functions with respect to origin is defined as:

for details, see ^[5]. A function $\hat{g}(\xi)\in QC,$ the class of quasi-convex function if and only if there exists $\hat{h}(\xi)\in C$ such that $\Re \left(\frac{(\xi \hat{g}^{^{\prime }}(\xi))^{^{\prime }}}{ \hat{h}^{^{\prime }}(\xi)}\right) > 0.$ In 1952, Kaplan ^[6] introduced the class KC of close-to-convex function. A function is of the form (1.2) is in KC if and only if there exists $\hat{h}(\xi)\in \mathbb{S}^{\ast }$ such that $\Re \left(\frac{\xi \hat{g}^{^{\prime }}(\xi)}{\hat{h}(\xi)}\right) > 0$. Let $\hat{g}(\xi)$ is of the form (1.1) and $\hat{h}(\xi)$ is of the form

Then the Hadamard product(convolution) of $\hat{g}$ and $\hat{h}$ is defined as:

The $q$-derivative of a function $\hat{g}$ belonging to $\mathfrak{A}$ defined as:

for details, see ^[7], where $q\in \left(0, 1\right)$ and $\xi \in \mathbb{U}.$ For $\xi = 0,$ (1.5) can be written as $\hat{g}^{^{\prime }}(0)$ provided that the derivative exist$.$ By using (1.1) and (1.5) the Maclaurin's series representation of $D_{q}\hat{g}$ is given by

It can be noted from (1.5) that

For any non negative integer $t,$ the $q$-number shift factorial is given by

see ^[8]. For $y > 0,$ the $q$-genralized Pochammar symbol is defined as:

For $\mu > -1,$ we defined a function $\mathfrak{F}_{q, 1+\mu }^{-1}(\xi)$ such that

where

The study of operators plays an important role in the geometric function theory. Many differential and integral operators can be written in terms of convolution of certain analytic functions. In ^[8] $q$-analogue of Noor integral operator ${\large \Im }_{q}^{\mu }:\mathfrak{A\rightarrow A}$ is define as:

where

From (1.11) we can easily obtain the following identity

from (1.11). It can be seen that ${\large \Im }_{q}^{0}\hat{g}(\xi) = \xi D_{q}\hat{g}(\xi),$ ${\large \Im }_{q}^{1}\hat{g}(\xi) = \hat{g}(\xi)$ and

From (1.14), we can observe that by applying limit $q\rightarrow 1$, the operator defined in (1.11) reduces to well known Noor integral operator see (^[9,10,11,12]).

In ^[13,14], Kanas and Waniowska introduced the concept of a conic domain ${ \Xi }_{l}$ for $l\geq 0$ as:

This domain merely represent the right half plane for $l = 0$, a hyperbola for $0 < l < 1$, parabola for $l = 1$ and ellipse for $l > 1$. The extremal functions ${ \varpi }_{l}$ for this conic region ${ \Xi }_{l}$ is given by

where $U\left(\xi \right) = \frac{\xi -\sqrt{n}}{1-\sqrt{n}\xi },$ for all $\xi \in \mathbb{U},$ $0 < l < 1$ and $l = \cosh \left[\frac{\pi R^{^{\prime }}\left(n\right) }{4R\left(n\right) }\right]$ where $R\left(n\right)$ is Legendre's complete elliptic integral of first kind and $R^{^{\prime }}\left(n\right)$ is complementary integral of $R\left(n\right)$ for more details, see ^{[13,14,15,16]}. If we take ${ \varpi } _{l}\left(\xi \right) = 1+\delta \left(l\right) \xi +\delta _{1}\left(l\right) \xi ^{2}+\cdots$, then

Let $\delta _{1}\left(l\right) = \delta _{2}\left(l\right) \delta \left(l\right),$ where

Definition 1. ^[17] Let $p$ be a analytic function with $p(0) = 1$. Then $p\in \mathfrak{P} {(} \lambda, M {)}$ if and only if

In ^[17] it was shown that $p\in \mathfrak{P} {(} \lambda, M {)}$ if and only if there exists a function $p\in$ $\mathfrak{P}$ such that

Definition 2. ^[18] A function $\hat{g}\in \mathfrak{A}$ is in the class $k- ST_{q}(N, O)$ if and only if

where

and $k\geq 0,$ $-1\leq O < N\leq 1,$ $L _{1} = q+1$ and $L_{2} = 3-q.$ For $q\rightarrow 1,$ $k- ST_{q}(N, O)$ was discussed in ^[21].

2.   Set of lemmas

Lemma 1. ^[22] Suppose $d\left(\xi \right) = 1+\sum_{t = 1}^{\infty }c_{t}\xi ^{t}\prec 1+\sum_{t = 1}^{\infty }C_{t}\xi ^{t} = \mathbb{H}\left(\xi \right).$ If $\mathbb{H}\left(\mathbb{U}\right)$ is convex and $\mathbb{H}\left(\xi \right) \in \mathfrak{A, }$ then

Lemma 2. ^[18] Suppose $1+\sum_{t = 1}^{\infty }c_{t}\xi ^{t} = d\left(\xi \right) \in k-ST_{q}(N, O),$ then

where $\delta \left(l\right)$ is given by (1.17).

Lemma 3. ^[18] If $d\left(\xi \right) = \xi +\sum_{t = 1}^{\infty }b_{t}\xi ^{t}\in k- ST_{q}(N, O)$ for $\xi \in \mathbb{U}$ and $k\geq 0,$ then

where $\delta \left(l\right)$ is given by (1.17).

Lemma 4. If $d\in \mathbb{S}^{\ast },$ $\mathcal{G\in }$ $\mathfrak{S}$ and $\hat{g}\in C$ then

Where $c\bar{o}\left(\mathcal{G}\left(\mathbb{U}\right) \right)$ is the closed convex hull $\mathcal{G}\left(\mathbb{U}\right).$

Lemma 5. ^[18] A function $\hat{g}\in \mathfrak{A}$ will be in the class $k- ST_{q}(N, O),$ if

Motivated by the work of Mahmood et al. ^[18], Noor and Malik ^[21] and Arif et al. ^[8], we define a new subclasses of Janowski type $q$-starlike functions associated with $q$-conic domains as:

Definition 3. A function $\hat{g}(\xi)\in \mathfrak{A}$ is apparently in the function class $k- ST_{q}(\mu, N, O)$ if and only if

where

and $\ k\geq 0,$ $-1\leq O < N\leq 1,$ $\mu > -1,$ $L_{1} = 1+q$ and $L_{2} = 3-q.$

It is noted that, for $\mu = 1,$ the function class $k-ST _{q}(\mu, N, O)$ reduces to well known class $k-ST_{q}(N, O).$ Also $0-ST _{q}(1, N, O) = S^{\ast }(N, O)$ introduced by Srivastava et al. ^[19,23], further for $q\rightarrow 1,$ $k-ST_{q\rightarrow 1^{-}}(N, O) = k-ST(N, O)$ this class was studied by Noor and Malik ^[21] also see ^[20,26].

3.   Main results

Theorem 1. A function $\hat{g}(\xi)\in \mathfrak{A}$ and of the form (1.1) is in the class $k-ST_{q}(\mu, N, O)$, if it fulfill the following restriction

where $\Lambda _{t} = \left \{ 2(k+1)L_{2}q\left[t-1, q\right] +\left \vert \left(OL_{1}+L_{2}\right) \left[t, q\right] -\left(NL_{1}+L_{2}\right) \right \vert \right \} \psi _{t-1}.$

Proof. Assume that (1.20) hold, then it is suffices to prove that

We consider,

The last inequalities is bounded above by $1$ if

which reduces to

This complete the proof.

For $\mu = 1$, we have the following corollary.

Corollary 1. ^[18] A function $\hat{g}(\xi)\in \mathfrak{A}$ and of the type (1.1) is considered to be in the function class $k-ST _{q}(1, N, O)$, if it fulfill the following criterion

If $q\rightarrow 1^{-}$, then corollary (3.2) reduces to:

Corollary 2. ^[21] A function $\hat{g}(\xi)\in \mathfrak{A}$ of the form (1.1) is considered to be in the class $k-ST(1, N, O),$ if it fulfill the following criterion

Further if we take $N = 1$, $O = -1$ then we have $k-ST(1, -1)$.

Corollary 3. ^[14] A function $\hat{g}(\xi)\in \mathfrak{A}$ of the form (1.1) is considered to be in the function class $k-ST(1, 1, -1),$ if it fulfill the following criterion

3.1.   Coefficient bound for the class $k-ST_{q}(\mu, N, O)$

Theorem 2. Let a function $\hat{g}\in k-ST_{q}(\mu, N, O)$ is of the form (1.1). Then

This result is sharp.

Proof. Since $\hat{g}\in k-ST_{q}(\mu, N, O),$ so let

where

If

then

Let $\wp \left(\xi \right) = 1+\sum_{t = 1}^{\infty }c_{t}\xi ^{t},$ then by Lemma 2.1 and relation (3.3), we get

Now from (3.2), we have

which implies that

By comparing coefficient of $\xi ^{t},$ we obtain

which yields

By using (3.4), above inequalities can be written as:

Next we need to show that

To derive (3.6), we will utilize the principle of mathematical induction.

For $t = 2,$ (3.5) become

Which shows that (3.1) is true for $t = 2$.

For $t = 3,$ (3.5) give us

This shows that (3.1) is true for $t = 3$. Now suppose that (3.6), for $t = m$ that is

On the other hand from (3.1), we have

Using induction hypothesis on (3.6), we have

As

Thus

which shows that inequality (3.8), is true for $t = m+1$ and hence we obtained the required result.

For $\mu = 1$, we have the following corollary.

Corollary 4. ^[18] Consider a function $\hat{g}\in k-ST_{q}(N, O)$ is of the form (1.1), then

If $k = 0$, then corollary (3.6) reduces to:

Corollary 5. ^[23] Consider a function $\hat{g}\in ST_{q}^{\ast }(N, O)$ is of the type (1.1), then

Further if we take $q\rightarrow 1^{-}$ then we have $ST (N, O)$.

Corollary 6. ^[21] Consider a function $\hat{g}\in ST (N, O)$ is of the type (1.1), then

If we take $N = 1$ and $O = -1$ then we have $k- ST _{q}(N, O).$

Corollary 7. ^[14] Consider a function $\hat{g}\in ST$ is of the type (1.1), then

Further if we take $N = 1-2\alpha$, $O = -1,$ $k = 0$ and$0\leq \alpha < 1$ then we have $S^{\ast }\left(\alpha \right).$

Corollary 8. ^[24] Let the function $\hat{g}\in S^{\ast }\left(\alpha \right)$ be of the form (1.1), then

Next we show that the class $k-P_{q}(\mu, N, O)$ is closed under convolution with convex function.

Theorem 3. If $\hat{g}\in k-P_{q}(\mu, N, O)$ and ${ \chi }\in { C, }$ then $\hat{g}\ast { \chi }\in k -P_{q}(\mu, N, O).$

Proof. We want to prove that

It can be easily seen that

where, $\frac{\xi D_{q}\left({\large \Im }_{q}^{\mu }\hat{g}(\xi)\right) }{ {\large \Im }_{q}^{\mu }\hat{g}(\xi)} = \Psi \left(\xi \right) \in k -P_{q}(\mu, N, O).$ By using Lemma 4, we obtain the required result.

For $\mu = 1$, we have the following corollary.

Corollary 9. ^[25] If $\hat{g}\in k-P_{q}(N, O)$ and ${ \chi }\in { C, }$ then $\hat{g}\ast { \chi \in } k-P_{q}(N, O).$

Theorem 4. If $\hat{g}\in k- ST _{q}(\mu, N, O)$ and ${ \Phi }\in { C, }$ then $\hat{g}\ast { \Phi }\in k - ST _{q}(\mu, N, O).$

Proof. We want to prove that

It can be easily seen that

where, $\frac{\xi D_{q}\left({\large \Im }_{q}^{\mu }\hat{g}(\xi)\right) }{ {\large \Im }_{q}^{\mu }\hat{g}(\xi)} = \Psi \left(\xi \right) \in k -P_{q}(\mu, N, O),$ by applying Lemma 4, we obtain the required result.

For $\mu = 1$, we have the following corollary.

Corollary 10. ^[25] If $\hat{g}\in k- ST _{q}(N, O)$ and ${ \Phi }\in { C, }$ then $\hat{g}\ast { \Phi }\in k- ST _{q}(N, O).$

3.2.   Linear combination

Linear combination for our defined classes are defined as following.

Theorem 5. Let $\hat{g}_{i}\in k- ST _{q}(\mu, N, O)$ and have the form

Then

Proof. By using (1.20), one can write

Therefore

however

3.3.   Weighted means

Theorem 6. If $\hat{g}$ and $\hat{h}$ belongs to $k- ST _{q}(\mu, N, O)$ where,

Proof. As

To prove that $h_{W}\left(\xi \right) \in$ $k- ST _{q}(\mu, N, O),$ we need to show

For this, consider

Corollary 11. If we take $\mu = 1,$ if $\hat{g}$ and $\hat{h}$ belongs to $k- ST _{q}(1, N, O) = k- ST _{q}(N, O),$ then their weighted mean $h_{W}$ is also in $k- ST _{q}(N, O).$

Further for $q\rightarrow 1^{-},$ then their weighted mean $h_{W}$ is also in $k- ST (N, O).$ Where,

3.4.   Arithmetic means

Theorem 7. Let $\hat{g}_{i}\in k- ST _{q}(\mu, N, O)$ where $i = 1, 2, \cdots, \nu$ then the arithmetic mean

also belongs to the class $k- ST _{q}(\mu, N, O).$

Proof. As $A_{M}\left(\xi \right) = \frac{1}{\nu }\sum_{i = 1}^{\nu }\hat{g} _{i}\left(\xi \right)$ and $\hat{g}_{i}\left(\xi \right) = \xi +\sum_{t = 2}^{\infty }a_{t, \; i}\xi ^{t}$ then we have

Since $\hat{g}_{i}\in k- ST _{q}(\mu, N, O)$ for every $i = 1, 2, \cdots, \nu,$ so by using (1.20) and (3.9), we get

i.e.

This complete the proof.

Corollary 12. If we take $\mu = 1,$ $\hat{g}_{i}\in k- ST _{q}(N, O)$ with $i = 1, 2, \cdots, \nu$ then the arithmetic mean

this belongs to the class $k- ST _{q}(N, O).$

Further, for $\mu = 1, \hat{g}_{i}\in k- ST _{q\rightarrow 1}(1, N, O) = k- ST (N, O)$ where $i = 1, 2, \cdots, \nu$ then the arithmetic mean $A_{M}\left(\xi \right)$ also belongs to the class $k- ST (N, O).$

3.5.   Radii of starlikeness

Theorem 8. Let $\hat{g}\in k- ST _{q}(\mu, N, O),$ then $\hat{g}$ will belongs to the family ${ S}^{\ast }\left(\alpha \right)$ called starlike functions of order $\alpha \left(0\leq \alpha < 1\right)$ for $\left \vert \xi \right \vert < \mathfrak{r}_{1},$ where

Proof. Let $\hat{g}\in k- ST _{q}(\mu, N, O).$ To prove $\hat{g}\in { S}^{\ast }\left(\alpha \right),$ we need to show

Using values of $\hat{g}\left(\xi \right)$ along with some staightforward calculations, we have

Since $\hat{g}\in k- ST _{q}(\mu, N, O)$, so from (1.20), we can easily obtain

The inequality (3.10), holds if the following relation are true

which implies that

Which completes the proof.

Corollary 13. If we take $\mu = 1,$ if $\hat{g}\in k- ST _{q}(N, O),$ then $\left \vert \xi \right \vert < \mathfrak{r}_{2},$ where

Further, $\hat{g}\in k- ST _{q\rightarrow 1}(1, N, O) = k- ST (N, O),$ $\hat{g}$ then $\left \vert \xi \right \vert < \mathfrak{r}_{3},$ where

3.6.   Growth and distortion theorems

Theorem 9. If $\hat{g}\in k- ST _{q}(\mu, N, O)$ has the form (1.1), then

where

Proof. Consider

This implies

Similarly,

It can be easily observed that

By using (1.20), we obtain

which gives

now using this relation in (3.11) and (3.12), we get

As required.

Corollary 14. If we take $\mu = 1,$ and $\hat{g}\in k- ST _{q}(N, O),$ has the form (1.1), then

where

Further, $\hat{g}\in k- ST _{q\rightarrow 1}(1, N, O) = k- ST (N, O),$ has the form (1.1), then

where,

Corollary 15. If $\hat{g}\in k- ST _{q}(N, O),$ has the form (1.1), then

where,

Further, $\hat{g}\in k- ST _{q\rightarrow 1}(1, N, O) = k- ST (N, O),$ has the form (1.1), then

where,

4.   Conclusions

By using $q$-analogue of Noor integral operator, we studied various properties such as necessary and sufficient conditions, coefficient bounds, convolution properties, linear combinations, weighted means, arithmetic means, distortion and covering theorems and radii of starlikenss, for a newly define class of analytic functions in conic regions. We also pointed out many special cases in the form of corollaries by specializing the parameters.

Acknowledgments

The work here is supported by FRGS/1/2019/STG06/UKM/01/1.

Conflict of interest

The authors declare that there is no conflict of interests in this paper.