Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions

1.   Introduction and definitions

We denote by $\mathcal{A}(p)$, the class of all functions having the following form:

which are analytic and multivalent ($p$-valent) in the open unit disk

It should be noted that

Furthermore, we denote by $\mathcal{S}\subset \mathcal{A}$ the class of univalent functions in $\mathbb{U}.$

The function class $\mathcal{S}^{\ast }(p)$ of $p$-valently starlike functions in $\mathbb{U}$ consists of functions $f\in \mathcal{A}(p)$ along with the following condition:

It is easily seen that

where, by $\mathcal{S}^{\ast}$, we mean the class of starlike functions with respect to the origin.

Next, by the notation ${\mathcal{C}}(p)$, we mean the class of $p$-valently convex functions which have the functions $f\in \mathcal{A}(p)$ that satisfy each of the following conditions:

and

It should be noted that

where, by ${\mathcal{C}}$, we mean the well-known class of convex functions in $\mathbb{U}$.

For some recent investigations about analytic and multivalent ($p$-valent) functions, we may refer to ^[16,33].

Also we let $\mathcal{P}$ denote the class of Carathéodory functions $\psi$, which are analytic in the open unit disk $\mathbb{U}$ and normalized by

such that

Definition 1.1. For two analytic functions $f, g\in\mathbb{U},$ the function $f$ is said to be subordinate to the function $g$ and written as follows:

if there exists a Schwarz function $w$, which is analytic in $\mathbb{U}$ with

such that

Moreover, if the function $g$ is univalent in $\mathbb{U}$, then it follows that

Definition 1.2. A function $h$ with $h\left(0\right) = 1$ is said to belong to the class $\mathcal{P}\left[A, B\right]$ if and only if

The analytic functions class $\mathcal{P}\left[A, B\right]$ was introduced by Janowski ^[14], he showed that $h\left(z\right) \in \mathcal{P} \left[A, B\right]$ if and only if there exist a function $\psi \in \mathcal{ P}$ such that

Definition 1.3. A function $f\in \mathcal{A}$ is said to belong to the class $\mathcal{S}^{\ast}\left[A, B\right]$ if and only if

We now recall some concept details and definitions of the $q$-difference calculus which will play vital role in our presentation. Throughout this article it should be understood that, unless otherwise stated, we presume that $0 < q < 1$ and

Definition 1.4. Let $q\in \left(0, 1\right)$ and define the $q$-number $\left[\lambda\right]_{q}$ by

Definition 1.5. Let $q\in \left(0, 1\right)$ and define the $q$-factorial $\left[n\right]_{q}!$ by

Definition 1.6. (see ^[12] and ^[13]) The $q$-derivative (or the $q$-difference) operator $D_{q}$ of a function $f$ is defined, in a given subset of $\mathbb{C},$ by

provided that $f^{\prime}\left(0\right)$ exists.

From Definition 1.6, we can observe that

for a differentiable function $f$ in a given subset of $\mathbb{C}$. It is also readily seen from $\left(1.1\right)$ and $\left(1.6\right)$ that

where $\left(D_{q}^{\left(p\right)}f\right)\left(z\right)$ is the $p$-th order $q$-derivative of $f\left(z\right).$

Recently, the study of the $q$-calculus has fascinated the intensive devotion of researchers. The great concentration is because of its advantages in many areas of mathematics and physics. The significance of the $q$-derivative operator $D_{q}$ is quite obvious by its applications in the study of several subclasses of analytic functions. Initially, in the year 1990, Ismail et al. ^[11] 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 details, ^[28]). After that, remarkable studies have been done by numerous mathematicians, which offer a momentous part in the advancement of Geometric Function Theory. In particular, Srivastava et al. ^[32] also considered some function classes of $q$-starlike functions related with conic region. Moreover, Srivastava et al. (see, for example, ^{[26,31,35,36]}) 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, a recently-published survey-cum-expository review article by Srivastava ^[29] is potentially useful for researchers and scholars working on these topics. In this survey-cum-expository review article ^[29], 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 ^{[3,4,15,18,19,21,22,23,24,30,34]} see also (^[2,7,8,17]).

In this paper, we propose to generalize the work of Srivastava et al. ^[31]. By applying higher-order $q$-derivative operator, we first define a new general version of the definition presented in ^[31]. We then derive some coefficient inequalities and a sufficient condition for the general function class which we introduce here. We also indicate a number of other related works on this subject.

Definition 1.7. (see ^[11]) A function $f\in \mathcal{A}$ is said to belong to the class $\mathcal{S}_{q}^{\ast}$ if

and

It is readily observed that, as $q\rightarrow 1-$, the closed disk

becomes the right-half plane and the class $\mathcal{S}_{q}^{\ast}$ of $q$-starlike functions reduces to the familiar class $\mathcal{S}^{\ast}$. Equivalently, by using the principle of subordination between analytic functions, we can rewrite the conditions in (1.10) and (1.11) as follows (see ^[37]):

Remark 1.8. For functions $f$ in ${\mathcal{A}},$ the Alexander theorem ^[6] was used by Baricz and Swaminathan ^[5] for defining the class ${\mathcal{C}}_{q}$ of $q$-convex functions in the usual way as follows:

Now, making use of the principle of subordination between analytic functions and the above-mentioned $q$-calculus, we have the following definition.

Definition 1.9. A function $f\in {\mathcal{A}}$ is said to belong to the class ${\mathcal{S}}_{q}^{\ast}\left[p, A, B\right]$ if and only if

which, by using the subordination principle, can be written as follows:

where

and $\left(D_{q}^{\left(p\right)}f\right) \left(z\right)$ is the $p$-th order $q$-derivative of $f\left(z\right).$

Remark 1.10. First of all, it is easy to see that

where ${\mathcal{S}}^{\ast}\left[A, B\right]$ is the function class introduced and studied by Janowski ^[14]. Secondly, we have

where ${\mathcal{S}}_{q}^{\ast}\left[A, B\right]$ is function class introduced and studied by Srivastava et al. ^[31]. Thirdly, we can see that

where ${\mathcal{S}}_{q}^{\ast}$ is the class of $q$-starlike functions, which is already given in Definition 1.7. Furthermore, in ${\mathcal{S}}_{q}^{\ast }$, if we let $q\rightarrow 1-$ then we get the well-known class of starlike functions. Finally, when

the function class ${\mathcal{S}}_{q}^{\ast}\left[p, A, B\right]$ reduces to the function class ${\mathcal{S}}_{q}^{\ast}\left(\alpha \right),$ which was introduced and studied by Agrawal and Sahoo ^[1]. One can also see that

where ${\mathcal{S}}^{\ast}\left(\alpha \right)$ is the function class which was introduced and studied by Silverman (see ^[27]).

By using the idea of the above-mentioned Alexander theorem, the class ${\mathcal{C}}_{q}\left[p, A, B\right]$ can be defined in the following way.

Definition 1.11. Just as in Remark 1.8, by using the idea of the Alexander theorem ^[6], the class of ${\mathcal{C}}_{q}\left[p, A, B\right]$ of $q$-convex functions can be defined by

2.   Preliminaries

Lemma 2.1. (see ^[20]) Let the function $\psi \left(z\right)$ given by

is in the class $\mathcal{P}$ of functions positive real part in $\mathbb{U}$, then, for any complex number $\upsilon,$

When $\upsilon < 0$ or $\upsilon > 1,$ the equality holds true in $\left(2.1\right)$ if and only if $\psi(z)$ is given by

or one of its rotations. If $0 < \upsilon < 1,$ then the equality holds true in $(2.1)$ if and only if

or one of its rotations. If $\upsilon = 0,$ the equality holds true in $\left(2.1\right)$ if and only if

or one of its rotations. If $\upsilon = 1,$ then the equality in $\left(2.1 \right)$ holds true if $\psi(z)$ is a reciprocal of one of the functions such that the equality holds true in the case when $\upsilon = 0.$

Lemma 2.2. (see ^[25]) Let the function $\psi\left(z\right)$ given by

be subordinate to the function $H\left(z\right)$ given by

If $H(z)$ is univalent in $\mathbb{U}$ and $H(\mathbb{U})$ is convex, then

Lemma 2.3. Suppose that the sequence $\left\{a_{k}\right\}_{k = 0}^{\infty}$ is defined by

and

Then

Proof. By virtue of ($2.2$), we easily get

and

Combining (2.3) and (2.4), we obtain

Similarly, we can deduce the following result:

The proof of Lemma 2.3 is evidently completed.

3.   Main results

In this section, we will prove our main results. Throughout our discussion, we assume that

Theorem 3.1. Let the function $f\in {\mathcal{S}}_{q}^{\ast}\left[p, A, B \right]$ be of the form given by $\left(1.1\right)$. Then

where

and

Each of the above results is sharp.

Proof. If $f\in {\mathcal{S}}_{q}^{\ast}\left[p, A, B\right],$ then it follows from $\left(1.12\right)$ that

where the function $\phi\left(z\right)$ is given by $\left(1.13 \right)$.

We now define a function $\psi$ by

It is clear that $\psi \in \mathcal{P}$. This implies that

Thus, by applying $\left(1.12\right)$, we have

with

Now

Thus, if

then we find after some simplification that

Similarly, we can find that

Therefore, we have

and

where

Thus, clearly, we find that

where

with $\kappa_{1}\left(q\right)$ given by $\left(3.3\right)$. By an application of Lemma 2.1 in $\left(3.4\right)$, we get the result as demonstrated by Theorem 3.1.

Remark 3.2. If we put $p = 1$ in Theorem 3.1, we arrive at a result which was already proved by Srivastava et al. ^[31].

If, in Theorem 3.1, we set

and let $q\rightarrow 1-,$ we have the following corollary.

Corollary 3.3. (see [10,Corollary 3]) Let the function $f$ be in the class $\mathcal{S}^{\ast}.$ Then

We next state and prove Theorem 3.4 below.

Theorem 3.4. Let the function $f\in {\mathcal{S}}_{q}^{\ast}\left[p, A, B\right]$ be of the form given by $\left(1.1\right)$. Then

Proof. By definition, for $f\in {\mathcal{S}}_{q}^{\ast}\left[p, A, B\right]$, we have

where

Since

then, by Lemma 2.2, we have

Now, from $\left(3.6\right)$, we have

which implies that

Equating the coefficients of $z^{n+1}$ on both sides, we have

This last equation implies that

By using $\left(3.7\right)$, we find that

Finally, in order to prove the result asserted by Theorem 3.4, we use Lemma 2.3 and so we get

The proof of Theorem 3.4 is now completed.

Remark 3.5. First of all, if we put $p = 1$ in Theorem 3.4, we deduce the result which was already proved by Srivastava et al. ^[31]. Secondly, if we put $p = 1$ and let $q\rightarrow 1-,$ then we get a result which was proved earlier by Janowski ^[14]. Thirdly, if we set

and let $q\longrightarrow 1-$, then Theorem 3.4 yields the following known result proved by Silverman in ^[9].

Corollary 3.6. (see ^[9]) Let the function $f\in \mathcal{A}$ be in the class ${\mathcal{S}}^{\ast}\left(\alpha \right)$. Then$,$ for $n\geqq 2,$

Theorem 3.7. Let the function $f\left(z\right)\in{\mathcal{C}}_{q}\left[p, A, B\right]$ be of the from $\left(1.1\right)$. Then

Proof. The proof of Theorem 3.7 follows immediately by using Theorem 3.4 and Definition $1.11$.

The following equivalent form of Definition 1.9 is potentially useful in further investigation of the function class ${\mathcal{S}}_{q}^{\ast}\left[p, A, B\right]$:

Theorem 3.8. A function $f\in {\mathcal{A}}\left(p\right)$ of the form given by $\left(1.1\right)$ is in the class ${\mathcal{S}}_{q}^{\ast} \left[p, A, B\right]$ if it satisfies the following condition$:$

Proof. Assuming that $\left(3.10\right)$ holds true, it suffices to show that

Indeed we have

The last expression in $\left(3.11\right)$ is bounded above by $\frac{1}{1-q}$ if

which completes the proof of Theorem 3.8.

Remark 3.9. If we put $p = 1$ in Theorem 3.4, we deduce the result which was already proved by Srivastava et al. ^[31].

Remark 3.10. If we set

and let $q\longrightarrow 1-$, then we have the following result proved by Silverman ^[27].

Corollary 3.11. (see ^[27]) A function $f\in {\mathcal{A}}$ of the form $\left(1.1\right)$ with $p = 1$ is in the class ${\mathcal{S}}^{\ast}(\alpha)$ if it satisfies the following condition$:$

Theorem 3.12. A function $f\in {\mathcal{A}(p)}$ of the form $\left(1.1\right)$ is in the class ${\mathcal{C}}_{q}\left[A, B\right]$ if it satisfies the following condition$:$

Proof. The proof of Theorem 3.12 follows easily when we apply Theorem 3.8 in conjunction with Definition $1.11$.

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 ^[29]. Here we have introduced and studied systematically some interesting subclasses of multivalent ($p$-valent) $q$-starlike functions in the open unit disk $\mathbb{U}$. We have also provided relevant connections of the various results, which we have demonstrated in this paper, with those derived in many earlier works cited here.

Acknowledgments

The work here is supported by UKM Grant: GUP-2019-032.

Conflict of interest

The authors declare that they have no competing interests