Citation: Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, H. M. Srivastava, Nazar Khan, Maslina Darus, Bilal Khan. Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions[J]. AIMS Mathematics, 2021, 6(2): 1110-1125. doi: 10.3934/math.2021067
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We denote by , the class of all functions having the following form:
(1.1) |
which are analytic and multivalent (-valent) in the open unit disk
It should be noted that
Furthermore, we denote by the class of univalent functions in
The function class of -valently starlike functions in consists of functions along with the following condition:
(1.2) |
It is easily seen that
where, by , we mean the class of starlike functions with respect to the origin.
Next, by the notation , we mean the class of -valently convex functions which have the functions that satisfy each of the following conditions:
and
(1.3) |
It should be noted that
where, by , we mean the well-known class of convex functions in .
For some recent investigations about analytic and multivalent (-valent) functions, we may refer to [16,33].
Also we let denote the class of Carathéodory functions , which are analytic in the open unit disk and normalized by
(1.4) |
such that
Definition 1.1. For two analytic functions the function is said to be subordinate to the function and written as follows:
if there exists a Schwarz function , which is analytic in with
such that
Moreover, if the function is univalent in , then it follows that
Definition 1.2. A function with is said to belong to the class if and only if
The analytic functions class was introduced by Janowski [14], he showed that if and only if there exist a function such that
Definition 1.3. A function is said to belong to the class if and only if
(1.5) |
We now recall some concept details and definitions of the -difference calculus which will play vital role in our presentation. Throughout this article it should be understood that, unless otherwise stated, we presume that and
Definition 1.4. Let and define the -number by
Definition 1.5. Let and define the -factorial by
Definition 1.6. (see [12] and [13]) The -derivative (or the -difference) operator of a function is defined, in a given subset of by
(1.6) |
provided that exists.
From Definition 1.6, we can observe that
for a differentiable function in a given subset of . It is also readily seen from and that
(1.7) |
(1.8) |
(1.9) |
where is the -th order -derivative of
Recently, the study of the -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 -derivative operator 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 -starlike functions. Nevertheless, a firm foothold of the usage of the -calculus in the context of Geometric Function Theory was effectively established, and the use of the generalized basic (or -) 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 -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 -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 -calculus and the fractional -derivative operators in Geometric Function Theory was systematically investigated. For other recent investigations involving the -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 -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 is said to belong to the class if
(1.10) |
and
(1.11) |
It is readily observed that, as , the closed disk
becomes the right-half plane and the class of -starlike functions reduces to the familiar class . 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 in the Alexander theorem [6] was used by Baricz and Swaminathan [5] for defining the class of -convex functions in the usual way as follows:
Now, making use of the principle of subordination between analytic functions and the above-mentioned -calculus, we have the following definition.
Definition 1.9. A function is said to belong to the class if and only if
which, by using the subordination principle, can be written as follows:
(1.12) |
where
(1.13) |
and is the -th order -derivative of
Remark 1.10. First of all, it is easy to see that
where is the function class introduced and studied by Janowski [14]. Secondly, we have
where is function class introduced and studied by Srivastava et al. [31]. Thirdly, we can see that
where is the class of -starlike functions, which is already given in Definition 1.7. Furthermore, in , if we let then we get the well-known class of starlike functions. Finally, when
the function class reduces to the function class which was introduced and studied by Agrawal and Sahoo [1]. One can also see that
where 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 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 of -convex functions can be defined by
Lemma 2.1. (see [20]) Let the function given by
is in the class of functions positive real part in , then, for any complex number
(2.1) |
When or the equality holds true in if and only if is given by
or one of its rotations. If then the equality holds true in if and only if
or one of its rotations. If the equality holds true in if and only if
or one of its rotations. If then the equality in holds true if is a reciprocal of one of the functions such that the equality holds true in the case when
Lemma 2.2. (see [25]) Let the function given by
be subordinate to the function given by
If is univalent in and is convex, then
Lemma 2.3. Suppose that the sequence is defined by
and
(2.2) |
Then
Proof. By virtue of (), we easily get
(2.3) |
and
(2.4) |
Combining (2.3) and (2.4), we obtain
Similarly, we can deduce the following result:
The proof of Lemma 2.3 is evidently completed.
In this section, we will prove our main results. Throughout our discussion, we assume that
Theorem 3.1. Let the function be of the form given by . Then
where
and
Each of the above results is sharp.
Proof. If then it follows from that
where the function is given by .
We now define a function by
It is clear that . This implies that
Thus, by applying , we have
with
Now
Thus, if
then we find after some simplification that
Similarly, we can find that
Therefore, we have
(3.1) |
and
(3.2) |
where
(3.3) |
Thus, clearly, we find that
(3.4) |
where
with given by . By an application of Lemma 2.1 in , we get the result as demonstrated by Theorem 3.1.
Remark 3.2. If we put 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 we have the following corollary.
Corollary 3.3. (see [10,Corollary 3]) Let the function be in the class Then
We next state and prove Theorem 3.4 below.
Theorem 3.4. Let the function be of the form given by . Then
(3.5) |
Proof. By definition, for , we have
(3.6) |
where
Since
then, by Lemma 2.2, we have
(3.7) |
Now, from , we have
which implies that
Equating the coefficients of on both sides, we have
This last equation implies that
By using , we find that
(3.8) |
Finally, in order to prove the result asserted by Theorem 3.4, we use Lemma 2.3 and so we get
(3.9) |
The proof of Theorem 3.4 is now completed.
Remark 3.5. First of all, if we put in Theorem 3.4, we deduce the result which was already proved by Srivastava et al. [31]. Secondly, if we put and let then we get a result which was proved earlier by Janowski [14]. Thirdly, if we set
and let , then Theorem 3.4 yields the following known result proved by Silverman in [9].
Corollary 3.6. (see [9]) Let the function be in the class . Then for
Theorem 3.7. Let the function be of the from . Then
Proof. The proof of Theorem 3.7 follows immediately by using Theorem 3.4 and Definition .
The following equivalent form of Definition 1.9 is potentially useful in further investigation of the function class :
Theorem 3.8. A function of the form given by is in the class if it satisfies the following condition
(3.10) |
Proof. Assuming that holds true, it suffices to show that
Indeed we have
(3.11) |
The last expression in is bounded above by if
which completes the proof of Theorem 3.8.
Remark 3.9. If we put 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 , then we have the following result proved by Silverman [27].
Corollary 3.11. (see [27]) A function of the form with is in the class if it satisfies the following condition
Theorem 3.12. A function of the form is in the class 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 .
Our present investigation is motivated by the well-established potential for the usages of the basic (or -) calculus and the fractional basic (or -) 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 (-valent) -starlike functions in the open unit disk . 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.
The work here is supported by UKM Grant: GUP-2019-032.
The authors declare that they have no competing interests
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