Abstract
In this study, we familiarise a novel class of Janowski-type star-like functions of complex order with regard to -symmetric points based on quantum calculus by subordinating with pedal-shaped regions. We found integral representation theorem and conditions for starlikeness. Furthermore, with regard to -symmetric points, we successfully obtained the coefficient bounds for functions in the newly specified class. We also quantified few applications as special cases which are new (or known).
1. Definitions and Preliminaries
The set of all analytic functions constructed on the unit disc is symbolised by . Also, indicates the subclass of that has a Taylor series representation:
The family of functions that are univalent in is represented by . This is well established that if , assume by (1), is in , then ( is a positive integer) is consequently in .
Definition 1. (see [1], Definition 3). Assume is a positive integer. A domain is known to be -fold symmetric if a rotation of about the origin through an angle carries onto itself. For , a function is said to be -fold symmetric if and only if for each in represents the family including all -fold symmetric functions.
The concept of -symmetrical function was protracted to so-called -symmetrical function by Liczberski and Połubiński in [2]. To be specific, a function is reported for being -symmetrical ifwhere is a fixed integer, and . The family of -symmetrical functions will indeed be indicated by . We believe that , , and are quite well groups of odd functions, even functions, and -symmetrical functions. Consider the subsequent equivalence demarcate as well
It is evident that is a linear operator from into . If is an integer, then the subsequent assumptions result directly from (4):
Let the function provided by (1) and of the form , the Hadamard product (or convolution) of these two functions is indicated by
Using Hadamard product, various authors studied the univalent function theory in dual with the theory of special functions, see [3–5] and references provided therein. Throughout this whole article, we will assume that , , andwhere
From (7), we, thus, have
The investigation of q-calculus (q stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [6, 7] was among the key contributors of all the scientists who introduced and developed the q-calculus theory. Just like q-calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. Though it is the first article in which a link was established between certain geometric nature of the analytic function and the q-derivative operator and the usage of q-calculus in function theory, a solid and comprehensive foundation is given in [8] by Srivastava. After this development, many researchers introduced and studied some useful operators in q-analog with the applications of convolution concepts. For example, Kanas and Raducanu [9] established the q-differential operator and then examined the behavior of this operator in function theory. For more applications of this operator, see [10, 11].
For assumed by (1) and , the Jackson’s -derivative operator or -difference operator for is specified under (see [12–14])
From (10), if is assumed as in (1), we can effortlessly see thatfor , provided the -integer number is represented byand take into consideration . During our study, we let signify
The -Jackson integral is defined by (see [6])
If the -series converges, further witness thatwhere the second equality grasps if is continuous at .
Let the classes of star-like functions of order and convex functions of order are symbolised by and , respectively. In , we categorize the collection of functions with and . The functions in the class are not univalent.
With , let be analytic. The function is said to be subordinate to in if the Schwarz function exists in such that and , as shown through . Whenever is univalent in , consequently the subordination is identical to and .
Using the concept of subordination for holomorphic functions, Ma and Minda [15] proposed the classes:where with maps onto a region star-like with respect to 1 and symmetric with respect to the real axis. By making a choice to map unit disc on to some specific regions such as cardioid, parabolas, lemniscate of Bernoulli, and booth lemniscate in the right-half of the complex plane, various interesting subclasses of star-like and convex functions could be gained well.
Lots of fascinating subclasses of star-like and convex functions may be constructed by using to map unit disc on to particular areas such as cardioid, parabolas, lemniscate of Bernoulli, and booth lemniscate on the right-half of the complex plane.
For arbitrary fixed numbers , , we express through the family of functions analytic in the unit disc and if and only ifwhere is the Schwarz function. Geometrically, if and only if and lies inside an open disc centred with center on the real axis having radius with diameter end points . On observing that for , we have if and only if for some
For detailed study on the class of Janowski functions, we refer [16]. The class of Janowski star-like functions and Janowski convex functions is defined as follows:
Inspired by the theory familiarized by Sakaguchi [17], and the study on analytic functions with respect to -symmetrical points by various authors (see [18–22]), under this article, we formulate new subclasses listed in Definition 2.
Definition 2. For , , and be defined as in (7). We say that if satisfies the subordination condition:where and is given by
Remark 1. Here, we list few exceptional cases of the defined class .(1)If we let , , and , then [19] and [19](2)Fixing , , and , then reduces to the class ([18], Definition 5)
For completeness, we will now define -analogue of the as follows.
Definition 3. For , and be defined as in (7). We say that if holds the subordination condition:where and is defined as in (21).
By letting in , we havewhere , is analytic in , and , .
Remark 2. The impact of Janowski functions on a particular conic region was initiated by Noor and Malik [23] and was subsequently studied by various authors (see [11, 24, 25] and references provided therein).
2. Inclusion Relationships and Integral Representations of the Classes and
Let us begin with the following.
Theorem 1. Let . If , then
Proof. From the definition of and (18), we have
Replacing by in (25), then for all , we have
Suppose in (27), respectively, and summing them, we arrive ator equivalently,
Hence the proof.
Now, by using the following two equivalent forms (see ([14], page 3)) of product rule of the -difference operator,we can establish the following result by retracing the steps as in Theorem 1.
Theorem 2. Let , where . If , then
Theorem 3. Let , thenwhere is given by (7), is analytic in , and , .
Proof. Let . In view of (20), we havewhere is analytic in and , . Substituting by in equality (33) and ensuing the steps as in Theorem 1, we getFrom this equality, we getUpon integration, we getor equivalently,This concludes the proof of Theorem 3.
Theorem 4. Let , then we havewhere defined by equality (7), is analytic in , and , .
Proof. Let . In sight of Theorems 2 and 3, we havewhere is analytic in and , . For and , we obtain (see [10])where is the Jackson -integral, defined as in (14). Integrating the above equality, we getor equivalently,
This concludes the proof of Theorem 3.
By fixing , , , and in Theorem 3, we state the subsequent result.
Corollary 1 (see ([19], Theorems 3 and 4)). Let be assumed as in (4).(i)If , then(ii)If , thenwhere is analytic in and , .
Corollary 2. Let be assumed as in (4). If , thenwhere is analytic in and , .
3. Coefficient Inequalities for and
The coefficient estimate of the defined function classes is determined in this section.
Theorem 5. Let and ’s be real. If , then for ,
Proof. By the definition of , we havewhere and satisfies the condition .
Equivalently, (47) can be rewritten asOn equating the coefficient of , we getFrom Lemma 6 of [26], we have , . On computation, we haveTaking , in (50), we getOn substituting in (46), we can see the hypothesis is true for . Now, taking in (50), we getIf we let , in (46), we haveHence, the hypothesis of the theorem is true for . Now, let us suppose (46) is valid for . On using triangle inequality in (46), we getBy induction hypothesis, we haveFrom the above inequality, we havewhich implies that inequality (46) is true for . Hence the proof of the theorem.
Theorem 6. Let for all and be real. If , then for ,
Proof. From the definition of , we haveEquating the coefficient of and retracing the steps as in Theorem 5, we getNow, by repeating the processes in Theorem 5, we acquire the required outcome. □
If we let in Theorem 6, we have the following.
Corollary 3. Let and be real, then for ,
If we let , , , and in Theorem 5, we have the following.
Corollary 4. (see [18], Theorem 2) Let and be real, then for ,
4. Conclusion
Very few studies have been showed on analytic functions with regard to -symmetric points. Since we have articulated the problem differently so as to deviate from the similar studies, only few special cases could be discussed. Furthermore, by swapping the ordinary differentiation with quantum differentiation, we have tried at the discretization of some of the familiar findings.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.