Abstract
A class of p-valent functions of complex order is defined with the primary motive of unifying the concept of prestarlike functions with various other classes of multivalent functions. Interesting properties such as inclusion relations, integral representation, coefficient estimates and the solution to the Fekete–Szegő problem are obtained for the defined function class. Further, we extended the results using quantum calculus. Several consequences of our main results are pointed out.
1. Introduction
Let denote the class of all analytic functions of the form
and let . Further, let denote the class of functions analytic in and satisfy for all in . Aouf ([], Equation (1.4)) defined the class if and only if
where is the Schwartz function. The class is an extension of the famous Janowski class of functions []. Further, it was proved, in [] (Theorem 5), that, if is in , then
Throughout this paper, we let and , which has a power series expansion of the form
Letting for some in (2), we have the following relation (see [], Equation (1.6)):
From (4), we see that
The first and second terms of infinite series (1) are convergent to p and , provided that . Hence, (4) can be rewritten as
For the functions and that are analytic in , we say that the function is subordinate to if there exits a function w, analytic in with and , , such that . We denote this subordination by or . In particular, if the function is univalent in , the above subordination is equivalent to (see [,]) and . For the functions of the form (1) and , the Hadamard product (or convolution) of and is defined by .
We let and denote the familiar subclasses of consisting of functions which are respectively p-valent starlike of order and p-valent convex of order in . In addition, we let denote the class of p-valent starlike functions of order satisfying the condition
The extremal function for the class is given by
with , . A function is said to be p-valent prestarlike of order if
We denote by the class of all p-valent prestarlike functions of order . The class of univalent prestarlike functions was introduced by Ruscheweyh ([], Section 2). The so-called class of prestarlike functions was further extended and studied by various authors; refer to [,,].
In the present section, we define a new differential operator motivated by the concept of convex combination of analytic functions and we use the operator to define presumably a new class of multivalent functions of complex order with respect to symmetric points. We focus on the coefficient estimates, inclusion results and solution to the Fekete–Szegő problem of the defined function class. In the subsequent section, we have extended the study using quantum calculus.
For , we now define following operator by
If , then, from (8) and (9), we may easily deduce that
where and . For , is a special case of the operator (see [], Equation (5)). If we let and in (10), then reduces to , the well-known Sălăgean differential operator [].
Unless otherwise mentioned,
Definition 1.
For , we say that the function χ belongs to the class if it satisfies the subordination condition
where “≺” denotes subordination and is defined as in (3).
Remark 1.
In the literature, for , numerous study of Janowski starlike and convex functions of complex order with respect to symmetric points can be found. Here, we give some recent studies as special cases of .
- 1.
- If we let , , and , the class reduces toThe class was recently introduced by Arif et al. in [].
- 2.
- If we let , , , and in Definition 1, then reduces toLetting in , we obtain the class of all prestarlike functions of order with respect to symmetric points.
For studies pertaining to the classes of functions with respect to symmetric points, refer to [] and references provided therein.
2. Inclusion Relationship and Initial Coefficient Estimates
Throughout this section, we let
We use the following results to obtain the solution of the Fekete–Szegő problem for the functions that belong to those classes we define in the first section.
Lemma 1
([], p. 41). If , then for all and the inequality is sharp for , .
Lemma 2
([]). If and v is complex number, then
and the result is sharp for the functions
If , then, by Definition 1, we have
Replacing by in (13),
Subtracting (13) and (14), we have the following after-simplification:
with . Integrating the equality (15), we obtain
or, equivalently,
On summarizing the above discussion, we have the following.
Theorem 1.
Let and be defined as in (12), then we have
where the odd function is defined by the equality , is analytic in Ω and , .
Remark 2.
Letting , , and in Theorem 1, we can obtain the integral representation for the odd function in the class .
Theorem 2.
If , then, for odd values of p, we have
and
Additionally, for all , we have
where is given by
The inequality is sharp for each .
Proof.
As , by (11), we have
Thus, let be of the form and defined by
On computation, we have
The right-hand side of (19):
Hence, the proof of (18).
Now, to prove the Fekete–Szegő inequality for the class , we consider
Further, if , from (24), we deduce
An examination of the proof shows that the equality for (25) holds if , . Equivalently, by Lemma 2, we have . Therefore, the extremal function of the class is given by
Similarly, the equality for (25) holds if . Equivalently, by Lemma 2, we have . Therefore, the extremal function in is given by
and the proof of the theorem is complete. □
3. Subclasses of Analytic Functions Using Quantum Derivative
In this section, we define a q-analogue of the operator defined in Section 1. The study of Geometric Function Theory in dual with quantum calculus was initiated by Srivastava []. For recent developments and applications of quantum calculus in Geometric Function Theory, refer to the recent survey-cum-expository article by Srivastava [] and references provided therein.
Now, we give a very brief introduction of the q-calculus. We let
For , the Jackson’s q-derivative operator or q-difference operator for a function is defined by
From (27), if , we can easily see that , for and note that . The q-Jackson integral is defined by (see [])
provided the q-series converges. Further, we observe that
where the second equality holds if is continuous at .
The class of q-starlike functions introduced by Ismail et al. in [] is defined as the class of functions which satisfies the condition
Here, we let denote the class of q-starlike functions. Equivalently, a function , if and only if the subordination condition (see ([], Definition 7))
holds.
The q-analogue of the function defined as in (7) is given by
with , . Srivastava et al. [,] introduced function classes of q-starlike functions related with conic regions and also studied the impact of Janowski functions on those conic regions. Inspired by the aforementioned works on q-calculus, we now define the q-analogue of the operator as follows:
The function plays the role of those extremal functions related to the conic domain and is given by
where and t is chosen such that , where is Legendre’s complete elliptic integral of the first kind and is complementary integral of . Clearly, is in , with the expansion of the form
we obtain
Instead of defining the same class of functions defined in Definition 1 involving quantum derivative, we define a class (motivated by the study of [] (Definition 1.2)) involving additional parameters.
Definition 2.
For , with , , let the class consist of a function in satisfying the subordination condition
where and .
Remark 3.
Unlike the function class , the presence of u and v in the function class unifies the various subclasses of analytic functions and classes of functions with respect to symmetric points. Now, we list some special cases:
For other special cases of our classes, see [] (p. 264).
Coefficient Estimates of
We need the following results to establish our main results.
Lemma 3
([], Theorem VII). Let be analytic in Ω and be analytic and convex in Ω. If , then for .
Lemma 4
Proof.
If the function has the power series expansion (3), then, from (1), we have
Since the subordination relation is invariant under translation, the assumption (35) is equivalent to
Further, because the convexity of ℵ implies the convexity of , from Lemma 3, the conclusion follows (36). □
Theorem 3.
Let and Ψ be chosen so that is convex in Ω. If , then, for ,
with , .
Proof.
Let us consider
where is analytic in and satisfies the subordination condition .
On computation, we have
The hypothesis is true for . Now, let in (39); we obtain
If we let in (37), we have
Hence, the hypothesis of the theorem is true for . Following the steps as in [] (Theorem 2), we can obtain the desired result using mathematical induction. □
If we let , and in Theorem 3, we obtain the coefficient estimate of the class (see Definition 1).
Corollary 1.
Let and Ψ be chosen so that is convex in Ω. If , then, for ,
with , .
Letting , , , and in Theorem 3, we obtain the following corollary.
Corollary 2
If , , and in Theorem 3, we obtain the following result.
Corollary 3
If we choose , , , and in Theorem 3, we obtain the following corollary.
Corollary 4
([], Theorem 2.6). For a function defined as in (31) with , let satisfy the condition
Letting , , and in Theorem 3, we obtain the following corollary.
Corollary 5.
For a function defined as in (31). Let satisfy the condition
By putting , , , , and , as in Theorem 3, we obtain the coefficient bounds for the class , defined by Shams et al. [].
Corollary 6.
Let . Then,
The inequality (40) is better than the result obtained by Owa et al. [].
Letting , , , , and .
Corollary 7
([], Theorem 2.1). Let the function satisfy the condition
Then, for ,
4. Conclusions
Using the Hadamard product, we define a new family of multivalent differential operator involving the convex combinations of analytic functions. Using the newly defined operator, the family of multivalent functions of complex order with respect to symmetric points is defined to unify the study of various classes of p-valent functions. Inclusion relationship and solution to the Fekete–Szegő problem for the defined function class are here established.
Further, a more comprehensive class of multivalent functions involving quantum calculus is introduced. Srivastava, in [] (Equation (9.4)), showed that all the results investigated using quantum derivative (q-derivative) can be translated into the corresponding so called post-quantum analogues (-derivative) using a straightforward parametric and argument variation of the following types:
Hence, the additional parameter r is unnecessary; therefore, here, we restrict our study with a q-derivative rather than extending it to a -derivative. Numerous q-results obtained by various authors are shown as special cases of our main results.
Author Contributions
Conceptualization, D.B., K.R.K. and A.S.; methodology, D.B., K.R.K. and A.S.; software, D.B., K.R.K. and A.S.; validation, D.B., K.R.K. and A.S.; formal analysis, D.B., K.R.K. and A.S.; investigation, D.B., K.R.K. and A.S.; resources, D.B., K.R.K. and A.S.; data curation, D.B., K.R.K. and A.S.; writing—original draft preparation, K.R.K. and A.S.; writing—review and editing, D.B., K.R.K. and A.S.; visualization, D.B., K.R.K. and A.S.; supervision, D.B., K.R.K. and A.S.; project administration, D.B., K.R.K. and A.S. All authors have read and agreed to the published version of the manuscript.
Funding
This research study received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
The authors thank the reviewers for their valuable remarks, comments and advice that helped to improve the quality of the paper.
Conflicts of Interest
All the authors declare that they have no conflict of interest.
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