Abstract
To consolidate or adapt to many studies on meromorphic functions, we define a new subclass of meromorphic functions of complex order involving a differential operator. The defined function class combines the concept of spiral-like functions with other studies pertaining to subclasses of multivalent meromorphic functions. Inclusion relations, integral representation, geometrical interpretation, coefficient estimates and solution to the Fekete-Szegö problem of the defined classes are the highlights of this present study. Further to keep up with the present direction of research, we extend the study using quantum calculus. Applications of our main results are given as corollaries.
1. Introduction
Let be the class of function of the form which are analytic in the unit disc . Also let denote the class of functions which are univalent in . The subclasses of consisting of functions which map unit disc onto a star-like and convex domain will be symbolized by and , respectively. Also let denote the class of functions analytic in the unit disc, given by and satisfies , . For , we let to denote the class of functions of the form which are analytic in . Shi et al. [1] defined the class if and only if where and . Here, denotes the usual subordination of analytic function. The class is the meromorphic analogue of the class of -valent spiral-like functions defined by Uyanik et al. in [2]. Similarly, we let to denote the class of function in satisfying the condition
Extending the class of Janowski function ([3]), Aouf [4] (Equation (4)) (also see [5]) defined the class if and only if where is the Schwartz function. Motivated by the recent study of Breaz et al. [5] and in view generalizing the superordinate function in (4), Cotîrl and Karthikeyan in [6] defined and studied the following relation where and .
It is well-known that the function maps the unit disc onto the right half plane. For an admissible choice of the parameter , , and , maps unit disc onto a domain which is convex with respect to point if (see Figure 1). Similarly, the function which is related to the class of functions associated with leaf-like domain (see [7–9]) gets rotated and translated on the impact of (see Figure 2) for a choice of the parameter , , and .
Remark 1. The purpose to study was mainly motivated by the study of Karthikeyan et al. [10] and Noor and Malik [11]. Here, we will list some recent studies. (1)If we let in (7), then, reduces toThe function was defined and studied by Breaz et al. in [5]. (2)If we let and in (7), then, reduces to (see the superordinate function in (4)).
It is well-known that if given by (1) is in , then, the -symmetrical function , ( is a positive integer) is also in . Let be a positive integer and . For , let
The function is said to be star-like with respect to -symmetric points if it satisfies the condition
Here, we will let to denote the class of star-like functions with respect to -symmetric points. The class was introduced by Sakaguchi [12] in which he showed that all functions in are univalent. Note that .
A function is said to be -symmetrical if for each
For , Equation (9) can be defined by the following equality
Now, we extend the operator defined by Selvaraj and Karthikeyan in [13]. Using Hadamard product (or convolution), we define a operator for functions as follows: where is the Pochhammer symbol defined by
For convenience, we shall henceforth denote
Note that in [13], was defined for . Here, we skip the discussion on the necessity of using differential or integral operator, refer to [13–17] and reference provided therein for detailed properties of .
Throughout this paper, we assume that , , and
1.1. Short Introduction to Quantum Calculus
For , the Jacksons -derivative operator is defined by (see [18, 19])
From (18), if has the power series expansion (3), we can easily see that , for , where the -integer number is defined by and note that . Throughout this paper, we let denote
The -Jackson integral is defined by (see [20]) provided the -series converges. Further observe that where the second equality holds if is continuous at . For details pertaining to the significance of univalent function theory in dual with quantum calculus, refer to [21, 22] (also see [23–26]).
Meromorphic multivalent functions have been extensively studied by various authors, but motivation and references of this study are [1, 13, 27–36].
Definition 2. For , , , and defined as in (13), a function belongs to the class if it satisfies where denotes subordination and is defined as in (2).
Now, we will define a class replacing ordinary derivative with a quantum derivative in .
Definition 3. For , , , and defined as in (13), a function belongs to the class if where is the analogue of , which is defined by
Remark 4. We note that in the definition of , the operator and are the same as used in . We have not used the -analogue operator as it would require the reader to contend with additional set of parameters.
2. Preliminaries and some Supplementary Results
Here, we will discuss the results which would help us to obtain our main results.
We note that everything in classical calculus cannot be generalized to quantum calculus, notably the chain rule needs adaptation. Hence, logarithmic differentiation needs some application of analysis. In [37], Agrawal and Sahoo obtained the following result on logarithmic differentiation. For and , we have where is the Jackson -integral, defined as in (21). Similarly, we can see that
If is an integer, then the following identities follow directly from (16):
Since -derivative satisfies the linearity condition, (29) holds if the classical derivative is replaced with quantum derivative. That is,
We now state the following result which will be used to establish the coefficient inequalities.
Lemma 5 (see [38]). Let and also let be a complex number, then the result is sharp for functions given by
The Maclaurin series for the function (see [6]) for the function is given by
If we define the function by
We note that and . Using (34), we have
For some , we have
3. Integral Representations and Closure Properties
We begin with the following.
Theorem 6. Let , then for , we get for And for , we have for where is defined by equality (16) and is analytic in with and .
Proof. Let . In view of (23), we have where is analytic in and , . Substituting by in the equality (39), respectively, (), we have Using (28) in (40), we get Using the equality (29) in (42), we can get Let in (42), respectively, and summing them we get
Case 1. Let in (43). We need to integrate from to to find . But from (43), we notice the presence of the first-order pole at the origin, the difficulty to integrate the above equality is avoided by integrating from to with , and then, let . Therefore, on applying integration, we get Hence, the proof of (37).
Case 2. If , (43) can be rewritten as On integrating the above expression we obtain (38). Hence, the proof of Theorem 6.
Theorem 7. Let , then for , we get And for , we have where is defined by equality (16) and is analytic in with and .
Proof. In view of (24), (30), and (43), we have
Case 1. Let in (48). Using the definition of logarithmic differentiation for -derivative operator (see (26)) in (48), we get where the integral is -Jackson integral. Hence, the proof of (37).
Case 2. If , using chain rule (see (27)) for the -difference operator defined in the previous section, (43) can be rewritten as On applying -Jackson integral in the above expression, we obtain (47).
Corollary 8 (see [1, Theorem 1]). Let , then where is analytic in with and .
Proof. Letting , , , , , , and in Theorem 6, then (43) reduces to the form Retracing the steps as in Theorem 6, we can establish the assertion of the corollary.
Setting , , , , and in Theorem 6, we get the following
Corollary 9. Let , then, for , we get for And for , we have for where is defined by equality (12) and is analytic in with and .
Letting , , and in Corollary 9, we get the following result.
Corollary 10. (see [1]). Let , then for where is analytic in with and .
4. Fekete-Szegö Inequality of and
Very few researchers have attempted at finding solution to the Fekete-Szegö problem for class of functions with respect to -symmetric points, as it is computational tedious. Notable among those works on coefficient inequalities of classes of functions with respect to -symmetric points were done by Aouf et al. [39].
Throughout this section, we let
Theorem 11. If , then, we have for all where is given by
The inequality is sharp for each .
Proof. As , by (23), we have Thus, let be of the form and defined by On computation, we have The right hand side of (58) From the left hand side of (58) is given by From (61) and (62), we obtain Now we consider On applying Lemma 5, we get the assertion.
To demonstrate the applications of our results, here, we provide the most simple special case of our result. Note that the following result was obtained [[40], Theorem 6] for functions in .
Corollary 12. If satisfies and , with , , then for all we have The inequality is sharp for the function given by
Proof. In Theorem 11, taking , , , , , , and , we get the inequality
Analogous to Theorem 11, we can prove the following.
Theorem 13. If , then, we have for all where is given by The inequality is sharp.
5. Conclusions
The defined function class though familiar with so called pseudo-star-like functions required lots of adaptation since it involves functions with a removable singularity of order at the origin. Integral representation and Fekete-Szegö inequalities have been established. Further, we extend the class by replacing the classical derivative with -derivative. Since all the results involving classical derivative does not get translated to the results involving -derivative, we used some modified conditions to obtain our main results. We note that these adaptation are essential for future research.
Data Availability
Not applicable.
Conflicts of Interest
Authors declare that they have no conflict of interest.
Authors’ Contributions
All authors contributed equally to this work. All the authors have read and agreed to the published version of the manuscript.
Acknowledgments
Authors thank the special issue editors and the referee for their helpful comments and suggestions, which helped us to improve the results and its presentation.