Abstract
In recent years, the usage of the -derivative and symmetric -derivative operators is significant. In this study, firstly, many known concepts of the -derivative operator are highlighted and given. We then use the symmetric -derivative operator and certain -Chebyshev polynomials to define a new subclass of analytic and bi-univalent functions. For this newly defined functions’ classes, a number of coefficient bounds, along with the Fekete–Szegö inequalities, are also given. To validate our results, we give some known consequences in form of remarks.
1. Introduction and Definitions
Let denote the class of functions which are analytic in the open unit disk:
Let be the subclass of functions , which satisfy the normalization condition given bythat is, which are represented by the following Taylor–Maclaurin series expansion:
Also, let be the class of functions in , which are univalent in .
It is well known that every function has an inverse defined bywhere
A function is said to be bi-univalent in if both and are univalent in .
Let denote the class of bi-univalent function in given by (3). Example of functions in the class is
However, the familiar Koebe function is an example of the class . Other common examples of functions in , such asare also not members of .
Lewin [1] investigated a bi-univalent functions class and showed that . Subsequently, Brannan and Clunie [2] conjectured that . Netanyahu [3], on the contrary, showed that
The coefficient for each of the Taylor–Maclaurin coefficients is presumably still an open problem.
Similar to the familiar subclasses and of star-like and convex functions of order , respectively, Brannan and Taha [4] introduced certain subclasses of the bi-univalent function class , namely, and of bi-star-like functions and bi-convex functions of order , respectively. For each of the function classes and , they found nonsharp bounds on the first two Taylor–Maclaurin coefficients and .
Furthermore, let and be analytic functions in open unit disc ; then, the function is subordinated to and symbolically denoted asif there occurs an analytic function with properties that
Suppose holomorphic in , such that
If the function is univalent in , then the above condition is equivalent to
Jackson [5] introduced and studied the -derivative operator of a function as follows:and . In case , for is a positive integer, the -derivative of is given bywhere . For more details on the concepts of -derivative, see [6, 7].
The quantum (or -) calculus is an essential tool for studying diverse families of analytic functions, and its applications in mathematics and related fields have inspired researchers. Srivastava [8] was the first person to apply it in the context of univalent functions. Numerous scholars conducted substantial work on -calculus and examined its various applications due to the applicability of -analysis in mathematics and other domains. For example, with the help of certain higher-order -derivative operators, Khan et al. [7] defined and studied a number of subclasses of -star-like functions. Also, Shi et al. [9] (see also [10]) used the -differential operator and defined a new subclass of Janowski-type multivalent -star-like functions. In [7, 9], a number of sufficient conditions and some other interesting properties have been examined. More importantly, the convolution theory enables us to investigate various properties of analytic functions. Due to the large range of applications of -calculus and the importance of -operators instead of regular operators, many researchers have explored -calculus in depth. In addition, Srivastava [11] recently published survey-cum-expository review paper which is useful for researchers and scholars (see, for example, [12, 13]) working on these subjects. Also, Srivastava’s recent survey-cum-expository review article [11] further motivates the use of the -analysis in geometric function theory, as well as commenting on the triviality of the so-called -analysis involving an insignificant and redundant parameter (see p. 340 of [11]).
Utilizing the idea of -derivative operator, in 2013, Brahim et al. introduced and studied the symmetric -derivative operator for a function as follows:
It is easy to see thatwhere
The relation between -derivative operator and symmetric -derivative operator is given by
Suppose is the inverse of ; then,
Al Salam and Ismail [14] discovered a family of polynomials that can be understood as -analogues of the second-order bivariate Chebyshev polynomials. In 2012, Johann Cigler introduced and studied the -Chebyshev polynomials as follows.
Definition 1 (see [15]). The polynomialsare called -Chebyshev polynomial of the second kind.
Theorem 1 (see [15]). The-Chebyshev polynomials of the second kind satisfywith initial values
Remark 1. It is clear thatwhere is the classical Chebyshev polynomial of the second kind.
Now, making use -Chebyshev polynomials, we define the following.
Definition 2. Let be defined as follows:
By using the principal of subordination and the symmetric -derivative operator , we define the following subclasses of analytic and bi-univalent functions.
Definition 3. A function given by (3) is said to be in the class if the following conditions are satisfied:
We note from (25) thatwhere and .
Also, from (22), we have the following:
The goal of this research is to investigate -Chebyshev polynomial expansions in order to derive initial coefficient estimates for some subclasses of analytic and bi-univalent functions defined by the symmetric -derivative operator. In addition, Fekete–Szegö inequalities for the class are established.
Lemma 1 (see [16]). Let the functionbe given bybe in the class of functions with positive real part. Then,
This last inequality is sharp.
2. Coefficients Bounds for
Theorem 2. Let. Then,
Proof. Let given by (3) be in the class . Then,Let be defined asIt follows that, from (37) and (38),From (39) and (40), applying as given in (25), we see thatFrom (35), (41) and (36), (42), we haveAdding (43) and (46), we haveAlso, adding (44) and (47) and applying (49) yieldsApplying (49) in (50) givesPutting (52) into (51) with some calculations, we haveApplying triangular inequality and Lemma 1, we haveSubtracting (47) from (44) with some calculations, we haveApplying triangular inequality and Lemma 1, we haveSubtracting (48) from (45), we haveApplying triangular inequality and Lemma 1, we have
3. Fekete–Szegö Inequalities for the Function Class
The coefficient of a function class is well known to be restricted by , and the coefficient limits give information about the functions geometric characteristics. The famous problem solved by Fekete–Szegö [17] is to determine the greatest value of the coefficient functional over the class for each , which was demonstrated using the Loewner technique. In this section, we aim to determine the upper bounds of the coefficient functional for the function class .
Theorem 3. Let. Then, for some,where
Proof. From (51) and (55), we havewhereApplying Lemma 1, we have
Remark 2. Taking and in Theorem 2 and Theorem 3, we have the results obtained by Altinkaya and Yalcin [18].
4. Conclusion
Recently, the -derivative and symmetric -derivative operators are particularly applicable in many diverse areas of mathematics and physics. In this study, firstly, many known concepts of the -derivative operator have been highlighted and given. We have then used the symmetric -derivative operator and certain -Chebyshev Polynomials and have defined a new subclass of analytic and bi-univalent functions. For these newly defined functions’ classes, a number of coefficients bonds, along with the Fekete–Szeg ö inequalities, have also been given. To validate our results, we have given some known consequence in the form of Remarks.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors jointly worked on the results, and they have read and approved the final manuscript.