Abstract
In this paper, we introduce and investigate a class of biunivalent functions, denoted by , that depends on the Ruscheweyh operator and defined by means of Horadam polynomials. For functions in this class, we derive the estimations for the initial Taylor–Maclaurin coefficients and . Moreover, we obtain the classical Fekete–Szegö inequality of functions belonging to this class.
1. Introduction
Let be the family of all analytic functions that are defined on the open unit disk and normalized by the conditions and . Any function has the following Taylor–Maclaurin series expansion:
Let denote the class of all functions that are univalent in . Let the functions and be analytic in , we say the function is subordinate by the function in , denoted by for all , if there exists a Schwarz function , with and for all , such that for all . In particular, if the function is univalent over , then equivalent to and . For more information about the subordination principle, we refer the readers to the monographs [1–3].
It is well known that univalent functions are injective functions. Hence, they are invertible, and the inverse functions may not be defined on the entire unit disk . In fact, the Koebe one-quarter theorem tells us that the image of under any function contains the disk of center 0 and radius . Accordingly, every function has an inverse which is defined as
Moreover, the inverse function is given by
For this reason, we define the class as follows. A function is said to be biunivalent if both and are univalent in . Therefore, let denotes the class of all biunivalent functions in which are given by equation (1). For example, the following functions belong to the class :
However, Koebe function, and , does not belong to the class . For more information about univalent and biunivalent functions, we refer the readers to the articles [4–6], the monographs [7–9], and the references therein.
Recently, many researchers have studied the geometric function theory in complex analysis, and the typical problem in this field is studying a functional made up of combinations of the initial coefficients of the functions . For a function in the class , it is well known that is bounded by . Moreover, the coefficient bounds give information about the geometric properties of those functions. For instance, the bound for the second coefficients of the class gives the growth and distortion bounds for the class. In addition, the Fekete–Szegö functional arises naturally in the investigation of univalency of analytic functions. In the year 1933, Fekete and Szegö [10] found the maximum value of , as a function of the real parameter for a univalent function . Since then, the problem of dealing with the Fekete–Szegö functional for with any complex is known as the classical Fekete–Szegö problem. There are many researchers investigated the Fekete–Szegö functional and the other coefficient estimates problems, for example, see the articles [4–6, 10–20] and the references therein.
2. Preliminaries
In this section, we present some information that are curial for the main results of this paper. In the year 1965, for , Horadam [21] introduced the sequence that is defined by the following recurrence relation:with the initial values and . The characteristic equation of this sequence is given by
In addition, the generating function of Horadam sequence is
The Horadam sequences generalize many famous sequences such as Fibonacci, Lucas, Pell, Pell-Lucas, and Jacobsthal sequences. These sequences have been studied for a long time. For more information about these sequences, we refer the readers to the articles [22, 23] and the monograph [24].
In the year 1985, Horadam and Mahon defined the Horadam polynomials by the following recurrence relation:with initial values
Moreover, the generating function of Horadam polynomials is given by
For particular values of , and , the Horadam polynomials lead to many known polynomials. In the following, we list some special cases of Horadam Polynomials.(i)If , we get Fibonacci polynomials whose recurrence relation is(ii)If and , we get Lucas polynomials whose recurrence relation is(iii)If and , we get Pell polynomials whose recurrence relation is(iv)If and , we get Pell–Lucas polynomials whose recurrence relation is(v)If and , we get Jacobsthal polynomials whose recurrence relation is(vi)If , and , we get Jacobsthal–Lucas polynomials whose recurrence relation is(vii)If and , and , we get Chebyshev polynomials of the second kind whose recurrence relation is(viii)If and , and , we get Chebyshev polynomials of the first kind whose recurrence relation is
For more information about Horadam polynomials and its special interesting cases, we refer the readers to the articles [5, 6, 11, 22, 25–29], the monograph [24], and the references therein.
In the year 1975, Ruscheweyh [30] introduced the operator which is defined, using the Hadamard product, as follows:where , , and real number . For , we get the Ruscheweyh derivative of order of the function :
Moreover, the Taylor–Maclaurin series of is given by
We say that a function in the subclass if it fulfills the subordination conditions, associated with the Horadam polynomials, for all , and for :and
The following lemma (see for details, [20]) is a well-known fact, so we omit its proof.
Lemma 1. Let , , and , . If and ,
Our investigation in this paper is motivated by the work of the researchers presented in the papers [31, 32]. In this presenting paper, we investigate a subclass of biunivalent functions in the open unit disk , which we denote by with , , and . For functions in this subclass, we derive upper bounds for the initial Taylor–Maclaurin coefficients and . Furthermore, we examine the corresponding Fekete–Szegö functional problem for functions belong to this subclass.
3. Initial Coefficient Estimates for the Function Class
In this section, we provide bounds for the initial Taylor–Maclaurin coefficients for the functions belong to the class which are given by equation (1).
Theorem 2. Let the function given by (1) be in the class . Then,and
Proof. Let belong to the class . Then, using (5) and (6), we can find two analytic functions and on the unit disk such thatandwhere the analytic functions and are given bysuch thatand for all , Moreover, it is well known that (see, for details, [7]) for all ,Now, upon comparing the coefficients in both sides of (28) and (29), we obtain the following equations:andIn view of equations (34) and (36), we get the following equations:andMoreover, if we add equations (35) and (37), we getIn view of equation (39), equation (40) can be written asUsing equations (22) and (41) becomesUsing the facts and , we get inequality (26), which is the desired estimate of .
Next, we turn our attention to find an upper bound for . Subtracting equation (37) from equation (35), we getIn view of equations (38) and (39), we obtainFinally, using equation (22) and the facts and , we getwhich gives the estimate (27). This completes the proof of Theorem 2.
Taking , we get the following corollary of Theorem 2. This proof is similar to the proof of previous theorem, so we omit the proof’s details.
Corollary 3. Let the function given by (1) be in the class . Then,
4. Fekete–Szegö Problem for the Function Class
In this section, we consider the classical Fekete–Szegö problem for our presenting class .
Theorem 4. Let the function given by (1) be in the class . Then, for some ,where
Proof. For some real number , using equations (39) and (43), we haveNow, using equation (41), we obtainThe last expression can be written as follows:whereUsing Lemma 1, we get the following equation:Simplifying the last inequality, we get the desired inequality (47), and this completes the proof of Theorem 4.
The following corollary is just a consequence of Theorem 4. Taking , we get the following Fekete–Szegö inequality.
Corollary 5. Let the function given by (1) be in the class . Then, for some ,where
5. Conclusion
This research paper has investigated a new subclass of biunivalent functions, defined in terms of the Ruscheweyh derivative of order , associated with the Horadam polynomials. For functions belong to this function class, the author has derived estimates for the Taylor–Maclaurin initial coefficients and Fekete–Szegö functional problem. The work presented in this paper will lead to many different results for subclasses defined by the means of special cases of Horadam polynomials, such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, and Chebyshev polynomials of first and second kinds. Moreover, the presented work in this paper will inspire researchers to extend its concepts to harmonic functions and symmetric -calculus.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The author would like to express his sincerest thanks to the referees for their valuable comments and various useful suggestions.