A Comprehensive Family of Biunivalent Functions Defined by -Fibonacci Numbers

Abstract

By using -Fibonacci numbers, we present a comprehensive family of regular and biunivalent functions of the type in the open unit disc . We estimate the upper bounds on initial coefficients and also the functional of Fekete-Szegö for functions in this family. We also discuss few interesting observations and provide relevant connections of the result investigated.

1. Introduction and Notations

Let be the set of all complex numbers and the disc be symbolized by . Let and be the collection of all real numbers. We denote the set of all normalized regular functions in that have the series of the formby and the symbol stands for set of all functions of that are univalent (or Schlicht) in . As per the popular Koebe theorem (see [1]), every function has an inverse function given bysuch that and .

A function of is called biunivalent (or bi-Schlicht) in if both and are univalent (or Schlicht) in . Let stands for the set of biunivalent (or bi-Schlicht) functions having the form (1). Historically, investigations of the family begun five decades ago by Lewin [2] and Brannan and Clunie [3]. Later, Tan [4] found some coefficient estimates for biunivalent functions. In 1986 [5], Brannan and Taha introduced certain well-known subfamilies of in . Many interesting results related to initial bounds for some special families of have appeared in [6–8].

In 2007, the concept of -Fibonacci number sequence , was examined by Falcón and Plaza [9] and is given bywhere and

is the well-known Fibonacci number sequence.

Özgür and Sokól in 2015 [10] proved that ifthen,where is as in (4) and . Further, if , then, we have

Fibonacci polynomials, Pell-Lucas polynomials, Gegenbauer polynomials, Chebyshev polynomials, Horadam polynomials, Fermat-Lucas polynomials, and generalizations of them are potentially important in many branches such as architecture, physics, combinatorics, number theory, statistics, and engineering. Additional information is associated with these polynomials one can go through [11–13]. More details about the very popular functional of Fekete-Szegö for biunivalent functions based on -Fibonacci numbers can be found in [14–20].

The recent research trends are the outcomes of the study of functions in based on any one of the above-mentioned polynomials, which can be seen in the recent papers [21–28]. Generally, interest was shown to estimate the first two coefficient bounds and the functional of Fekete-Szegö for some subfamilies of .

For functions and regular in , is said to subordinate , if there is a Schwarz function in , such that , , and . This subordination is indicated as . In particular, if , then and .

Inspired by the recent articles and the new trends on functions in , we present a comprehensive family of defined by using -Fibonacci numbers as given by (3) with as in (4).

Throughout this paper, is as in (2), , is as in (4), and is as in (5).

Definition 1. A function having the power series (1) is said to be in the family , ifwhere .

Remark 2. The function families and were investigated by Frasin et al. [29].
It is interesting to note that (i) , (ii) , and (iii) lead the family to various subfamilies, as illustrated in the following:(1) is the family of functions satisfying(2) is the set of functions satisfying(3) is the collection of functions satisfying

Remark 3. We note that (i) and (ii) .

Remark 4. (i)When , the family was introduced by Frasin et al. [30](ii)The family was mentioned by Güney et al. [18], when and (iii)For , the class was investigated by Magesh et al. [31]

We now state the following lemma, which we will be using in the proof of our theorem.

Lemma 5 (see [32]). If , where is the collection of regular functions in , satisfying , with , then , for each .

In the next section, we derive the estimates for , and obtain the Fekete-Szegö [33] inequalities for functions in the class .

2. Coefficient Bounds and Fekete-Szegö Functional

In this section, we offer to get the upper bounds on initial coefficients and find the functional of Fekete-Szegö for functions .

Theorem 6. Let , and . If of the form (1) in the family , thenand for ,where

Proof. Let the function . Then, from Definition 1, we haveLet and . Then, there exists a regular function with in and . Therefore, the function is in the class , whereSo it follows thatSimilarly, it follows thatwhere is a regular function such that in such that and the function is in the class , whereBy virtue of (14), (15), (18), and (19), we obtainFrom (21) and (23), we getand also,If we add (26) and (24), then we obtainSubstituting the value of from (26) in (27), we getwhich gets (10), on using Lemma 5.
On using (25) in the subtraction of (24) from (26), we arrive atThen, in view of Lemma 5 and equation (28), (29) reduces to (11).
From (28) and (29), for , we can easily compute thatwhereIn view of (4), we find thatwhich enable us to conclude (12) with as in (13). Theorem 6 is proved.

Remark 7. By taking in the above theorem, we obtain a result of Frasin et al. ([29], Corollary 3.4) and if we let in the above theorem, we get another result of Frasin et al. ([29], Corollary 3.7).

Remark 8. Allowing and in the above theorem, we have Theorem 2.3 of Magesh et al. [31].

Remark 9. Letting and in the Theorem 6, we obtain two results of Güney et al. ([18], Corollary 10 and Corollary 23). Further, if we take , we get results of Güney et al. ([17], Corollary 1 and Corollary 4).

In Section 3, few interesting consequences and relevant observations of the main result are mentioned.

3. Outcome of the Main Result

By setting (i) , (ii) , and (iii) in our main theorem, we obtain the following results, respectively.

Corollary 10. If the function , thenand for ,where

Remark 11. (i)By taking and in the above corollary, we obtain Theorem 2.3 of Magesh et al. [31](ii)By allowing and in the above corollary, we get two results Güney et al. ([18], Corollary 10 and Corollary 23)

Corollary 12. If the function , thenand for some ,where

Remark 13. For , Corollary 12 reduces to a result of Frasin et al. ([30], Corollary 3.6). Further, allowing , we get Corollary 10 of Altnkaya [22].

Corollary 14. If the function , thenand for ,where

4. Conclusion

A comprehensive family of biunivalent (or bi-Schlicht) functions is introduced by using -Fibonacci numbers. Bounds of the first two coefficients and and the celebrated Fekete-Szegö functional have been fixed for this family. Through corollaries of our main result, we have highlighted many interesting new consequences.

A comprehensive family examined in this research paper could inspire further research related to other aspects such as a comprehensive family using -derivative operator, a meromorphic biunivalent function family associated with Al-Oboudi differential operator, and a comprehensive family using integrodifferential operator.

Data Availability

There is no data used for this manuscript.

Conflicts of Interest

The authors confirm that there are no competing interests regarding the publication of this manuscript.

Authors’ Contributions

The authors contributed equally in the preparation of this manuscript and have approved the final version of the manuscript.