Fekete–Szegö Inequality for Bi-Univalent Functions Subordinate to Horadam Polynomials

Abstract

Making use of Horadam polynomials, we propose a special family of regular functions of the type which are bi-univalent (or bi-schlicht) in the disc . We find estimates on the coefficients and and the functional of Fekete–Szegö for functions in this subfamily. Relevant connections to existing results and new observations of the main result are also presented.

1. Preliminaries

Let and be the sets of real numbers and positive integers, respectively. Let be the set of all complex numbers, and let denote the disc . We denote by , the set of all regular functions in that has the series of the formand be the set of all members of that are univalent in . According to the well-known Koebe theorem (see [1]), every univalent function has an inverse defined bysatisfying and .

A function of is said to be bi-univalent (or bi-schlicht) in if both and are univalent in . Let stand for the set of bi-univalent functions having form (1). Lewin [2] investigated the family and proved that . Brannan and Clunie [3] claimed that . Later, Tan [4] obtained initial coefficient estimates for bi-univalent functions. Subsequently, Brannan and Taha [5] examined certain well-known subfamilies of in . The momentum on the study of bi-univalent function family was gained recently, which is due to the work of Srivastava et al. [6]. This article has revived the topic apparently, and many researchers have investigated several interesting special families of (see [7–10]).

Recently, Hörçum and Koçer [11] (see also Horadam and Mahon [12]) examined the Horadam polynomials (or ), which is defined by the recurrence relationwhere , and are real constants. It is seen from (3) that . The generating function of the sequence , , is as follows (see [11]):where is such that , .

Few particular cases of are(1)(2)(3)(4)(5),(6)

The estimates on and the very popular Fekete–Szegö functional were determined for bi-univalent functions linked with certain polynomials like Lucas polynomials, Fibonacci polynomials, Chebyshev polynomials, Horadam polynomials, and Gegenbauer polynomials. It is well-known that these polynomials play a potentially important role in architecture, approximation theory, physics, statistics, mathematical, and engineering sciences.

The recent research trend is the study of functions in linked with any of the abovementioned polynomials. Generally, interest was shown to obtain the initial coefficient bounds and the celebrated inequality of Fekete–Szegö for the special subfamilies of . Recently, the Horadam polynomial was used by Abirami et al. [13] to find coefficient estimates for the families of bi-Bazilevic and -bi-starlike function, Frasin et al. [14] obtained coefficient estimates and Fekete–Szegö inequalities for certain subfamilies of Al-Oboudi-type bi-univalent functions related to k-Fibonacci numbers involving modified activation function, initial coefficient bounds for certain subsets of bi-univalent functions family subordinate to Horadam polynomials were obtained in [15, 16], Shaba and Wanas [17] obtained coefficient bounds which are sharp, for a family of bi-univalent functions using -Lucas polynomials, Srivastava et al. [18] have proposed a methodology to estimate coefficient bounds and Fekete–Szegö problem for certain subsets of bi-univalent function family linked with Horadam polynomials, and Swamy [19] and Swamy et al. [20, 21] have initiated the study of some subfamilies of bi-univalent function family subordinate to Horadam polynomials involving modified activation function. Swamy and Sailaja [22] have used Horadam polynomials to investigate coefficient estimates for two families of bi-univalent functions, Swamy et al. [23] have introduced some subfamilies of Sălăgean type bi-univalent functions subordinate to -Lucas polynomials and found initial coefficients, and Wanas and Alina [24] have fixed the Fekete–Szegö problem for Bazilevic bi-univalent function class linked with Horadam polynomials.

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

Inspired by the article [25] and the recent trends on functions in , we present a comprehensive family of associated with Horadam polynomials as in (3) having the generating function (4).

Throughout this paper, the inverse function is as in (2) and is as in (4).

Definition 1. A function in having the power series (1) is said to be in the set , and , if

The family is of interest as it contains many existing as well as new subfamilies of for particular choices of , and , as illustrated as follows:(1) and , is the collection of functions satisfying(2) and , is the collection of functions satisfying(3) and , is the collection of functions satisfying(4)The function classes and were investigated by the author in [19].

Remark 1. We note that(i)  and (ii) and

Remark 2. (i)For , the family was investigated by Swamy and Sailaja [22](ii) was due to Abirami et al. [13](iii) was introduced by Magesh et al. [16]

Remark 3. (i)For and , the class was studied by Alamoush [15](ii)For and , the family was introduced by Srivastava et al. [6]In Section 2, we derive the estimates for and and the inequality of Fekete and Szegö [26] for functions in the class . In Section 3, relevant connections to the existing results and few interesting consequences of the main result are presented.

2. Bi-Univalent Function Class

We determine the initial coefficients bounds and the inequality of Fekete–Szegö for functions in , in the following theorem.

Theorem 1. Let the function defined by (1) be in the family and let and . Then,and for ,where

Proof. Let . Then, on account of Definition 1, we getwhereare some regular functions in with , and . It follows from (13)–(16) with (4) thatFrom (17) and (18), in view of (3), we findIt is known that if , thenSimilarly, if , thenComparing (19) and (20), we haveFrom (23) and (25), we easily obtainand alsoIf we add (24) and (26), then we obtainSubstituting the value of from (28) in (29), we getwhich yields (10) on using (21) and (22).
After subtracting (26) from (24) and then using (27), we obtainThen, in view of (28), (31) becomeswhich yields (10) using (21) and (22).
From (30) and (31), for , we get in view of (3) thatwhereClearly,from which we conclude (9) with as in (12). Thus, Theorem 1 is proved.

Remark 4. By taking in Theorem 1, we get a result of the author (Corollary 2.4 [19]), and by letting in Theorem 1, we obtain another result of the author (Corollary 3.3 in [19]).

3. Outcome of the Main Result

Theorem 1 would yield the following outcome when .

Corollary 1. Let the function defined by (1) be in the family , and let and . Then,and for some ,where

Remark 5. For , Corollary 1 reduces to Corollary 2.1 of Swamy and Sailaja [22]. Also, we obtain Theorem 2.2 of Alamoush [15] from Corollary 1 when and .

Allowing in Theorem 1, we obtain the following.

Corollary 2. Let the function defined by (1) be in the set , and let and . Then,and for ,where

Remark 6. (i)Corollary 2 coincides with Theorem 2.1 of Abirami et al. [13] when (ii)Corollary 2 further coincides with Theorem 2.1 of Magesh et al. [16] when (iii)We obtain Corollary 1 and Corollary 3 of Srivastava et al. [18] from Corollary 1 when and Setting  in Theorem 1, we have the following.

Corollary 3. Let the function defined by (1) be in the class and let , and . Then,and for ,where

By choosing appropriate values for parameters , and in Theorem 1 and above corollaries, we have effectively shown the similarity between many subfamilies of bi-univalent functions.

4. Conclusion

A subfamily of bi-univalent (or bi-schlicht) functions is examined by using Horadam polynomial. Bounds of the first two coefficients and the Fekete–Szegö functional have been fixed for this subfamily. We have presented relevant connections to the existing results and few interesting consequences of the main result.

A subfamily examined in this article could inspire researchers to focus on other aspects such as a family bi-univalent functions using -derivative operator [27, 28], a family using -integral operator [29], meromorphic bi-univalent function family based on Al-Oboudi differential operator [30], regular bi-univalent function family based on Frasin operator [31], and a family using integro-differential operator [32].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.