A Subclass of Bi-Univalent Functions Defined by Generalized Sãlãgean Operator Related to Shell-Like Curves Connected with Fibonacci Numbers

Abstract

The aim of this paper is to study certain subclasses of bi-univalent functions defined by generalized Sãlãgean differential operator related to shell-like curves connected with Fibonacci numbers. We find estimates of the initial coefficients and and upper bounds for the Fekete-Szegö functional for the functions in this class. The results proved by various authors follow as particular cases.

1. Introduction and Preliminaries

Let be the class of functions of the formwhich are analytic in the unit disc with normalization . By , we denote the class of functions and univalent in .

Let us denote by the class of bounded or Schwarz functions satisfying and which are analytic in the open unit disc and of the formConsider two functions and analytic in . We say that is subordinate to (symbolically ) if there exists a bounded function for which . This result is known as principle of subordination.

By , we denote the class of starlike functions which satisfies the following condition:By , we denote the class of convex functions which satisfies the following condition:A function is said to be -convex if it satisfies the inequalityThe class of -convex functions is denoted by and was introduced by Mocanu [1]. In particular and .

For and , Al-Oboudi [2] introduced the following differential operator:and in general,or equivalent towith . It is obvious that, for , the operator is equivalent to the Sãlãgean operator introduced in [3]. So the operator is named as the Generalized Sãlãgean operator.

The inverse functions of the functions in the class may not be defined on the entire unit disc although the functions in the class are invertible. However using Koebe-one quarter theorem [4] it is obvious that the image of under every function contains a disc of radius . Hence every univalent function has an inverse , defined byandwhereA function is said to be bi-univalent in if both and are univalent in U.

By , we denote the class of bi-univalent functions in defined by (1).

Lewin [5] discussed the class of bi-univalent functions and obtained the bound for the second coefficient. Brannan and Taha [6] investigated certain subclasses of bi-univalent functions, similar to the familiar subclasses of univalent functions consisting of strongly starlike, starlike and convex functions. They introduced bi-starlike functions and bi-convex functions and obtained estimates on the initial coefficients.

Sokól [7] introduced the class of shell-like functions defined as below.

Definition 1. A function given by (1) is said to be in the class of starlike shell-like functions if it satisfies the following condition:where .

It should be observed that is a subclass of the class of starlike functions.

Later Dziok et al. [8] introduced the class of convex functions related to a shell-like curve as below.

Definition 2. A function given by (1) is said to be in the class of convex shell-like functions if it satisfies the condition thatwhere .

Again Dziok et al. [9] defined the following class of -convex shell-like functions.

Definition 3. A function given by (1) is said to be in the class of -convex shell-like functions if it satisfies the condition thatwhere .

Obviously and .

The function is not univalent in , but it is univalent in the disc . For example, and , and it may also be noticed thatwhich shows that the number divides such that it fulfils the golden section. The image of the unit circle under is a curve described by the equation given bywhich is translated and revolved trisectrix of Maclaurin. The curve is a closed curve without any loops for . For , it has a loop, and for , it has a vertical asymptote. Since satisfies the equation , this expression can be used to obtain higher powers as a linear function of lower powers, which in turn can be decomposed all the way down to a linear combination of and . The resulting recurrence relationships yield Fibonacci numbers :Also the subclasses of bi-univalent functions related to shell-like curves were studied by various authors [10–12].

The earlier work on bi-univalent functions related to shell-like curves connected with Fibonacci numbers motivated us to define the following subclass.

To avoid repetition, throughout the paper we assume that , and .

Definition 4. A function given by (1) is said to be in the class if it satisfies the following conditions:andwhere and .

The following observations are obvious:

(i) , the class of bi-univalent functions defined by Sãlãgean operator related to shell-like curves.

(ii) , the class of bi-univalent -convex shell-like functions studied by Güney et al. [13].

(iii) .

(iv) .

In this paper, we study the class and obtain estimates of the initial coefficients and and upper bounds for the Fekete-Szegö functional for the functions in this class.

2. Coefficient Bounds for the Function Class

For deriving our results, we need the following lemma.

Lemma 5 (see [14]). If is family of all functions analytic in for which and has the form for , then for each .

Theorem 6. If , thenand

Proof. As , so by Definition 4 and using the principle of subordination, there exist Schwarz functions and such thatandwhere and .
On expanding, it yieldsandAgainandUsing (24) and (26) in (22) and equating the coefficients of and , we getandAgain using (25) and (27) in (23) and equating the coefficients of and , we getandFrom (28) and (30), it is clear thatandAdding (29) and (31), it yieldsPutting (33) in (34), we getUsing Lemma 5 and on applying triangle inequality in (35), (20) can be easily obtained.
Now subtracting (31) from (29), we obtainApplying triangle inequality and using Lemma 5 and (35) in (36), it yieldsFrom (37), result (21) is obvious.

For , Theorem 6 gives the following result.

Corollary 7. If , thenandFor , , Theorem 6 gives the following result due to Güney et al. [13].

Corollary 8. If , thenand

For , Theorem 6 agrees with the following result proved by Güney et al. [13] (orollary 1).

Corollary 9. If , thenand

For , , , Theorem 6 agrees with the following result proved by Güney et al. [13] (orollary 2).

Corollary 10. If , thenand

3. Fekete-Szegö Inequality for the Function Class

Theorem 11. Let , then

Proof. From (35) and (36), it yieldsEquation (47) can be expressed aswhereTaking modulus, we obtainSo (46) can be easily obtained from (50).

For , Theorem 11 gives the following result.

Corollary 12. Let , then

For , , Theorem 11 gives the following result due to Güney et al. [13].

Corollary 13. Let , then

For , , , Theorem 11 agrees with the following result proved by Güney et al. [13] (orollary 4).

Corollary 14. If , then

For , , , Theorem 11 agrees with the following result proved by Güney et al. [13] (orollary 5).

Corollary 15. If , then

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.