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

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 |a2| and |a3| and upper bounds for the Fekete-Szegö functional for the functions in this class. The results proved by various authors follow as particular cases.


Introduction and Preliminaries
Let  be the class of functions of the form ( Consider 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: Re ( (  ()) The class of -convex functions is denoted by () and was introduced by Mocanu [1].In particular (0) ≡  * and (1) ≡ .
For  ≥ 1 and  ∈ , Al-Oboudi [2] introduced the following differential operator: and in general, or equivalent to with    (0) = 0.It is obvious that, for  = 1, 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 Koebeone quarter theorem [4] it is obvious that the image of  under every function  ∈  contains a disc of radius 1/4.Hence every univalent function  has an inverse  −1 , defined by and where A function  ∈  is said to be bi-univalent in  if both  and  −1 are univalent in U.By Σ, we denote the class of bi-univalent functions in  defined by (1).
Lewin [5] discussed the class Σ of bi-univalent functions and obtained the bound for the second coefficient.Brannan and Taha [6] investigated certain subclasses of biunivalent 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  = (1 − √5)/2 ≈ −0.618.
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 that where  = (1 − √ 5)/2 ≈ −0.618.
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 that where  = (1 − √ 5)/2 ≈ −0.618.
Obviously  0 ≡  and  1 ≡ .The function p is not univalent in , but it is univalent in the disc || < (3 − √ 5)/2 ≈ 0.38.For example, p(0) = p(−1/2) = 1 and p( ∓ arccos(1/4) ) = √ 5/5, and it may also be noticed that which shows that the number || divides [0, 1] such that it fulfils the golden section.The image of the unit circle || = 1 under p is a curve described by the equation given by which is translated and revolved trisectrix of Maclaurin.The curve p(  ) is a closed curve without any loops for 0 <  ≤  0 = (3 − √5)/2 ≈ 0.38.For  0 <  < 1, it has a loop, and for  = 1, it has a vertical asymptote.Since  satisfies the equation  2 = 1+, 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 1.The resulting recurrence relationships yield Fibonacci numbers   : Also the subclasses of bi-univalent functions related to shelllike curves were studied by various authors [10][11][12].
The earlier work on bi-univalent functions related to shell-like curves connected with Fibonacci numbers motivated us to define the following subclass.

Coefficient Bounds for the Function
Class   ,Σ (, p()) For deriving our results, we need the following lemma.