Coefficient Inequalities using Lucas Polynomials

: The aim of this paper is to apply Lucas polynomial, in order to obtain the initial coefficients on 2 a and 3 a belonging to the new subclass ( )   ,  L of analytic, univalent and Sakaguchi functions as defined in the open unit disc  . Furthermore, the Fekete – Szego inequality is also investigated for this subclass. 2010 Mathematical Subject Classification


Introduction
In ( [4], [5]), for any variable quantity x , Lucas polynomials ) (x L n are explained recursively as follows: Various authors have analyzed the properties of the Lucas polynomials and obtained many fruitful results.It is widely accepted that many number and polynomial sequences can be generated by recurrence relations of second order.Of these the important sequences remain celebrated sequences of Lucas.These sequences of polynomials and numbers are of great importance in different branches such as physics, engineering, architecture, nature, art, number theory, combinatory and numerical analysis.These sequences have been carefully considered in several papers from a theoretical point of view (see, [7,8,10,16,17]).
Let A be the family of functions f that are analytic in the open unit disk   Let S mean the subclass of A consisting of univalent functions in  .It is well known (refer [1]) that every function of

S f 
virtually possesses an inverse of f, defined by When the function A f  is bi univalent, both f and 1 − f are univalent in  .Let  be the class of bi univalent functions in  given by (2).In fact, Srivastava et al. [15] have revived the study of analytic and bi univalent functions in recent years.Many researchers investigated and propounded various subclasses of bi univalent functions and fixed the initial coefficients 2 a and 3 a (refer [3, 6, 9, 12, 13, and 14]).
For analytic functions f and g , f is said to be subordinate to g, denoted ( ) ( ) , if there is an analytic is said to be Bazilevic function if it satisfies (see [12]): : Frasin [2] investigated the inequalities of coefficient for certain classes of Sakaguchi type functions that satisfy geometrical condition as for complex numbers s, t but s ≠ t and α (0 ≤ α < 1) .The objective of this paper is to introduce convolution in Sakaguchi type of new subclass and find estimates on the coefficient 2 a and 3 a for functions in the new subclass.The Fekete -Szego functional which were studied by Sahsene Altinkaya et al [11] when the parameters k = 0 and µ = 1.Remark 2.2: For 0 =  , the function , if the following subordination conditions are satisfied :  that are studied by Sahsene Altinkaya et al [11].
. Then there are two analytic functions  , such as ( ) ( ) , which can be written as From the equalities ( 6) and (7).it is obtained that   Comparing the coefficients in ( 8) and ( 9), It follows from ( 11) and ( 13) that By summing ( 12) and ( 14), it is obtained that By substituting the values of ( ) 16) in the right side of ( 17), it is deduced that Moreover by computations using (1), ( 10) and (18), it is found that . By subtracting ( 12) and ( 14) , it is obtained that . which was studied by Sahsene Altinkaya et al [11] when the parameters k = 0 and µ = 1.By putting λ = 0, in Theorem 1, it is obtained that, Remark 2.1 : If f is belongs to LΣ,, then Proof : From ( 14) and (15), it is observed that  x a a  − .which was studied by Sahsene Altinkaya et al [11].
polynomials the Lucas numbers are deduced.The ordinary generating function of the Lucas polynomials is

z
This class of the function was denoted by  B .Consequently when 0 =  , the class of starlike function is obtained.

,
if the following subordination conditions are satisfied :

.
In this section, it is attempted to provide Fekete -Szego inequalities for functions in the class are given in the following theorem.Theorem 3.1: For