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In recent research, working on coefficient bounds is very popular and useful to deal with geometric properties of the underlying functions. In this work, two new subclasses of Sakaguchi type functions with respect to symmetric points through subordination are considered. Moreover, the initial coefficients and the sharp upper bounds for the functional $|\rho_{2k+1}-\mu \rho_{k+1}^{2}|$ corresponding to $k^{th}$ root transformation belong to the above classes are obtained and thoroughly investigated.

The aim of this paper is to apply Lucas polynomial, in order to obtain the initial coefficients on and  belonging to the new subclass   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: 30C45, 30C50.