In this paper,we define a class of analytic functions F (β,λ) (H, α, δ, µ) , satisfying the following subordinate condition associated with Chebyshev polynomialswhereWe obtain initial coefficients |a2| and |a3| for this subclass by means of Chebyshev polynomials expansions of analytic functions in D. Furthermore, we solve Fekete-Szegö problem for functions in this subclass.We also provide relevant connections of our results with those considered in earlier investigations. The results presented in this paper improve the earlier investigations.