In this present work, the authors obtain Fekete-Szegö inequality for certain normalized analytic function $f(z)$ defined on the open unit disk for which $frac{(1-alpha )z(D_{lambda, mu }^{m}f(z))^{prime }+alpha z(D_{lambda, mu }^{m+1}f(z))^{prime }}{% (1-alpha )D_{lambda, mu }^{m}f(z)+alpha D_{lambda, mu }^{m+1}f(z)}$ $(lambda geq mu geq 0,$ $min {mathbb{N}}_{0},$} $alpha geq 0) $ lies in a region starlike with respect to 1 and is symmetric with respect to the real axis. Also certain applications of the main result for a class of functions defined by Hadamard product (or convolution) are given. As a special case of this result, Fekete-Szegö inequality for a class of functions defined through fractional derivatives is obtained. The motivation of this paper is to generalize the Fekete-Szegö inequalities obtained by Srivastava et al., Orhan et al. and Shanmugam et al, by making use of the generalized differential operator $D_{lambda, mu }^{m}.$

Keywords: Fekete-Szegö, problem, Analytic functions, Hadamard product, Starlike functions