In this present work, the authors obtain Fekete-Szeg$ddot{o}$ inequality for certain normalized analytic function $f(z)$ defined on the open unit disk for which $$frac{{lambda}{eta}z^3(L(a,c)f(z))^{{prime}{prime}{prime}}+(2{lambda}{eta}+{lambda}-{eta})z^2(L(a,c)f(z))^{{prime}{prime}}+z(L(a,c)f(z))^{{prime}}}{{lambda}{eta}z^2(L(a,c)f(z))^{{prime}{prime}}+({lambda}-{eta})z(L(a,c)f(z))^{prime}+(1-{lambda}+{eta})(L(a,c)f(z))};(0{leq}{eta}{leq}{lambda}{leq}1)$$ 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-Szeg$ddot{o}$ inequality for a class of functions defined through fractional derivatives are obtained.