We introduce and investigate in this paper a new subclass of bi-univalent functions associated with the Gegenbauer polynomials which satisfy subordination conditions defined in a symmetric domain, which is the open unit disc. For this new subclass, we obtain estimates for the Taylor–Maclaurin coefficients $\left|a_2\right|,$$\left|a_3\right|$ and the Fekete–Szegö inequality $\left|a_3-\mu a_2^2\right|.$

By using Dziok-Srivastava operator a new subclass of analytic functions generalized k-parabolic starlike functions, denoted by k–SPl,m (α1;γ), is introduced. For this class the Fekete-Szegö problem is completely solved. Various known or new special cases of our results are also point out.

For , , , , and , a new class of analytic functions defined by means of the differential operator is introduced. Our main object is to provide sharp upper bounds for Fekete-Szegö problem in . We also find sufficient conditions for a function to be in this class. Some interesting consequences of our results are pointed out.

The main goal of this investigation is to obtain sharp upper bounds for Fekete-Szegö functional and the third Hankel determinant for a certain subclass ${\mathcal{SL}}^{\ast }\left(u,v,\alpha \right)$ of holomorphic functions defined by the Carlson-Shaffer operator in the unit disk. Finally, for some special values of parameters, several corollaries were presented.