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 $\left|a_3-\mu a_2^2\right|.$

In this paper, we introduce and investigate new subclasses of bi-univalent functions with respect to the symmetric points in $U=\left\{z\in \mathbb{C}:\left|z\right|<1\right\}$ defined by Bernoulli polynomials. We obtain upper bounds for Taylor–Maclaurin coefficients $\left|a_2\right|,$$\left|a_3\right|$ and Fekete–Szegö inequalities $\left|a_3-\mu a_2^2\right|$ for these new subclasses.

Let the function $f\left( z \right) =z+\sum_{k=2}^{\infty}a{_{k}}z {^{k}}\in A$ be locally univalent for $z \in \mathbb{D}%:=\{z \in \mathbb{C}:{|}z {|}<1\}$ and $0\leq\alpha\alpha,\quad z \in \mathbb{D}.\end{equation*}%Due to their geometrical characteristics, this class has a significantimpact on the theory of geometric functions. In the article we obtain sharp bounds for the second Hankel determinant \begin{equation*}\left\vert H_{2}\left( 2\right) \left( f\right) \right\vert =\left\verta_{2}a_{4}-{a_{3}^{2}}\right\vert \end{equation*}and some Toeplitz determinants \begin{equation*}\left\vert {T}_{3}\left( 1\right) \left( f\right) \right\vert =\left\vert 1-2%{a_{2}^{2}}+2{a_{2}^{2}}a_{3}-{a_{3}^{2}}\right\vert,\ \\left\vert {T}_{3}\left( 2\right) \left( f\right) \right\vert =\left\vert {%a_{2}^{3}}-2a_{2}{a_{3}^{2}}+2{a_{3}^{2}}a_{4}-a_{2}{a_{4}^{2}}\right\vert \end{equation*}of a subclass of analytic functions $M(\alpha )$ in the open unit disk $%\mathbb{D}$.

In this study, we introduce a new class of bi-univalent functions using q-Ultraspherical polynomials. We derive the Fekete and Szegö functional problems for functions in this new subclass, as well as estimates for the Taylor-Maclaurin coefficients |α2| and |α3|. Furthermore, a collection of fresh outcomes is presented by customizing the parameters employed in our initial discoveries.