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We introduce a new class of bi-univalent functions defined in the open unit disc and connected with a $q$-convolution. We find estimates for the general Taylor-Maclaurin coefficients of the functions in this class by using Faber polynomial expansions and we obtain an estimation for the Fekete-Szegö problem for this class.

We obtain several fuzzy differential subordinations by using a linear operator ${ℐ}_{m,\gamma }^{n,\alpha }f\left(z\right)=z+\sum _{k=2}^{\infty }{(1+\gamma (k-1))}^n{m}^{\alpha }{(m+k)}^{-\alpha }{a}_k{z}^k$. Using the linear operator ${ℐ}_{m,\gamma }^{n,\alpha },$ we also introduce a class of univalent analytic functions for which we give some properties.