In this paper, we estimate the second and third Maclaurin's coefficients of certain subclass of bi-univalent functions in open unit disk defined by Caputo Fractional calculus operator and also indicate certain special cases.

In this work, we introduce and investigate two new subclasses of the bi-univalent functions in which both f and f−1 are m -fold symmetric analytic functions. For functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for |am+1| and |a2m+1| .

In this research contribution, we have constructed a subclass of analytic bi-univalent functions using Horadam polynomials. Bounds for certain coefficients and Fekete- Szegö inequalities been estimated.

For a certain subclass of Bazilevic functions, Faber polynomials expansions are used to obtain bi-univalent properties. Estimates on the $n$th Taylor-Maclaurin coefficients functions in this class found. Moreover, some special cases also indicated.

In this article, we establish the bounds for initial Taylor–Maclaurin coefficients |a2| and |a3| a new family G?(?,?,?;h) of holormorphic bi-univalent functions which involve prestarlike functions. Furthermore, investigate Fekete–Szegö type inequality, special cases consequences.