Recently, in their pioneering work on the subject of bi-univalent functions, Srivastava et al. cite{HMS-AKM-PG} actually revived the study of the coefficient problems involving bi-univalent functions. Inspired by the pioneering work of Srivastava et
al. cite{HMS-AKM-PG}, there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions. Motivated largely by Ali et al. cite{Ali-Ravi-Ma-Mina-class}, Srivastava et al. cite{HMS-AKM-PG} and G{u}ney et al. cite{HOG-GMS-JS-Fib-2018} in this paper, we consider certain classes of bi-univalent functions related to shell-like curves connected with Fibonacci numbers to obtain the estimates of second, third Taylor-Maclaurin coefficients and Fekete - Szeg{o} inequalities. Further, certain special cases are also indicated. Some interesting remarks of the results presented here are also discussed.
Making use of Chebyshev polynomials, we obtain upper bound estimate for the second Hankel determinant of a subclass $mathcal{N}_{sigma }^{mu}left( lambda ,tright) $ of bi-univalent function class $sigma.$
