For $\lambda\geq0$ and $0\leq\lambda<1<\beta$, we denote by $\mathcal K(\lambda;\alpha,\beta)$ the class of normalized analytic functions satisfying the two sided-inequality \[ lpha<\mathfrak R\Big(\frac{zf'(z)}{f(z)}+ambda\frac{z^2f''(z)}{f(z)}\Big)<\beta\quad(zı\Bbb U), \] where $\Bbb U$ is the open unit disk. Let $\cal K_\Sigma(\lambda;\alpha,\beta)$ be the class of bi-univalent functions such that $f$ and its inverse $f^{-1}$ both belong to the class $\cal K(\lambda;\alpha,\beta)$. In this paper, we establish bounds for the coeffcients, and solve the Fekete--Szegö problem, for the class $\cal K(\lambda;\alpha,\beta)$. Furthermore, we obtain upper bounds for the first two Taylor--Maclaurin coeffcients of the functions in the class $\cal K_\Sigma(\lambda;\alpha,\beta)$.