SUMMARY
Hankel matrices are related to a wide range of disparate determinant computations and algorithms and some very attractive computational properties are allocated to them. Also, the Hankel determinants are crucial factors in the research of singularities and power series with integral coefficients. It is specified that the Fekete-Szegö functional and the second Hankel determinant are equivalent to H1(2) and H2(2), respectively. In this study, the upper bounds were obtained for the second Hankel determinant of the subclass of bi-univalent functions, which is defined by subordination. It is worth noticing that the bounds rendered in the present paper generalize and modify some previous results.