The estimates for the second Hankel determinant a 2 a 4 − a 3 2 of the analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + ⋯ , for which either z f ′ ( z ) / f ( z ) or 1 + z f ″ ( z ) / f ′ ( z ) is subordinate to a certain analytic function, are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.

For two different choices of φ, the sharp bound for the fifth coefficient of a normalized analytic function f satisfying zf′(z)/f(z)≺φ(z) is obtained by using a bound for a polynomial in the coefficients of functions with positive real part. Our proof uses a characterization of functions with positive real part in terms of certain positive semi-definite Hermitian form and certain well-known coefficient inequalities for functions with positive real part are shown to follow easily from this characterization.