Sharpened forms of the generalized Schwarz inequality on the boundary

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function f(z) = 1 + c_pz^p + c_{p + 1}z^{p + 1}…, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account c_{p + 1} and zeros of the function f(z) ? 1. The sharpness of these inequalities is also proved.