Sharpened forms of the generalized Schwarz inequality on the boundary

Abstract

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(p)z(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.