A Sharp Generalization of Schwarz and Caratheodory Inequality on the Boundary

Abstract

In this paper, a boundary version of the Generalization of Schwarz and Caratheodory inequality are investigated. New inequalities of the Caratheodory's and Schwarz inequality at boundary are obtained by taking into account zeros of f(z) function which are different from zero. Also, we obtain sharp lower bounds on the angular derivative f'(z(0)) at the point z(0) where vertical bar z(0)vertical bar = vertical bar f(z(0))vertical bar = 1.