Partial sums of certain classes of analytic functions

In Geometric Function Theory it is well known that the familiar Koebe function f(z)=z/(1-z)(2) is the extremal function for the class S* of starlike functions in the open unit disk U and also that the function g(z)=z/(1-z) is the extremal function for the class K of convex functions in the open unit disk U. However, the partial sum f(n) (z) of f(z) is not starlike in U and the partial sum g(n) (z) of g(z) is not convex in U. The aim of the present paper is to investigate the starlikeness and convexity of these partial sums f(n) (z) and g(n) (z). Computational and graphical usages of Mathematica (Version 4.0) as well as geometrical descriptions of the image domains in several illustrative examples are also presented.