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Partial sums of certain classes of analytic functions. (English) Zbl 1060.30022
Summary: 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 \({\mathcal S}^*\) 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 \({\mathcal K}\) of convex functions in the open unit disk \(\mathbb{U}\). However, the partial sum \(f_n(z)\) of \(f(z)\) is not starlike in \(\mathbb{U}\) and the partial sum \(g_n(z)\) of \(g(z)\) is not convex in \(\mathbb{U}\). 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.

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
Full Text: DOI
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