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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.

MSC:
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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References:
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[6] DOI: 10.1155/S0161171201005099 · Zbl 0984.30008 · doi:10.1155/S0161171201005099
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