Partial sums of certain classes of analytic functions

Abstract

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.

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Keywords:

• Analytic functions
• Univalent functions
• Starlike functions
• Convex functions
• Alexander's theorem
• Partial sums
• Extremal functions
• Libera integral transform
• Image domains

Acknowledgements

The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

Notes

E-mail: [email protected]