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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:
 [1] Duren PL, Univalent Functions (1983) [2] Goodman AW, Univalent Functions (1983) [3] Szegö G, Math. Ann., 100 pp 188– (1928) [4] Padmanabhan KS, Ann. Polon. Math. 23 pp 83– (1970) [5] Li J-L, J. Math. Anal. Appl., 213 pp 444– (1997) [6] DOI: 10.1155/S0161171201005099 · Zbl 0984.30008 · doi:10.1155/S0161171201005099
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