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On partial sums of convex functions of order \(\alpha\). (English) Zbl 0596.30025
f(z)\(=z+a_ 2z^ 2+...\), regular in the open unit disc U, is convex of order \(\alpha\), \(0\leq \alpha \leq 1\), if \[ Re\{1+zf''(z)/f'(z)\}\geq \alpha,\quad for\quad z\quad in\quad U. \] The author studies the function \(J_ n(f(z))\), where \[ J_ n(f(z))=f(z)/P_ n(f(z)),\quad P_ n(f(z))=z+a_ 2z^ 2+...+a_ nz^ n. \] She shows for example, that \(Re\{J_ n(f(z))\}\geq (3-\alpha)/(4-2\alpha)\), \(n=1,2,..\). under additional conditions on f(z). Subordination is a main tool used in this work. She also shows that f(z)/z is subordinate to a function which is given explicitly (in terms of \(\alpha)\). Her work continues earlier explorations of similar questions by L. Brickman, D. J. Hallenbeck, T. H. MacGregor and D. R. Wilken [Trans. Am. Math. Soc. 185 (1973), 413-428 (1974; Zbl 0278.30021)] and T. Sheil- Small [Bull. Lond. Math. Soc. 2, 165-168 (1970; Zbl 0217.097)].
Reviewer: R.Libera

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