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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Radii problems for generalized sections of convex functions


Authors: Richard Fournier and Herb Silverman
Journal: Proc. Amer. Math. Soc. 112 (1991), 101-107
MSC: Primary 30C45; Secondary 30C50
DOI: https://doi.org/10.1090/S0002-9939-1991-1047000-3
MathSciNet review: 1047000
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Abstract: A classical theorem of Szëgo states that for functions $ f(z) = z + \Sigma _{k = 2}^\infty {a_k}{z^k}$ convex in $ \vert z\vert < 1$, the sequence of partial sums $ {f_n}(z) = z + \Sigma _{k = 2}^n{a_k}{z^k}$ must be convex in $ \vert z\vert < \frac{1}{4}$. For the more general family consisting of functions of the form $ z + \Sigma _{k = 2}^\infty {a_{{n_k}}}{z^{{n_k}}}$, where $ \left\{ {{n_k}} \right\}$ denotes an increasing (finite or infinite) sequence of integers $ ( \geq 2)$, we find the radius of convexity $ ( \approx 0.21)$ and the radius of starlikeness $ ( \approx 0.37)$. The extremal function in both cases is $ z + {z^2}/(1 - {z^2}) = z + \Sigma _{k = 1}^\infty {z^{2k}}$ associated with the convex function $ z/(1 - z) = z + \Sigma _{k = 2}^\infty {z^k}$.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1047000-3
Article copyright: © Copyright 1991 American Mathematical Society