Proceedings of the American Mathematical Society

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



On the partial sums of convex functions of order $ 1/2$

Author: Ram Singh
Journal: Proc. Amer. Math. Soc. 102 (1988), 541-545
MSC: Primary 30C45; Secondary 30C55
MathSciNet review: 928976
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Abstract: Let $ f\left( z \right) = z + {a_2}{z^2} + \ldots $ be regular and univalently convex of order $ 1/2$ in the unit disc $ U$ and let $ {s_n}\left( {z,f} \right)$ denote its $ n$th partial sum. In the present note we determine the radius of convexity of $ {s_n}\left( {z,f} \right)$, depending on $ n$, and generalize and sharpen a result of Ruscheweyh concerning the partial sums of convex functions. We also prove that for every $ n \geq 1,{\text{Re}}\left( {{s_n}\left( {z,f} \right)/z} \right) > 1/2$ in $ U$.

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Keywords: Univalent, starlike and convex functions, convex null sequences
Article copyright: © Copyright 1988 American Mathematical Society