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The Zalcman conjecture for close-to-convex functions

Author: Wan Cang Ma
Journal: Proc. Amer. Math. Soc. 104 (1988), 741-744
MSC: Primary 30C50; Secondary 30C45
MathSciNet review: 964850
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Abstract: Let $ S$ be the class of functions $ f(z) = z + \cdots $ analytic and univalent in the unit disk $ D$. For $ f(z) = z + {a_2}{z^2} + \cdots \in S$, Zalcman conjectured that $ \vert a_n^2 - {a_{2n - 1}}\vert\; \leq \;{(n - 1)^2}(n = 2,3, \ldots )$. This conjecture is verified for $ n \geq 4$ and close-to-convex functions.

References [Enhancements On Off] (What's this?)

  • [1] J. E. Brown and A. Tsao, On the Zalcman conjecture for starlike and typically real functions, Math. Z. 191 (1986), 467-474. MR 824446 (87h:30038)
  • [2] M. Fekete and G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85-89.
  • [3] L. Brickman, T. H. MacGregor and D. R. Wilken, Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc. 156 (1971), 91-107. MR 0274734 (43:494)

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