The Zalcman conjecture for close-to-convex functions
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- Proc. Amer. Math. Soc. 104 (1988), 741-744 Request permission
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 $|a_n^2 - {a_{2n - 1}}|\; \leq \;{(n - 1)^2}(n = 2,3, \ldots )$. This conjecture is verified for $n \geq 4$ and close-to-convex functions.References
- Johnny E. Brown and Anna Tsao, On the Zalcman conjecture for starlike and typically real functions, Math. Z. 191 (1986), no. 3, 467–474. MR 824446, DOI 10.1007/BF01162720 M. Fekete and G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, J. London Math. Soc. 8 (1933), 85-89.
- 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 274734, DOI 10.1090/S0002-9947-1971-0274734-2
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 741-744
- MSC: Primary 30C50; Secondary 30C45
- DOI: https://doi.org/10.1090/S0002-9939-1988-0964850-X
- MathSciNet review: 964850