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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On a multiplier conjecture for univalent functions


Authors: V. Gruenberg, F. Rønning and St. Ruscheweyh
Journal: Trans. Amer. Math. Soc. 322 (1990), 377-393
MSC: Primary 30C55
DOI: https://doi.org/10.1090/S0002-9947-1990-0991960-7
MathSciNet review: 991960
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Abstract: Let $ \mathcal{S}$ be the set of normalized univalent functions, and let $ \mathcal{D}$ be the subset of $ \mathcal{S}$ containing functions with the property:

$\displaystyle \left\vert {f''(z)} \right\vert \leq {\text{Re}}f'(z),\quad \left\vert z \right\vert < 1$

. We present and discuss the following conjecture: For $ f \in \mathcal{D},{\text{g,}}h \in \overline {{\text{co}}} (\mathcal{S})$,

$\displaystyle {\text{Re}}\frac{1}{z}(f * {\text{g}} * h)(z) > 0,\quad \left\vert z \right\vert < 1$

. In particular, we prove that the conjecture holds with $ \mathcal{S}$ replaced by $ \mathcal{C}$, the class of close-to-convex functions, and show its truth for a number of special members of $ \mathcal{D}$. These latter results are extensions of old ones of Szegà and Kobori about sections of univalent functions.

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DOI: https://doi.org/10.1090/S0002-9947-1990-0991960-7
Article copyright: © Copyright 1990 American Mathematical Society