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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
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.

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

  • [1] L. Bieberbach, Einführung in die konforme Abbildung, De Gruyter, Berlin, 1967. MR 0236365 (38:4661)
  • [2] L. de Branges, A Proof of the Bieberbach Conjecture, Acta Math. 154 (1985), 137-152. MR 772434 (86h:30026)
  • [3] P. L. Duren, Univalent functions, Springer-Verlag, 1983. MR 708494 (85j:30034)
  • [4] D. J. Hallenbeck and T. H. MacGregor, Linear problems and convexity techniques in geometric function theory, Pitman, 1984. MR 768747 (86c:30016)
  • [5] A. Kobori, Zwei Sätze über die Abschnitte schlichter Potenzreihen, Mem. Coll. Kyoto 17 (1934), 172-186.
  • [6] St. Ruscheweyh, Extension of Szegà's theorem on the sections of univalent functions, SIAM J. Math. Anal. 19 (1988), 1442-1449. MR 965264 (89m:30032)
  • [7] T. Sheil-Small, On the convolution of analytic functions, J. Reine Angew. Math. 258 (1973), 137-152. MR 0320761 (47:9295)
  • [8] G. Szegà, Zur Theorie der schlichten Abbildungen, Math. Ann. 100 (1928), 188-211. MR 1512482

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