On a multiplier conjecture for univalent functions
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- by V. Gruenberg, F. Rønning and St. Ruscheweyh PDF
- Trans. Amer. Math. Soc. 322 (1990), 377-393 Request permission
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: \[ \left | {f"(z)} \right | \leq {\text {Re}}f’(z),\quad \left | z \right | < 1\]. We present and discuss the following conjecture: For $f \in \mathcal {D},{\text {g,}}h \in \overline {{\text {co}}} (\mathcal {S})$, \[ {\text {Re}}\frac {1}{z}(f * {\text {g}} * h)(z) > 0,\quad \left | z \right | < 1\]. 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
- Ludwig Bieberbach, Einführung in die konforme Abbildung, Sammlung Göschen, Bd. 768/768a, Walter de Gruyter & Co., Berlin, 1967 (German). Sechste, neubearbeitete Auflage. MR 0236365
- Louis de Branges, A proof of the Bieberbach conjecture, Acta Math. 154 (1985), no. 1-2, 137–152. MR 772434, DOI 10.1007/BF02392821
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- D. J. Hallenbeck and T. H. MacGregor, Linear problems and convexity techniques in geometric function theory, Monographs and Studies in Mathematics, vol. 22, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 768747 A. Kobori, Zwei Sätze über die Abschnitte schlichter Potenzreihen, Mem. Coll. Kyoto 17 (1934), 172-186.
- Stephan Ruscheweyh, Extension of Szegő’s theorem on the sections of univalent functions, SIAM J. Math. Anal. 19 (1988), no. 6, 1442–1449. MR 965264, DOI 10.1137/0519107
- T. Sheil-Small, On the convolution of analytic functions, J. Reine Angew. Math. 258 (1973), 137–152. MR 320761, DOI 10.1515/crll.1973.258.137
- G. Szegö, Zur Theorie der schlichten Abbildungen, Math. Ann. 100 (1928), no. 1, 188–211 (German). MR 1512482, DOI 10.1007/BF01448843
Additional Information
- © Copyright 1990 American Mathematical Society
- 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