Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On dual sets generated by lacunary polynomials

Authors: Hans Dobbertin and Volker Kasten
Journal: Proc. Amer. Math. Soc. 111 (1991), 323-330
MSC: Primary 30C10; Secondary 30C15, 30C50
MathSciNet review: 935105
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The notion of dual sets of analytic functions has been developed by Ruscheweyh. In terms of this theory a well-known convolution theorem of Szegö states that the set of all polynomials $ 1 + {a_1}z + \cdots + {a_n}{z^n}$ nonvanishing in the unit disc $ {\mathbf{D}}$ is the dual hull of $ {(1 - z)^n}$. More general for $ T = \{ {m_1}, \ldots ,{m_n}\} $ let $ {\hat P_T}$ denote the set of all lacunary polynomials $ 1 + {a_{{m_1}}}{z^{{m_1}}} + \cdots + {a_{{m_n}}}{z^{{m_n}}}$ nonvanishing in $ {\mathbf{D}}$. In this paper we investigate whether the sets $ {\hat P_T}$ are generated in a similar way. Some necessary conditions are given, and the case $ \vert T\vert \leq 3$ is completely solved.

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

  • [1] H. Dobbertin, A remark on dual sets, with application to univalent trinomials, submitted.
  • [2] R. C. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1965. MR 0180696 (31:4927)
  • [3] V. Kasten and St. Ruscheweyh, On dual sets of analytic functions, Math. Nachr. 123 (1985), 277-283. MR 809350 (87c:30039)
  • [4] W. Marden, The geometry of the zeros of a polynomial in a complex variable, Amer. Math. Soc., Providence, RI, 1949. MR 0031114 (11:101i)
  • [5] St. Ruscheweyh, Convolutions in geometric function theory, Les Presses de l'Université de Montréal, Montréal, 1982. MR 674296 (84a:30029)
  • [6] G. Szegö, Bemerkungen zu einem Satz von J. H. Grace über die Wurzeln algebraischer Gleichungen, Math. Z. 13 (1922), 28-55. MR 1544526

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C10, 30C15, 30C50

Retrieve articles in all journals with MSC: 30C10, 30C15, 30C50

Additional Information

Keywords: Convolution, dual sets, lacunary polynomials, Szegö's Theorem
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society