On dual sets generated by lacunary polynomials

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 $|T| \leq 3$ is completely solved.