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 + ⋯ + a n z n 1 + {a_1}z + \cdots + {a_n}{z^n} nonvanishing in the unit disc D {\mathbf {D}} is the dual hull of ( 1 − z ) n {(1 - z)^n} . More general for T = { m 1 , … , m n } T = \{ {m_1}, \ldots ,{m_n}\} let P ^ T {\hat P_T} denote the set of all lacunary polynomials 1 + a m 1 z m 1 + ⋯ + a m n z m n 1 + {a_{{m_1}}}{z^{{m_1}}} + \cdots + {a_{{m_n}}}{z^{{m_n}}} nonvanishing in D {\mathbf {D}} . In this paper we investigate whether the sets P ^ T {\hat P_T} are generated in a similar way. Some necessary conditions are given, and the case | T | ≤ 3 |T| \leq 3 is completely solved.