A classical theorem due to Eneström and Kakeya gives some bounds for the moduli of the zeros of polynomials having a monotone sequence of non-negative (real) coefficients. Nowadays, one can find several modifications and generalizations of this result in the literature. The main subject of the paper is a study of links of coefficients which occur in some of these general results with a view to the recurrence relations fulfilled by systems of orthogonal polynomials on the unit circle. In particular, we discuss the question, if the relevant links are consistent with the recurrence relations or not. This leads to some new insight into the analyzed classes of polynomials.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: LA 1386/3–2
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