Polynomials with weighted sum

Abstract

In this paper, we study the equation zn = ∑ k=0n-1 akzk, where ∑k=0n-1 ak = 1, ak ≥ 0 f°r each k. We show that, given p > 1, there exist C(1/p)-polynomials with the degree of weighted sum n - 1. However, we obtain sufficient conditions for nonexistence of certain lacunary C(1/p)-polynomials. In case of the degree of weighted sum n - 2, we see that, by giving an example, our sufficient condition is best possible in a certain sense.