On the Kakeya–Eneström Theorem and Gegenbauer Polynomial Sums

An extension of the classical Kakeya–Eneström theorem is given. As an application we show that for $lambda geqq frac{1}{2}$, $ - 1 < x < 1$ and arbitrary nonincreasing sequences $a_k > 0$, $k = 0,1, cdots ,n$, we have [sum_{k = 0}^n {a_k frac{{C_k^{(lambda )} (x)}}{{C_k^{(lambda )} (1)}}z^k e 0,quad | z | leqq 1,} ] where $C_k^{(lambda )} $, are the Gegenbauer or ultraspherical polynomials. This extends an old result due to G. Szegö and settles two recent conjectures of R. Askey and J. Bustoz. Other related results are obtained as well.