According to the well known Eneström-Kakeya theorem, a polynomial
![$ p(z) = \sum_{j=0}^{n}a_jz^j$](img38.png)
of degree
![$ n$](img11.png)
with real coefficients satisfying
![$ a_n \geq a_{n-1} \geq a_{n-2} \geq \hdots \geq a_1 \geq a_0 > 0$](img39.png)
has all its zeros in
![$ \vert z\vert \leq 1$](img40.png)
. In this paper we obtain a refinement of this theorem in the special case that the coefficients of
![$ p$](img33.png)
satisfy the condition
![$ a_n > a_{n-1} > a_{n-2} > \hdots > a_1 > a_0 > 0$](img41.png)
.