Vinay K. Jain: A refinement of Eneström-Kakeya theorem (special type), 321-325

Abstract:

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

Key Words: zeros, polynomial, positive coefficients.

2010 Mathematics Subject Classification: Primary 30C15, Secondary 30C10.