An Eneström–Kakeya Theorem with Monotonicity Conditions on the Even- and Odd-Indexed Coefficients of a Polynomial

The classical Eneström–Kakeya theorem states that an n-degree polynomial p(z)=∑k=0nakzk with real coefficients satisfying 0≤a0≤a1≤⋯≤an has all of its zeros in |z|≤1 in the complex plane. Numerous generalizations of this result exist, many of them weakening the condition on the coefficients in order to be applicable to a larger class of polynomials. In this paper, a monotonicity condition on the real and imaginary parts of the even- and odd-indexed coefficients is imposed and bounds on the location of the zeros are established.