Consider the class of polynomials P (z) = (Formula Presented) with 0 ≤ a0 ≤ a1 ≤ · · · ≤ an. The classical Eneström–Kakeya Theorem states that any polynomial in this class has all its zeros in the unit disk |z| ≤ 1 in the complex plane. We introduce new classes of polynomials by imposing a monotonicity-type condition on the coefficients with all indices congruent modulo m for some given m ≤ n. We give the inner and outer radii of an annulus containing all zeros of such polynomials. We also give an upper bound on the number of zeros in a disk for polynomials in these classes.

The well-known Eneström–Kakeya Theorem states that a polynomial with real, nonnegative, monotone increasing coefficients has all its complex zeros in the closed unit disk in the complex plane. In this paper, we extend this result by showing that all quaternionic zeros of such a polynomial lie in the unit sphere in the quaternions. We also extend related results from the complex to quaternionic setting.