An Eneström–Kakeya Theorem for New Classes of Polynomials

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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.