Zeros of Quaternionic Polynomials with Incomplete Monotonicity Conditions on the Coefficients

The classical Eneström–Kakeya Theorem restricts the location of the complex zeros of polynomials with real, positive, monotone increasing coefficients. That is, for p(z)=∑v=0navzv, where 0≤a0≤a1≤…≤an, the zeros of p lie in the unit disk |z|≤1 in the complex plane. Following the introduction of an analytic theory of functions of a quaternionic variable, this result was extended to polynomials of a quaternionic variable. Numerous generalizations of both the complex and quaternionic versions of the Eneström–Kakeya Theorem have appeared which modify the monotonicity condition and extend results to complex and quaternionic coefficients. We give a related theorem which generalizes several of the known results and includes them as corollaries. We impose a type-of-monotonicity condition on some of the real and imaginary parts of the coefficients of the polynomial.