A Type of Eneström-Kakeya Theorem for Quaternionic Polynomials Involving Monotonicity with a Reversal

The Eneström-Kakeya theorem states that if $P(z)=\sum_{\ell =0}^n a_\ell z^\ell$ is a polynomial of degree $n$ with real coefficients satisfying $0\leq a_0\leq a_1\leq \cdots\leq a_n$, then all zeros of $P$ lie in $|z|\leq 1$ in the complex plane. Motivated by recent results concerning an Eneström-Kakeya "type" condition on the real and imaginary parts of complex coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients of a quaternionic polynomial. We give bounds on the moduli of quaternionic zeros of such polynomials.