A Type of Eneström-Kakeya Theorem for Quaternionic Polynomials Involving Monotonicity with a Reversal - Gardner, Robert B.; Gladin, Matthew (Armenian Journal of Mathematics, 2025)
The Eneström-Kakeya theorem states that if P(z)=_=0^n a_z^ is a polynomial of degree n with real coefficients satisfying 0≤ a_0≤ a_1≤ …≤ a_n, then all zeros of P lie in |z|≤ 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.