Generalizations of the Eneström–Kakeya Theorem Involving Weakened Hypotheses

The well-known Eneström–Kakeya Theorem states that, for P(z)=∑ℓ=0naℓzℓ, a polynomial of degree n with real coefficients satisfying 0≤a0≤a1≤⋯≤an, then all the zeros of P lie in |z|≤1 in the complex plane. Motivated by recent results concerning an Eneström–Kakeya “type” condition on real coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients and concerning the moduli of the coefficients. In this way, our results generalize the other recent results