Restrictions on the zeros of a polynomial as a consequence of conditions on the coefficients of even powers and odd powers of the variable

The classical Eneström–Kakeya Theorem states that if p(z)=∑v=0navzv is a polynomial satisfying 0⩽a0⩽a1⩽⋯⩽an, then all of the zeros of p(z) lie in the region |z|⩽1 in the complex plane. Many generalizations of the Eneström–Kakeya theorem exist which put various conditions on the coefficients of the polynomial (such as monotonicity of the moduli of the coefficients). We will introduce several results which put conditions on the coefficients of even powers of z and on the coefficients of odd powers of z. As a consequence, our results will be applicable to some polynomials to which these related results are not applicable.