Restrictions on the zeros of a polynomial as a consequence of conditions on the coefficients of even powers and odd powers of the variable (Q1811616)

By a theorem of Eneström-Kakeya, a polynomial \(f(z)=\sum_{h=0}^n a_hz^h\) and so that \(0\leq a_0\leq a_1\leq\cdots\leq a_n\) has all its zeros in the disc \(|z|\leq 1\); this result has been extensively generalised (for example so that ``monotonicity conditions'' on the real, respectively imaginary parts of complex coefficients suffice to place the zeros in some annulus). Denote by \(g\) and \(h\) the odd and even parts of \(f\). The present paper provides yet further gneralisation based on the observation that appropriate monotonicity conditions separately on the coefficients of \(g\) and of \(h\) already confine the zeros of \(f\) to some annulus in the complex plane.