Some generalizations of the Eneström-Kakeya theorem (Q1307310)

Page 137 of \textit{M. Marden}'s [Geometry of polynomials (1966; Zbl 0162.37101)] contains the following result, equivalent to the better known Eneström-Kakeya theorem: All the zeros of the polynomial \(f(z)=a_0+a_1z+\dots+a_n z^n\) having real positive coefficients \(a_j\) lie in a ring \(\rho_1\leq| z| \leq \rho_2\) where \(\rho_1=\min(a_k/a_{k+1})\), \(\rho_2=\max(a_k/a_{k+1})\) for \(k=0,\dots,n-1\). The main theorem of the present paper generalizes this result in several ways, and its corollaries strengthen more recent results. The coefficients are no longer assumed to be positive, or even real: \(a_j=\alpha_j+i\beta_j\), \(j=0,\dots,n\), \(a_n\neq 0\). It is assumed that there exists a positive number \(t\) and a nonnegative integer \(k\) such that \(\alpha_0\leq t\alpha_1\leq t^2\alpha_2\leq \dots \leq t^k\alpha_k\), and \(t^k\alpha_k\geq t^{k+1}\alpha_{k+1}\geq \dots\geq t^n\alpha_n\). Then there are two quantities \(M_1\) and \(M_2\) depending on \(t\), \(k\), and the \(a_j\) such that \(p(z)\) has all its zeros in \((t^2| a_0|)/M_1\leq | z| \leq \max(M_2/| a_n| ,1/5)\). The proof involves investigating \(P(z):=(t-z)p(z)\). Let \(G_2(z):=P(z)+a_n z^{n+1}\) and \(G_1(z):=P(z)-ta_0\). Estimates on these functions show that \(P(z)\neq 0\) outside the annulus.