Inclusion radii for the zeros of special polynomials (Q2895943)

The paper deals with complex-valued polynomials of the form NEWLINE\[NEWLINE f(z)=f_{n_1}(z) g_{n_2}(z)= \sum_{i=0}^{n_1} b_i z^i \sum_{i=0}^{n_2} c_i z^i NEWLINE\]NEWLINE where \(|b_{n_1}|>|b_i|\), \(0 \leq i \leq n_1-1\), and \(|c_{n_2}|>|c_i|\), \(0 \leq i \leq n_2-1\). The author obtains the following estimate for the radius of the disk in the complex plane centered at the origin containing all zeros of the polynomial: NEWLINE\[NEWLINER=\max\left \{\frac{1+\phi_1}{2}+\frac{\sqrt{(\phi_1-1)^2+4M_1}}{2},\;\;\frac{1+\phi_2}{2}+\frac{\sqrt{(\phi_2-1)^2+4M_2}}{2}\right\},NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\phi_1=\frac{b_{n_1-1}}{b_{n_1}}, \quad\phi_2=\frac{c_{n_2-1}}{c_{n_2}}, \quad M_1= \max_{0 \leq i \leq n_1-2}\frac{b_{i}}{b_{n_1}}, \quad M_2= \max_{0 \leq i \leq n_2-2}\frac{c_{i}}{c_{n_2}}.NEWLINE\]NEWLINE Therefore, all zeros lie in the disk \(|z|<2\).