On the zeros of a class of polynomials and related analytic functions (Q1355697)

Let \(p(z)= \sum^n_{j=0} \alpha_j z^j\) be a polynomial of degree \(n\) such that \(\alpha_j =a_je^{i\varphi} +b_j e^{i\psi}\), \(j=0,1, \dots, n,|\varphi -\psi |<\pi\) with real \(a_j, b_j\). If for some \(t_1> t_2\geq 0\) the following conditions are satisfied \[ a_k t_1t_2 +a_{k-1} (t_1-t_2) -a_{k-2} \geq 0, \quad b_kt_1 t_2+ b_{k-1} (t_1-t_2) -b_{k-2} \geq 0, \] for \(k=0,1, \dots, n+1\) with \(a_{-2} =a_{-1} =a_{n+1} =0= b_{n+1} =b_{-1} =b_{-2}\), then all the zeros of the polynomial \(p(z)\) lie in the disc \[ \begin{multlined} |z|<{1\over 2 |\alpha_n|} \Bigl[(a_n + b_n) (t_1-t_2) -a_{n-1} -b_{n-1}+ \\ \biggl\{\bigl((a_n+ b_n) (t_1-t_2) -a_{n-1} -b_{n-1} \bigr)^2 +4 |\alpha_n |t_1 \bigl((a_n+ b_n) t_2+ a_{n-1} +b_{n-1} \bigr) \biggr\}^{1/2} \Bigr]. \end{multlined} \] Some other results concerning the zeros of such polynomials and analytic functions are also investigated. These results generalize the known result of Eneström-Kakeya and the result of \textit{C. Karanicoloff} [Math. Lapok 14, 133-136 (1963; Zbl 0122.25203)].