[For the entire collection see Zbl 0545.00020.]
The paper is concerned with the location of the zeros of a complex polynomial expressed as a linear combination of the form \[ P_ n(z)=\Pi_ n(z)-i\theta_{n-1}\Pi_{n-1}(z) \] where \(\{\Pi_ n\}\) is a system of monic polynomials orthogonal with respect to an even weight function on (-a,a), \(0<a<\infty\), and \(\theta_{n-1}\) is a real constant. First, the following lemma is proved: for each \(z\in \partial D_ a\), \(| \Pi_ k(z)/\Pi_{k-1}(z)| \geq \Pi_ k(a)/\Pi_{k- 1}(a),\) \(k=1,2,..\). where \(D_ a=\{z:\quad | z| <a\}\) and \(\partial D_ a\) is the boundary of \(D_ a\). The proof is based on a recurrence relation. Then, using the following:
1) The above lemma, and 2) Rouché’s theorem related to the number of zeros of f and \(f+g\) inside \(\Gamma\) where \(\Gamma\) is a rectifiable Jordan curve inside an open region where f and g are analytic and \(| g(z)| <| f(z)|\) for all \(z\in \Gamma\) and 3) A result by Giroux concerning the location of the zeros of \(f(x)+icg(x)\) where \[ f(x)=\prod^{n}_{i=1}(x-x_ i),\quad g(x)=\prod^{n}_{i=1}(x-y_ i) \] and \(x_ 1<y_ 1<x_ 2<...<y_{m-1}<x_ n\), an interesting theorem becomes readily apparent. Namely, all the zeros of \(P_ n(z)\), under some given restrictions on \(\theta_{n-1}\), are contained in a half disc of radius a. An illustrative example is also provided.