On the zeros of polynomials and some related functions (Q1193056)

The author considers the zeros of a polynomial \(P(z)\) together with those of \((z-a)P'(z)-vP(z)\), where \(a\) and \(v\) are arbitrary complex constants, and extends some results obtained by Obrechkoff and Weisner on the relations between these sets of zeros. These results are applied to the zeros of certain quasi-trigonometric polynomials. Let \(F(z)=(z-a)P'(z)- vP(z)\). The theorem due to \textit{N. Obrechkoff} [C. R. Acad. Sci., Paris 208, 1270-1272 (1939; Zbl 0021.03604)] and \textit{L. Weisner} [Bull. Am. Math. Soc. 48, 283-286 (1942; Zbl 0060.052)] deals with the case of \(v=n/2\), where zeros of \(F(z)\) lie inside (on, outside) the same circle. \textit{T.Genchev} [C. R. Acad. Bulgare Sci. 28, 449-451 (1975; Zbl 0336.30002)] extended the above theorem to more general cases of \(v\). In Theorem I the author establishes six cases, in which, if the zeros of \(P(z)\) lie in a closed circular region \(G\), it follows that the zeros of \(F(z)\) lie in a second closed circular region \(G^*\). For Theorem II the function \(R(z)=P(z)/(z-a)^ v\) is defined and the six results of Theorem I are established for \(R^ 1(z)\). Theorem III deals with even powered polynomials and another rational function. Next the results are applied to functions \[ T(z)=e^{-ivz} \sum_{k=-n}^{k=n} a_ k e^{ikz} \] by means of the substitution \(w=e^{iz}\).