On the zeros of certain composite polynomials and an operator preserving inequalities

Abstract

If all the zeros of nth degree polynomials f(z) and $g(z) = \sum _{k=0}^{n}\lambda _k\left( {\begin{array}{c}n\\ k\end{array}}\right) z^k$ respectively lie in the cricular regions $|z|\le r$ and $|z| \le s|z-\sigma |$, $s>0$, then it was proved by Marden (Geometry of polynomials, Math Surveys, No. 3, American Mathematical Society, Providence, 1949, p. 86) that all the zeros of the polynomial $h(z)= \sum _{k=0}^{n}\lambda _k f^{(k)}(z) \frac{(\sigma z)^k}{k!}$ lie in the circle $|z| \le r ~ \max (1,s)$. In this paper, we relax the condition that f(z) and g(z) are of the same degree and instead assume that f(z) and g(z) are polynomials of arbitrary degree n and m, respectively, $m\le n,$ and obtain a generalization of this result. As an application, we also introduce a linear operator which preserves Bernstein type polynomial inequalities.