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.