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

If all the zeros of $n$th degree polynomials $f(z)$ and $g(z)={\sum }_{k=0}^n{\lambda }_k\left(\genfrac{}{}{0px}{}{n}{k}\right)z^k$ respectively lie in the cricular regions $\mathrm{\mid }z\mathrm{\mid }\le r$ and $\mathrm{\mid }z\mathrm{\mid }\le s\mathrm{\mid }z-\sigma \mathrm{\mid }$, $s>0$, then it was proved by Marden \cite[p. 86]{mm} 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 $\mathrm{\mid }z\mathrm{\mid }\le r\max (1,s)$. 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 preserve Bernstein type polynomial inequalities.