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.

如果第n次多项式f（z）和\（g（z）= \ sum _ {k = 0} ^ {n} \ lambda _k \ left（{\ begin {array} {c} n \ \ k \ end {array}} \ right）z ^ k \）分别位于环形区域\（| z | \ le r \）和\（| z | \ le s | z- \ sigma | \）中，\（s> 0 \），然后由Marden（多项式的几何，Math Surveys，No.3，American Mathematical Society，Providence，1949，p.86）证明了多项式\（h（z ）= \ sum _ {k = 0} ^ {n} \ lambda _k f ^ {（（k）}（z）\ frac {（\ sigma z）^ k} {k！} \）位于圆\（ | z | \ le r〜\ max（1，s）\）。在本文中，我们放宽了f（z）和克（Ž）是相同程度，并代之以假定˚F（Ž）和克（Ž）是任意程度的多项式Ñ和米分别\（M \了N，\） ，将获得的这一个概括结果。作为应用程序，我们还引入了保留伯恩斯坦类型多项式不等式的线性算子。