The Ramanujan Journal ( IF 0.6 ) Pub Date : 2020-07-20 , DOI: 10.1007/s11139-020-00261-2 N. A. Rather , Ishfaq Dar , Suhail Gulzar
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.
中文翻译:
关于某些复合多项式的零点和一个保留不等式的算子
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。