On a Composition Preserving Inequalities between Polynomials

Abstract

The Schur-Szegö composition of two polynomials \(f\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{z^j}} \) and \(g\left( z \right) = \sum\nolimits_{j = 0}^n {{B_j}{z^j}} \), both of degree n, is defined by \(f * g\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{B_j}{{\left( {\begin{array}{*{20}{c}}
n \\
j
\end{array}} \right)}^{ - 1}}{z^j}} \). In this paper, we estimate the minimum and the maximum of the modulus of f * g(z) on z = 1 and thereby obtain results analogues to Bernstein type inequalities for polynomials.