Inequalities concerning $B$-operators

Let $\mathcal{P}_{n}$ be the class of polynomials of degree at most $n.$ Rahman introduced the class $\mathcal {B}_{n}$ of operators $B$ that map $\mathcal {P}_{n}$ into itself. In this paper we prove some results concerning such operators and thereby obtain generalizations of some well known polynomial inequalities.