An integral estimate for an operator preserving inequalities between polynomials

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