  If $P(z)$ be a polynomial of degree at most $n$ which does not vanish in $|z|
< 1$, it was recently formulated by Shah and Liman \cite[\textit{Integral
estimates for the family of $B$-operators, Operators and Matrices,}
\textbf{5}(2011), 79 - 87]{wl} that for every $R\geq 1$, $p\geq 1$,
  \[\left\|B[P\circ\sigma](z)\right\|_p
\leq\frac{R^{n}|\Lambda_n|+|\lambda_{0}|}{\left\|1+z\right\|_p}\left\|P(z)\right\|_p,\]
where $B$ is a $ \mathcal{B}_{n}$-operator with parameters $\lambda_{0},
\lambda_{1}, \lambda_{2}$ in the sense of Rahman \cite{qir}, $\sigma(z)=Rz$ and
$\Lambda_n=\lambda_{0}+\lambda_{1}\frac{n^{2}}{2}
+\lambda_{2}\frac{n^{3}(n-1)}{8}$. Unfortunately the proof of this result is
not correct. In this paper, we present a more general sharp $L_p$-inequalities
for $\mathcal{B}_{n}$-operators which not only provide a correct proof of the
above inequality as a special case but also extend them for $ 0 \leq p <1$ as
well.
