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Lp mean estimates for an operator preserving inequalities between polynomials

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 $R\geq 1$, $p\geq 1$, \[\left\|B[P\circσ](z)\right\|_p \leq\frac{R^{n}|Λ_n|+|λ_{0}|}{\left\|1+z\right\|_p}\left\|P(z)\right\|_p,\] where $B$ is a $ \mathcal{B}_{n}$-operator with parameters $λ_{0}, λ_{1}, λ_{2}$ in the sense of Rahman \cite{qir}, $σ(z)=Rz$ and $Λ_n=λ_{0}+λ_{1}\frac{n^{2}}{2} +λ_{2}\frac{n^{3}(n-1)}{8}$. 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.