Lp mean estimates for an operator preserving inequalities between polynomials (1306.0714v1)

Abstract: 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|BP\circ\sigma\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.