Lp mean estimates for an operator preserving inequalities between polynomials

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 $R\ge 1$, $p\ge 1$, \[\left\|B[P\circσ](z)\right\|_p \leq\frac{R^{n}|Λ_n|+|λ_{0}|}{\left\|1+z\right\|_p}\left\|P(z)\right\|_p,\] where $B$ is a ${\mathcal{B}}_n$-operator with parameters ${\lambda }_0,{\lambda }_1,{\lambda }_2$ 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 $0\le p<1$ as well.

16 Pages