Lp mean estimates for an operator preserving inequalities between polynomials

TL;DR

This paper establishes sharp $L_p$-inequalities for $ ext{B}_n$-operators acting on polynomials, correcting and generalizing previous results to include the case where $0 \,\leq\, p < 1$, with applications to polynomial inequalities.

Contribution

It provides a correct and more general proof of $L_p$-inequalities for $ ext{B}_n$-operators, extending previous results to all $p \geq 0$ and correcting earlier inaccuracies.

Findings

Corrected proof of polynomial inequalities for $ ext{B}_n$-operators.

Extended inequalities to include $0 \leq p < 1$.

Unified framework for $L_p$-estimates in polynomial operator theory.

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 $R\geq 1$, $p\geq 1$, \[\left\|B[P\circ\sigma](z)\right\|_p \leq\frac{R^{n}|\Lambda_n|+|\lambda_{0}|}{\left\|1+z\right\|_p}\left\|P(z)\right\|_p,\] 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.