$L^q$ Inequalities and Operator Preserving Inequalities

Abstract

Let $\mathbb{P}_n$ be the class of polynomials of degree at most $n$. Rather and Shah [15] proved that if $P\in \mathbb{P}_n$ and  $P(z)\neq 0$ in $|z| < 1$, then for every $R  > 0$ and 0 $\leq q < \infty, $ $$| B[P(Rz)]|_q \leq  \frac{| R^{n}B[z^n] +\lambda_0 |_{q}}{| 1+z^n|_q} | P(z)|_q,$$where $B$ is a $ B_{n}$-operator.
In this paper, we prove some generalization of this result which in particular yields some known polynomial inequalities as special. We also consider an operator $D_{\alpha}$ which maps a polynomial $P(z)$ into $D_{\alpha} P(z) := n P(z) + ( \alpha - z ) P' (z)$ and obtain extensions and generalizations of a number of well-known $L_{q}$ inequalities.