Anal. Theory Appl., 30 (2014), pp. 377-386.

Published online: 2014-11

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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.

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.

*Analysis in Theory and Applications*.

*30*(4). 377-386. doi:10.4208/ata.2014.v30.n4.5