arrow
Volume 30, Issue 4
$L^q$ Inequalities and Operator Preserving Inequalities

M. Bidkham & S. Ahmadi

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

Published online: 2014-11

Export citation
  • 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.

  • AMS Subject Headings

30C10, 30C15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{ATA-30-377, author = {}, title = {$L^q$ Inequalities and Operator Preserving Inequalities}, journal = {Analysis in Theory and Applications}, year = {2014}, volume = {30}, number = {4}, pages = {377--386}, 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2014.v30.n4.5}, url = {http://global-sci.org/intro/article_detail/ata/4502.html} }
TY - JOUR T1 - $L^q$ Inequalities and Operator Preserving Inequalities JO - Analysis in Theory and Applications VL - 4 SP - 377 EP - 386 PY - 2014 DA - 2014/11 SN - 30 DO - http://doi.org/10.4208/ata.2014.v30.n4.5 UR - https://global-sci.org/intro/article_detail/ata/4502.html KW - Complex polynomial, polar derivative, $B$-operator AB -

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.

M. Bidkham & S. Ahmadi. (1970). $L^q$ Inequalities and Operator Preserving Inequalities. Analysis in Theory and Applications. 30 (4). 377-386. doi:10.4208/ata.2014.v30.n4.5
Copy to clipboard
The citation has been copied to your clipboard