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 - <p style="text-align: justify;">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 &nbsp;$P(z)\neq 0$ in $|z| &lt; 1$, then for every $R &nbsp;&gt; 0$ and 0 $\leq q &lt; \infty, $ $$| B[P(Rz)]|_q \leq &nbsp;\frac{| R^{n}B[z^n] +\lambda_0 |_{q}}{| 1+z^n|_q} | P(z)|_q,$$where $B$ is a $ B_{n}$-operator. <br/>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 $D_{\alpha} P(z) := n P(z) + ( \alpha - z ) P&#39; (z)$ and obtain extensions and generalizations of a number of well-known $L_{q}$ inequalities.</p>