$L^q$ Inequalities and Operator Preserving Inequalities
M. Bidkham, S. Ahmadi · Analysis in Theory and Applications · 2014
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 $P(z) eq 0$ in $|z| 0$ and 0 $\leq q 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.