$L^q$ Inequalities and Operator Preserving Inequalities (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 $|z| < 1$, then for every $R > 0$ and 0 $≤ q < ∞, $ $$| B[P(Rz)]|_q ≤ \frac{| R^{n}B[z^n] +λ_0 |_{q}}{| 1+z^n|_q} | P(z)|_q,$$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 $D_{α}$ which maps a polynomial $P(z)$ into $D_{α} P(z) := n P(z) + ( α - z ) P' (z)$ and obtain extensions and generalizations of a number of well-known $L_{q}$ inequalities.
Keywords: Complex polynomial; polar derivative; $B$-operator