$L^p$ Inequalities and Admissible Operator for Polynomials (Analysis in Theory and Applications, 2012)
Let $p(z)$ be a polynomial of degree at most $n$. In this paper we obtain some new results about the dependence of$$\Bigg\|p(Rz)−β p(rz)+α\Big\{\frac{R+1}{r+1}\Big)^n-|β|\Big\} p(rz)\Bigg\|_s$$ on $\|p(z)\|_s$ for every $α$, $β \in C$ with $|α| ≤ 1$, $|β| ≤ 1$, $R > r \ge 1$, and $s > 0$. Our results not only generalize some well known inequalities, but also are variety of interesting results deduced from them by a fairly uniform procedure.