Let $P\left(z\right)={\sum }_{\nu =0}^n{a}_{\nu }{z}^{\nu }$ be a polynomial of degree at most $n$ which does not vanish in the disk $\left|z\right|<1$, then for $1\le p<\infty$ and $R>1$, Boas and Rahman proved $${∥P\left(Rz\right)∥}_p\le ({∥R^n+z∥}_p/{∥1+z∥}_p){∥P∥}_p.$$ In this paper, we improve the above inequality for $0\le p<\infty$ by involving some of the coefficients of the polynomial $P\left(z\right)$. Analogous result for the class of polynomials $P\left(z\right)$ having no zero in $\left|z\right|>1$ is also given.

The estimate ${∥D^{k-1}u∥}_{L^{n/(n-1)}}\le {∥A\left(D\right)u∥}_{L^1}$ is shown to hold if and only if $A\left(D\right)$ is elliptic and canceling. Here $A\left(D\right)$ is a homogeneous linear differential operator $A\left(D\right)$ of order $k$ on ${ℝ}^n$ from a vector space $V$ to a vector space $E$. The operator $A\left(D\right)$ is defined to be canceling if ${\bigcap }_{\xi \in {ℝ}^n\setminus \left\{0\right\}}A\left(\xi \right)\left[V\right]=\left\{0\right\}$. This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential...