Let $p(z)$ be a polynomial of degree at most $n$. In this paper we obtain some new results about the dependence of$$\Bigg\|p(Rz)−\beta p(rz)+\alpha\Big\{\frac{R+1}{r+1}\Big)^n-|\beta|\Big\} p(rz)\Bigg\|_s$$ on $\|p(z)\|_s$ for every $\alpha$, $\beta \in C$ with $|\alpha| \leq 1$, $|\beta| \leq 1$, $R > r \ge 1$, 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.