Refinement of Erdös-Lax inequality for $\mathrm{N}$-operator

Let $\mathcal{P}_n$ be the space of all polynomials of degree less than or equal to $n$. In this paper, we establish a refinement of Erdös-Lax inequality in which the classical derivative (as an operator on $\mathcal{P}_n$) is replaced by a $B_n$ operator. The result obtained includes some interesting inequalities as special cases.