Estimation of the norm of the derivative of a monotone rational function in the spaces $L_p$

Abstract

We show that the derivative of an arbitrary rational function $R$ of degree $n$ that increases on the segment $[−1, 1]$ satisfies the following equality for all $0 < ε < 1$ and $p, q > 1$: $$∥R′∥_{L_p[−1+ε,1−ε]} ≤ C⋅9^{n(1−1/p)}ε^{1/p−1/q−1}∥R∥_{L_q[−1,1]},$$ where the constant $C$ depends only on $p$ and $p$. The degree of a rational function $R(x) = P(x)/Q(x)$ is understood as the largest degree among the degrees of the polynomials $P$ and $Q$.