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

  • M. S. Vyazovs’ka


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 $Q$.
How to Cite
Vyazovs’ka, M. S. “Estimation of the Norm of the Derivative of a Monotone Rational Function in the Spaces $L_p$”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 12, Dec. 2009, pp. 1713–1719,
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