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

• Date: 2009
• Main Authors: Vyazovs’ka, M. S., Вязовська, M. С.
• Format: Article
• Language: Ukrainian; English
• Published: Institute of Mathematics, NAS of Ukraine 2009
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/3132
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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...