Estimation of the norm of the derivative of a monotone rational function in the spaces L _p

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:

$$ {\left\| {R^{\prime}} \right\|_{{L_p}\left[ { - 1 + {\upvarepsilon },1 - {\upvarepsilon }} \right]}} \leq C \cdot {9^{n\left( {1 - {{1} \left/ {p} \right.}} \right)}}{{\upvarepsilon }^{{{1} \left/ {p} \right.} - {{1} \left/ {q} \right.} - 1}}{\left\| {R} \right\|_{{L_q}\left[ { - 1,1} \right]}}, $$

where the constant C depends only on p and q. 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.