Some estimates of derivatives of rational functions in integral metrics

Estimates are established in the metric of $L_q$ $($0<q<p/(p+1)$, $0<р\le\infty$) of derivatives of a~rational function in terms of the norm of the function itself in the metric of $L_p$. Local norms in $L_p$ are also estimated of Taylor remainders of a~rational function in terms of its norm in $L_p$.