Exact estimates of the best rational approximations of functions with derivative of generalized finite variation

This paper is devoted to exact (in the sense of the order of smallness) estimates of the best rational approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. The obtained results were announced in the authors' paper in 2014.
They are analogous to the results of the first author, where A. Khatamov establishes exact (in the sense of the order of smallness) estimates of the best spline approximations of functions with derivative of generalized finite variation on a finite segment of a straight line in uniform and integral metrics. Results announced by the authors in 2014 generalize those obtained by N.Sh. Zagirov in 1982, namely, exact (in the sense of the order of smallness) estimates of rational approximations of functions with generalized finite variation in the integral metric, to the best rational approximations of functions with derivative of generalized finite variation on a finite segment in uniform and integral metrics. Generally speaking, the calculation of exact (in the sense of the order of smallness) estimates for the best approximations for any class of functions in any metric is a difficult problem.