Comparison of the Best Uniform Approximations of Analytic Functions in the Disk and on Its Boundary

Denote by $C_A$ the set of functions that are analytic in the disk $|z|<1$ and continuous on its closure $|z|\le 1$; let $\mathcal {R}_n$, $n=0,1,2,\dots$, be the set of rational functions of degree at most $n$. Denote by $R_n(f)$ ($R_n(f)_A$) the best uniform approximation of a function $f\in C_A$ on the circle $|z|=1$ (in the disk $|z|\le 1$) by the set $\mathcal {R}_n$. The following equality is proved for any $n\ge 1$: $\sup \{R_n(f)_A/R_n(f)\colon f\in C_A\setminus \mathcal {R}_n\}=2$. We also consider a similar problem of comparing the best approximations of functions in $C_A$ by polynomials and trigonometric polynomials.