On the Asymptotics of the Rows of the Padé Table of Analytic Functions with Logarithmic Branch Points

For the functions $f(z)=\sum_{n=0}^\infty z^{l_n}/a_n$, where $l_n$ and $a_n$ are arithmetic progressions and their Padé approximants $\pi_{n,m}(z;f)$, we establish an asymptotics of the decrease of the difference $f(z)-\pi_{n,m}(z;f)$ for the case in which $z\in D=\{z:|z|<1\}$, $m$ is fixed, and $n\to\infty$. In particular, we obtain proximate orders of decrease of best uniform rational approximations to the functions $\ln(1-z)$ and $\operatorname{arctan}z$ in the disk $D_q=\{z:|z|\le q<1\}$.