On approximation of the rational functions, whose integral is single-valued on $\mathbb{C}$, by differences of simplest fractions

We study a uniform approximation by differences $\Theta_1-\Theta_2$ of simplest fractions (s.f.'s), i. e., by logarithmic derivatives of rational functions on continua $K$ of the class $\Omega_r$, $r>0$ (i. e., any points $z_0, z_1\in K$ can be joined by a rectifiable curve in $K$ of length $\le r$). We prove that for any proper one-pole fraction $T$ of degree $m$ with a zero residue there are such s.f.'s $\Theta_1,\Theta_2$ of order $\le (m-1)n$ that $\|T+\Theta_1-\Theta_2\|_K\le 2r^{-1}A^{2n+1}n!^2/(2n)!^2$, where the constant $A$ depends on $r$, $T$ and $K$. Hence, the rate of approximation of any fixed individual rational function $R$, whose integral is single-valued on $\mathbb{C}$, has the same order. This result improves the famous estimate $\|R+\Theta_1-\Theta_2\|_{C(K)}\le 2e^r r^n/n!$, given by Danchenko for the case $\|R\|_{C(K)}\le 1$.