Some properties of rational approximations of degree $(k,1)$ in the Hardy space $H_2(\mathscr D)$

We prove that the well-known interpolation conditions for rational approximations with free poles are not sufficient for finding a rational function of the least deviation. For rational approximations of degree $(k,1)$, we establish that these interpolation conditions are equivalent to the assertion that the interpolation point $c$ is a stationary point of the function $\Omega_k(c)$ defined as the squared deviation of $f$ from the subspace of rational functions with numerator of degree $\leq k$ and with a given pole $1/\overline c$. For any positive integers $k$ and $s$, we construct a function $g\in H_2(\mathscr D)$ such that $R_{k,1}(g)=R_{k+s,1}(g)>0$. where $R_{k,1}(g)$ is the least deviation of $g$ from the class of rational function of degree $\leq (k,1)$.