Best uniform polynomial approximation of some rational functions

In this research paper using the Chebyshev expansion, we explicitly determine the best uniform polynomial approximation out of $P_{qn}$ (the space of polynomials of degree at most $qn$) to a class of rational functions of the form $1/(T_q\left(a\right)\pm T_q\left(x\right))$ on $?1,1]$, where $T_q\left(x\right)$ is the first kind of Chebyshev polynomial of degree $q$ and $a^2>1$. In this way we give some new theorems about the best approximation of this class of rational functions. Furthermore we obtain the alternating set of this class of functions.