  Let $r(z)$ be a rational function with at most $n$ poles, $a_1, a_2, \ldots,
a_n,$ where $|a_j| > 1,$ $1\leq j\leq n.$ This paper investigates the estimate
of the modulus of the derivative of a rational function $r(z)$ on the unit
circle. We establish an upper bound when all zeros of $r(z)$ lie in $|z|\geq
k\geq 1$ and a lower bound when all zeros of $r(z)$ lie in $|z|\leq k \leq 1.$
In particular, when $k=1$ and $r(z)$ has exactly $n$ zeros, we obtain a
generalization of results by A. Aziz and W. M. Shah [Some refinements of
Bernstein-type inequalities for rational functions, Glas. Mat., {\bf 32}(52)
(1997), 29--37.].
