Bounds of the Derivative of Some Classes of Rational Functions

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.].