On the Bernstein inequality for rational functions with a prescribed zero

We extend some results of Giroux and Rahman (Trans. Amer. Math. Soc. 193 (1974), 67-98) for Bernstein-type inequalities on the unit circle for polynomials with a prescribed zero at z=1 to those for rational functions. These results improve the Bernstein-type inequalities for rational functions. The sharpness of these inequalities is also established. Our approach makes use of the Malmquist-Walsh system of orthogonal rational functions on the unit circle associated with the Lebesgue measure.