Bernstein type inequalities for rational functions on analytic curves and arcs

Borwein and Erd\'elyi proved a Bernstein type inequality for rational functions on the unit circle and on the real line. Here we establish asymptotically sharp extensions of their inequalities for rational functions on analytic Jordan arcs and curves. In the proofs key roles are played by Borwein-Erd\'elyi inequality on the unit circle, Gonchar-Grigorjan type estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials.