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Borwein and Erd\'elyi proved a Bernstein type inequality for rationalfunctions on the unit circle and on the real line. Here we establishasymptotically sharp extensions of their inequalities for rational functions onanalytic Jordan arcs and curves. In the proofs key roles are played byBorwein-Erd\'elyi inequality on the unit circle, Gonchar-Grigorjan typeestimate of the norm of holomorphic part of meromorphic functions and Totik'sconstruction of fast decreasing polynomials.

Bernstein type inequalities for rational functions on analytic curves and arcs