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The Bernstein Markov Property, shortly BMP, is an asymptotic quan- titativeassumption on the growth of uniform norms of polynomials or rational functionson a compact set with respect to L {\mu} 2 -norms, where {\mu} is a positivefinite measure. We consider two variants of BMP for rational functions withrestricted poles and compare them with the polynomial BMP finding out somesufficient condi- tions for the latter to imply the former. Moreover, werecover a sufficient mass- density condition for a measure to satisfy therational BMP on its support.

Some results on the rational Bernstein Markov property in the complex plane