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Markov's property for kth derivative

Mirosław BaranBeata MilówkaPaweł Ozorka — 2012

Annales Polonici Mathematici

Consider the normed space ( ( N ) , | | · | | ) of all polynomials of N complex variables, where || || a norm is such that the mapping L g : ( ( N ) , | | · | | ) f g f ( ( N ) , | | · | | ) is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality | / z j P | | M ( d e g P ) m | | P | | , j = 1,...,N, P ( N ) , with positive constants M and m is equivalent to the inequality | | N / z . . . z N P | | M ' ( d e g P ) m ' | | P | | , P ( N ) , with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In...

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