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Essentially sharp Markov-type inequalities are known for various classes of polynomials with constraints including constraints of the coefficients of the polynomials. For and we introduce the class as the collection of all polynomials of the form , , , . In this paper, we prove essentially sharp Markov-type inequalities for polynomials from the classes on . Our main result shows that the Markov factor valid for all polynomials of degree at most on improves to for polynomials in the classes on .