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We prove a sharp Bernstein-type inequality for complex polynomials which arepositive and satisfy a polynomial growth condition on the positive real axis.This leads to an improved upper estimate in the recent work of Culiuc and Treilon the weighted martingale Carleson embedding theorem with matrix weights. Inthe scalar case this new upper bound is optimal.

A sharp Bernstein-type inequality and application to the Carleson embedding theorem with matrix weights