Abstract
We prove a sharp Bernstein-type inequality for complex polynomials which are positive 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 Treil (Int. Math. Res. Not. 2019: 3301–3312, 2019) on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.
Highlights
Remark 1 The method we use for the proof of Lemma 1.1 can be used to improve the bound (1.4) given by [1, Lemma 2.2], which holds for any polynomial p : C → C satisfying (1.1)
In order to find such an estimate, it turns out to be essential that the auxiliary function f is real and positive (i.e., ≥ 0) on |z| = 1
We can apply as above the inequality of Lax which leads to max |g (z)| ≤ N · max |g(z)| ≤ n. This brings us in a position to apply Corollary 14.2.8 in [5] for the polynomial g which has degree ≤ 2n − 1
Summary
Lemma 1.1 Let n be a positive integer and p : C → C a polynomial such that p(s) ≥ 0 for all s ≥ 0 and. The proof of Theorem A in [1] requires the estimate (1.4) only for polynomials p : C → C with degree n = 2d, which satisfy (1.1) and are real and positive on the positive real axis. Remark 1 The method we use for the proof of Lemma 1.1 can be used to improve the bound (1.4) given by [1, Lemma 2.2], which holds for any polynomial p : C → C satisfying (1.1).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
