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

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 on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.


RESULT
Here, T n (x) = cos(n arccos x) is the n-th Chebyshev polynomial of the first kind.
The source of motivation for Lemma 1.1 has been the recent work of Culiuc and Treil [1] on the Carleson embedding theorem with matrix weights. In fact, Lemma 2.2 in [1], which they attribute to F. Nazarov

Moreover, the best constants A and B satisfy B ≤
In fact, 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. Therefore Lemma 1.1 implies that one can take In the scalar case (d = 1) this new upper bound produces the upper estimate A ≤ 4B, which is known to be optimal [4, Theorem 3.3].
Remark 1. The method we use for the proof of Lemma 1.1 can also 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). This leads to (1.5) |p(0)| ≤ 2n 2 − n , see the next section for the proof. The estimate (1.5) is presumably not best possible.

PROOFS
The idea is to view both estimates, (1.2) and (1.5), as Bernstein-type estimates. Recall that for a polynomial h of degree N the classical Bernstein inequality says that Proof of Lemma 1.1. By assumption, p : C → C is a polynomial satisfying (1.1) and p(s) ≥ 0 for all s ≥ 0. Then q(z) := zp(z) is polynomial of degree at most n with q(0) = 0, p(0) = q ′ (0), and q(s) ≥ 0 for all s ≥ 0. We define the auxiliary function a Laurent polynomial of degree ≤ n. It is not difficult to see that the growth condition (1.1) for p implies the uniform bound | f (z)| ≤ 1 for all |z| = 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. To see this just note that is the Koebe function, familiar from the classical theory of univalent functions, which maps the unit circle |z| = 1 onto the half-line [1/4, +∞). Hence, on |z| = 1, f (z) is the product of two real and positive functions.
We are thus in a position to apply the Fejér-Riesz theorem [2] for the Laurent polynomial f . This gives us a complex polynomial P of degree ≤ n with no zeros in |z| < 1 such that Clearly, |P(z)| ≤ 1 for all |z| = 1. We can therefore apply a sharpening of Bernstein's inequality due to P. Lax [3] (confirming an earlier conjecture of Erdös) which asserts that In particular, proving (1.2). Clearly, the polynomial P n (z) = (z n − 1)/2 has the property |P ′ n (1)| = n/2, so | f ′′ n (1)| = n 2 /2 for f n (z) := P n (z)P n (1/z). It is easy to see that for a polynomial q n of degree at most n with q n (0) = 0, and it is straightforward to check that p n (z) := q n (z)/z has the form (1.3).
Proof of (1.5). By assumption, p : C → C is a polynomial satisfying (1.1). Then q(z) := zp(z) is polynomial of degree at most n with q(0) = 0 and p(0) = q ′ (0). We define, closely following the proof of [1, Lemma 2.2], the auxiliary function , a polynomial of degree N ≤ 2n. As before, the polynomial g has the property that |g(z)| ≤ 1 for all |z| = 1 .

Now note that
Hence, we could apply the classical Bernstein inequality twice, first for g ′ and then for g ′′ , but this would result in which is not particularly good. However, as observed in [1, Proof of Lemma 2.2] we can assume without loss of generality that g has no zeros in |z| < 1. We can therefore apply as above the inequality of Lax which leads to This brings us in a position to apply Corollary 14.2.8 in [5] for the polynomial g ′ which has degree ≤ 2n − 1. Hence Taking z = 1 and noting that g ′ (1) = nq(0) = 0, gives 2|g ′′ (1)| ≤ n(2n − 1), as required.

REMARK
The polynomials p which occur in the proof of Theorem A in [1]

ACKNOWLEDGEMENT
The authors would like to thank Stefanie Petermichl for various helpful discussions and remarks.