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This article is cited in 2 scientific papers (total in 2 papers) Bernstein-Szegő inequality for the Riesz derivative of trigonometric polynomials in A. O. Leont'eva Ural Federal University named after the first President of Russia B. N. Yeltsin, Ekaterinburg, Russia
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     § 1. Introduction1.1. Notation. The statement of the problemLet Tn=Tn(C) be the class of all trigonometric polynomials with complex coefficients. For a parameter p, 0⩽p⩽∞, consider the following functionals on Tn: these functionals define a norm only for 1 \leqslant p\leqslant \infty.In 1917 Weyl [1] introduced fractional derivatives of a periodic function. On the class \mathscr{T}_n, the fractional (or Weyl) derivative of real order \alpha\geqslant 0 is defined by
\begin{equation*}
\begin{aligned} \, D^\alpha f_n(t) &=\sum_{k=1}^n k^\alpha\biggl(a_k \cos \biggl(kt+\frac{\pi\alpha}{2}\biggr) +b_k \sin \biggl(kt+\frac{\pi\alpha}{2}\biggr)\biggr) \\ &=\sum_{k=-n}^n c_k |k|^\alpha e^{(i\pi\alpha/2)\operatorname{sign} k} e^{ikt}. \end{aligned}
\end{equation*}
\notag
For any positive integer \alpha the Weyl derivative coincides with the classical derivative: D^\alpha f_n =f_n^{(\alpha)}. In the case when \alpha=0 the operator D^0 removes the constant term of the polynomial: D^0 f_n(t)=f_n(t)-c_0. The Weyl derivatives have the following semigroup property: D^\beta D^\alpha=D^{\alpha+\beta}, \alpha,\beta\geqslant 0. Along with the polynomial (1.1) we consider the conjugate polynomial
\begin{equation*}
\widetilde{f}_n(t)=\sum_{k=1}^n (b_k\cos kt-a_k\sin kt) =i\sum_{k=-n}^n c_k (\operatorname{sign} k) e^{ikt};
\end{equation*}
\notag
note that this definition of the conjugate polynomial, in which we follow [2], differs in sign from the classical definition (see, for example, [3], Vol. 1, § 2.5). Given a real \theta, consider the Weyl-Szegő operator
\begin{equation}
\begin{aligned} \, \notag D^\alpha_\theta f_n(t) &=f_n^{(\alpha)}(t)\cos\theta+\widetilde{f}_n^{(\alpha)}(t)\sin\theta \\ \notag &=\sum_{k=1}^n k^\alpha \biggl(a_k \cos\biggl(kt+\frac{\pi\alpha}{2}+\theta\biggr) +b_k\sin\biggl(kt+\frac{\pi\alpha}{2}+\theta\biggr)\biggr) \\ &=\sum_{k=-n}^n c_k |k|^\alpha e^{i(\pi\alpha/2+\theta) \operatorname{sign} k} e^{ikt}. \end{aligned}
\end{equation}
\tag{1.2}
For \theta=-\pi\alpha/2 we have the operator
\begin{equation}
D^\alpha_{-\pi\alpha/2} f_n(t)=\sum_{k=1}^n k^\alpha(a_k \cos kt+ b_k \sin kt) =\sum_{k=-n}^n c_k |k|^\alpha e^{ikt},
\end{equation}
\tag{1.3}
and for \theta=\pi(1-\alpha)/2, the operator
\begin{equation}
D^\alpha_{\pi(1-\alpha)/2} f_n(t)=\sum_{k=1}^n k^\alpha(b_k \cos kt -a_k \sin kt) =i\sum_{k=-n}^n c_k |k|^\alpha (\operatorname{sign} k) e^{ikt}.
\end{equation}
\tag{1.4}
The operator (1.3) is known as the Riesz derivative; for some of its properties, see § 5.25.4 in [4]. The operator (1.4) will be called the conjugate Riesz derivative, since the polynomial D^\alpha_{\pi(1-\alpha)/2} f_n is conjugate to D^\alpha_{-\pi\alpha/2} f_n for any f_n\in\mathscr{T}_n. In what follows we write D^\alpha_R and \widetilde{D}^\alpha_R for D^\alpha_{-\pi\alpha/2} and D^\alpha_{\pi(1-\alpha)/2}, respectively. An important property of such operators is that the Riesz derivative of an even polynomial is an even polynomial again, and the conjugate Riesz derivative of an even polynomial is an odd one. Note that D^\alpha_R for even \alpha\in\mathbb N and \widetilde{D}^\alpha_R for odd \alpha\in\mathbb N are the classical derivatives of order \alpha. On the other hand, D^\alpha_R for odd \alpha\in\mathbb N and \widetilde{D}^\alpha_R for even \alpha\in\mathbb N are the conjugate derivative operators of order \alpha. Also note that D^0_R=D^0, and \widetilde{D}^0_R is the conjugation operator. The Weyl-Szegő operator (1.2) can be written in a different way in terms of Riesz derivatives:
\begin{equation}
D^\alpha_\theta f_n(t)=D^\alpha_R f_n(t)\cos\tau+\widetilde{D}^\alpha_R f_n(t) \sin\tau, \qquad \tau=\frac{\pi\alpha}{2}+\theta.
\end{equation}
\tag{1.5}
We are interested in the norm of operator (1.2) on the set \mathscr{T}_n, that is, the least constant B_n(\alpha,\theta)_p in the inequality
\begin{equation}
\|D^\alpha_\theta f_n\|_p\leqslant B_n(\alpha,\theta)_p \|f_n\|_p, \qquad f_n\in \mathscr{T}_n.
\end{equation}
\tag{1.6}
Inequalities of this kind are called Bernstein-Szegő inequalities (Bernstein inequalities for \theta=0, and Szegő inequalities for \theta=\pi/2). It is easily checked that the constant B_n(\alpha,\theta)_p is \pi-periodic in \theta; so we assume in what follows that \theta\in[0,\pi]. Since D^\alpha_\theta is the convolution operator, for the constant B_n(\alpha,\theta)_p in (1.6) we have
\begin{equation}
B_n(\alpha,\theta)_p\leqslant B_n(\alpha,\theta)_\infty, \qquad 1\leqslant p\leqslant\infty,
\end{equation}
\tag{1.7}
and
\begin{equation}
n^\alpha=B_n(\alpha,\theta)_2\leqslant B_n(\alpha,\theta)_p \leqslant B_n(\alpha,\theta)_0, \qquad 0\leqslant p\leqslant\infty.
\end{equation}
\tag{1.8}
Inequality (1.7) is known; the first inequality in (1.8) is quite clear — the corresponding lower estimate is given by the polynomial e^{int}. The last inequality in (1.8) was proved by Arestov [5]. This inequality means that B_n(\alpha,\theta)_p assumes its largest value over p\in[0,\infty] at p=0; thus, the case p=0 is of great value in this field of research. In this paper, for all n\in\mathbb N we characterize the \alpha for which the constants B_n(\alpha,R)_p and \widetilde{B}_n(\alpha,R)_p in the inequalities and
\begin{equation}
\|\widetilde{D}^\alpha_R f_n\|_p\leqslant \widetilde{B}_n(\alpha,R)_p \|f_n\|_p
\end{equation}
\tag{1.10}
are equal to n^\alpha for all p, 0\leqslant p\leqslant\infty. In view of (1.8), to do this it suffices to investigate the inequalities and
\begin{equation}
\|\widetilde{D}^\alpha_R f_n\|_0\leqslant \widetilde{B}_n(\alpha,R)_0 \|f_n\|_0,
\end{equation}
\tag{1.12}
or, more precisely, to characterize n and \alpha for which the constants B_n(\alpha,R)_0 and \widetilde{B}_n(\alpha,R)_0 are n^\alpha. 1.2. Historical remarksInequalities of the form (1.6) have been studied for over 90 years. For a historical account, see [2], [6]–[12] and the books [13], Ch. 3, [14], § 8.1, and [15], §§ 6.1.2 and 6.1.7. For 1\leqslant p\leqslant\infty and \alpha\geqslant 1, for each \theta\in[0,\pi] the sharp inequality
\begin{equation}
\bigl\|f_n^{(\alpha)}\cos\theta+ \widetilde{f}_n^{(\alpha)}\sin\theta\bigr\|_p \leqslant n^\alpha \|f_n\|_p,\qquad f_n \in \mathscr{T}_n,
\end{equation}
\tag{1.13}
holds with the classical constant B_n(\alpha,\theta)_p=n^\alpha. For the first-order derivative, inequality (1.13) in the uniform norm is due to Bernstein, M. Riesz and Szegő. For \alpha\in\mathbb N and p\geqslant 1 it was established by Zygmund. For real \alpha\geqslant 1 and p\geqslant 1 inequality (1.13) is due to Lizorkin for \theta=0 and Kozko for all \theta\in[0,\pi]. Moreover, Kozko [16] examined conditions on the parameters \alpha and \theta under which (1.13) holds with the constant n^\alpha for all L_p, 1\leqslant p\leqslant\infty. For 0<p<1 even Bernstein’s inequality (for the first-order derivative) was quite a challenge. It was proved by Arestov, who created in [6] and [17] a new method for dealing with extremal problems for algebraic polynomials on the unit circle, and, as a consequence, with trigonometric polynomials on the period with respect to the norms generated by functions \varphi in the class \Phi^+, which he introduced. In particular, his method also works in the L_p-spaces, 0\leqslant p\leqslant\infty. Using this approach, for all 0\leqslant p\leqslant\infty he proved the sharp Bernstein inequality
\begin{equation}
\|f_n^{(r)}\|_p\leqslant n^r \|f_n\|_p, \qquad f_n \in \mathscr{T}_n,
\end{equation}
\tag{1.14}
on the class of all trigonometric polynomials \mathscr{T}_n with positive integer r. In 1994 Arestov [2] considered Szegő’s inequality for the derivatives of nonnegative integer order r for conjugate trigonometric polynomials in L_0, that is,
\begin{equation}
\|\widetilde{f}_n^{(r)}\|_0\leqslant B_n\biggl(r,\frac\pi2\biggr)_0 \|f\|_0, \qquad f_n\in \mathscr{T}_n.
\end{equation}
\tag{1.15}
He showed that, for a fixed nonnegative integer r, the constant in Szegő’s inequality behaves as
\begin{equation}
B_n\biggl(r,\frac \pi2\biggr)_0=4^{n+o(n)} \quad\text{as } n\to\infty.
\end{equation}
\tag{1.16}
Thus, the behaviour of B_n(r,\pi/2)_0 differs substantially from that of the constant B_n(r,0)_0=n^r in Bernstein’s inequality (1.14) for r\in\mathbb N in L_0. In the same paper [2] Arestov raised the problem of characterizing r and n such that inequality (1.15) for the derivative of the conjugate polynomial holds with the classical constant B_n(r,\pi/2)_0=n^r. He showed that this inequality holds with this constant if r\geqslant n\ln 2n. In 1994, on the basis of computer experiments, Arestov put forward the following conjecture regarding the constant B_n(r,\pi/2)_0 in (1.15). Conjecture A. A necessary and sufficient condition for Szegő’s inequality (1.15) to hold in L_0 for the derivative of order r\in \mathbb{N} of the conjugate polynomial of order n with constant n^r is r\geqslant 2n-2. In 2014 Arestov and Glazyrina [10] investigated the Bernstein-Szegő inequality for real \alpha\geqslant 0 and arbitrary real \theta. In this and more general settings, they examined conditions on n, \alpha and \theta under which the Bernstein-Szegő inequality in L_0 (and, as a consequence of (1.8), in all L_p-spaces, 0<p\leqslant\infty) holds with constant n^\alpha. They showed that for all \theta\in[0,\pi] a sufficient condition for this is \alpha\geqslant n \ln 2n. Let A_n(\theta) be the set of all \alpha\geqslant 0 such that B_n(\alpha,\theta)_0=n^\alpha. Arestov and Glazyrina made the following two conjectures. Conjecture 1. If \alpha\in\mathbb R and \alpha\geqslant 2n - 2, then the Bernstein-Szegő inequality in L_0 for the derivative of order \alpha of a polynomial of degree n holds with constant n^\alpha for each \theta. Conjecture 2. For \theta=0 Bernstein’s inequality holds with constant n^\alpha if and only if \alpha\in\mathbb N or \alpha\geqslant 2n-2, that is, Arestov and Glazyrina [10] proved these two conjectures for n=2. For each \theta, they described the set A_2(\theta). Namely, they showed that A_2(0)=\{1\}\cup[2,\infty) and A_2(\theta)=[\alpha^*(\theta),\infty) for \theta\in(0,\pi), where \alpha^*(\theta)\in(1,2) is a root of a certain equation. Popov announced (in his talks at several conferences of 2017–2021; see [18] and the references given there) that Conjecture 2 holds for n\leqslant 10 and \theta=0 (that is, in the case of Bernstein’s inequality). In 2022 this author [12] showed that, for any \theta\in[0,\pi], the condition {\alpha\geqslant 2n-2} ensures the Bernstein-Szegő inequality with the classical constant n^\alpha, that is, Conjecture 1 was confirmed for all n\in\mathbb N. 1.3. The main resultsIn this paper we prove the following results on the Riesz and conjugate Riesz derivatives. Theorem 1. A necessary and sufficient condition that inequality (1.9) hold for the Riesz derivative with constant B_n(\alpha,R)_p=n^\alpha for all 0\leqslant p\leqslant\infty is that Theorem 2. A necessary and sufficient condition that inequality (1.10) hold for the conjugate Riesz derivative with constant \widetilde{B}_n(\alpha,R)_p=n^\alpha for all 0\leqslant p\leqslant\infty is that For n\in\mathbb{N} let \alpha^*_n be the least nonnegative number such that B_n(\alpha,\theta)_0=n^\alpha for all \alpha\geqslant\alpha^*_n and all \theta\in[0,\pi]. The result of this author confirming Conjecture 1 (see [12]) means that \alpha^*_n\leqslant 2n-2. Theorem 1 implies that \alpha^*_n\geqslant 2n-2. Therefore, \alpha^*_n=2n-2. Below, in § 5, for each p, 0\leqslant p<\infty, we describe the set of trigonometric polynomials for which inequalities (1.9) and (1.10) for \alpha given by conditions (1.17) and (1.18), respectively, become equalities. § 2. The method2.1. Arestov’s method for extremal problemsBelow we attack the Bernstein-Szegő inequality using Arestov’s method (see [5], [6] and [17]), which is capable of dealing with extremal problems for algebraic polynomials on the unit circle of the complex plane, or, which is the same, for trigonometric polynomials on the period, in view of the formula Let \mathscr{P}_m=\mathscr{P}_m(\mathbb C) be the set of algebraic polynomials of degree at most m with complex coefficients. For a polynomial of degree s<m it is convenient to assume that it has a zero of multiplicity m-s at the point at infinity z=\infty. It is clear that P_m(e^{it})\in\mathscr{T}_m. Given a polynomial P_m\in \mathscr{P}_m, for 0\leqslant p\leqslant\infty, we write for brevity \|P_m\|_{p}=\|P_m(e^{it})\|_p; in particular,
\begin{equation}
\|P_m\|_{0}=\|P_m(e^{it})\|_0=\exp\biggl({\frac{1}{2\pi}\int_{0}^{2\pi}\ln |P_m(e^{it})|\, dt}\biggr).
\end{equation}
\tag{2.2}
For a polynomial P_m with nonzero leading coefficient c_m\ne 0 and zeros \{z_j\}_{j=1}^m, it follows from Jensen’s formula (see, for example, [19], Vol. 1, Problem 175) that Given two polynomials
\begin{equation*}
P_m(z)=\sum_{k=0}^m C_m^k a_k z^k \in \mathscr{P}_m\quad\text{and} \quad \Lambda_m(z)=\sum_{k=0}^m C_m^k \lambda_k z^k \in \mathscr{P}_m,
\end{equation*}
\notag
the polynomial is known as their Szegő composition. For some properties of the Szegő composition, see [19], Vol. 2, Pt. V, Ch. 2, and [20], Ch. 4. For fixed \Lambda_m the Szegő composition (2.4) is a linear operator in \mathscr{P}_m. The polynomial has the following property: for each \Lambda_m\in \mathscr{P}_m, In view of this I_m can be regarded as the ‘identity element’ for Szegő composition. The following result is due to Arestov [5]. Theorem A. For all polynomials \Lambda_m, P_m \in\mathscr{P}_m and 0\leqslant p\leqslant\infty, For p=0 inequality (2.7) is sharp for any \Lambda_m and is attained at the polynomial (2.5).2.2. Turning to the investigation of the (conjugate) Riesz derivative of the extremal polynomialLet us verify that the Riesz derivative operator D^\alpha_R and the conjugate Riesz derivative operator \widetilde{D}^\alpha_R, as defined by (1.3) and (1.4) on the set \mathscr{T}_n via (2.1), can be represented as the Szegő composition operator (see (2.4)) on the set \mathscr{P}_{2n} for some polynomials \Lambda_{2n}^\alpha and \widetilde{\Lambda}_{2n}^\alpha. More precisely,
\begin{equation}
D^\alpha_R f_n(t)=e^{-int}(\Lambda_{2n}^\alpha P_{2n})(e^{it}), \qquad f_n(t)=e^{-int}P_{2n}(e^{it}),
\end{equation}
\tag{2.8}
and
\begin{equation}
\widetilde{D}^\alpha_R f_n(t)=e^{-int} (\widetilde{\Lambda}_{2n}^\alpha P_{2n})(e^{it}), \qquad f_n(t)=e^{-int}P_{2n}(e^{it}).
\end{equation}
\tag{2.9}
The following notation will be adhered to in what follows: Proposition 1. For \alpha\geqslant 0, to the Riesz derivative D^\alpha_R on the class \mathscr{T}_n there corresponds via (2.8) the operator of Szegő composition with the polynomial
\begin{equation}
\Lambda_{2n}^\alpha(z) = \sum_{k=1}^n C_{2n}^{n+k} k^\alpha z^{n-k} +\sum_{k=1}^n C_{2n}^{n+k} k^\alpha z^{n+k} =z^n Q_n^\alpha\biggl(\frac1z\biggr)+z^n Q_n^\alpha(z)
\end{equation}
\tag{2.11}
on the set \mathscr{P}_{2n}, and to the conjugate Riesz derivative \widetilde{D}^\alpha_R there corresponds via (2.9) the operator of Szegő composition with the polynomial Proof. We write a trigonometric polynomial f_n \in \mathscr{T}_n as Using (1.3) and (1.4),
\begin{equation*}
D^\alpha_R f_n(t)=\sum_{k=-n}^n C_{2n}^{n+k} |k|^\alpha c_k e^{ikt}\quad\text{and} \quad \widetilde{D}^\alpha_R f_n(t)=\sum_{k=-n}^n C_{2n}^{n+k} |k|^\alpha(\operatorname{sign} k) c_k e^{ikt}.
\end{equation*}
\notag
The algebraic polynomial is defined from the polynomial f_n via (2.1), and, in a similar way, the polynomials
\begin{equation*}
R^\alpha_{2n}(z)= \sum_{k=-n}^n C_{2n}^{n+k} |k|^\alpha c_k z^{n+k}\quad\text{and} \quad \widetilde{R}^\alpha_{2n}(z)= \sum_{k=-n}^n C_{2n}^{n+k} |k|^\alpha (\operatorname{sign} k) c_k z^{n+k}
\end{equation*}
\notag
are defined from the polynomials D^\alpha_R f_n and \widetilde{D}^\alpha_R f_n, respectively. From (2.4) we obtain
\begin{equation*}
R^\alpha_{2n}=\Lambda_{2n}^\alpha P_{2n}\quad\text{and} \quad \widetilde{R}^\alpha_{2n}=\widetilde{\Lambda}_{2n}^\alpha P_{2n},
\end{equation*}
\notag
where the polynomials \Lambda_{2n}^\alpha and \widetilde{\Lambda}_{2n}^\alpha are defined by (2.11) and (2.12), respectively. This proves Proposition 1. In view of (1.8) we must look at inequalities (1.11) and (1.12), that is,
\begin{equation*}
\|D^\alpha_R f_n\|_0\leqslant B_n(\alpha,R)_0 \|f_n\|_0, \qquad f_n\in\mathscr{T}_n,
\end{equation*}
\notag
and
\begin{equation*}
\|\widetilde{D}^\alpha_R f_n\|_0\leqslant \widetilde{B}_n(\alpha,R)_0 \|f_n\|_0, \qquad f_n\in\mathscr{T}_n.
\end{equation*}
\notag
Using (2.8) and (2.7) we have
\begin{equation}
\|D^\alpha_R f_n\|_0=\|\Lambda^\alpha_{2n}P_{2n}\|_0 \leqslant\|\Lambda^\alpha_{2n}\|_0 \|P_{2n}\|_0 =\|\Lambda^\alpha_{2n}\|_0 \|f_n\|_0, \qquad P_{2n}(e^{it})=e^{int}f_n(t).
\end{equation}
\tag{2.13}
Inequality (2.13) is sharp; it becomes an equality for From this polynomial, via (2.1) we define
\begin{equation}
\begin{aligned} \, \notag h_n(t) &=e^{-int}I_{2n}(e^{it})=4^n\cos^{2n}\frac{t}{2}=2^n(1+\cos t)^n \\ &=\sum_{k=-n}^n C_{2n}^{n+k} e^{ikt} =C_{2n}^n+2\sum_{k=1}^n C_{2n}^{n+k} \cos kt. \end{aligned}
\end{equation}
\tag{2.15}
Note that \|h_n\|_0=\|I_{2n}\|_0=1 by (2.14) and (2.3). In view of (2.6) and (2.15) the sharp constant in (2.13) is
\begin{equation}
\begin{gathered} \, \notag \|\Lambda^\alpha_{2n}\|_0=\|\Lambda^\alpha_{2n}I_{2n}\|_0=\|D^\alpha_R h_n\|_0, \\ D^\alpha_R h_n(t)=2\sum_{k=1}^n C_{2n}^{n+k}k^\alpha\cos kt. \end{gathered}
\end{equation}
\tag{2.16}
Thus, the polynomial (2.15) is extremal in inequality (2.13) for p=0. The conjugate Riesz derivative is dealt with similarly. In this case we obtain the polynomial
\begin{equation}
\widetilde{D}^\alpha_R h_n(t)=2\sum_{k=1}^n C_{2n}^{n+k}k^\alpha\sin kt.
\end{equation}
\tag{2.17}
This establishes the following result. Proposition 2. The sharp constants in inequalities (1.11) and (1.12) satisfy andThe leading coefficients of the polynomials (2.11) and (2.12) are \lambda_{2n}=n^\alpha. Hence by Jensen’s formula (2.3) the equality \|\Lambda_{2n}^\alpha\|_0=n^\alpha (or the equality {\|\widetilde{\Lambda}_{2n}^\alpha\|_0=n^\alpha}) holds if and only if all the 2n zeros of the polynomial (2.11) (or of (2.12), respectively) lie in the closed unit disc |z|\leqslant 1. Since \Lambda_{2n}^\alpha(z)=z^{2n}\overline{\Lambda_{2n}^\alpha(1/\overline{z})} and since \widetilde{\Lambda}_{2n}^\alpha has the same property, this is possible if and only if all the zeros of \Lambda_{2n}^\alpha (or \widetilde{\Lambda}_{2n}^\alpha) lie on the unit circle. But this is equivalent to saying that all the 2n zeros of the polynomial D^\alpha_R h_n (of \widetilde{D}^\alpha_R h_n, respectively) lie on the period. Thus, we have established the following result. Proposition 3. For parameters n and \alpha, inequality (1.11) for the Riesz derivative in L_0 holds with constant n^\alpha if and only if all the 2n zeros of the polynomial (2.16) lie on the period. The same result also holds for the conjugate Riesz derivative and the polynomial (2.17). The result below follows from (1.8) and Proposition 3. Corollary. Inequality (1.9) (inequality (1.10)) holds with the constant n^\alpha for all 0\leqslant p\leqslant\infty if and only if all the zeros of the polynomial (2.16) (of (2.17), respectively) lie on the period. Thus, the above problem reduces to examining the position of the zeros of the polynomials (2.16) and (2.17) (or, which is the same in view of (2.1), of the zeros of the polynomials (2.11) and (2.12)). For this investigation we invoke the polynomials (see (2.10)), since
\begin{equation}
D^\alpha_R h_n(t)=2\operatorname{Re} Q_n^\alpha(e^{it})\quad\text{and} \quad \widetilde{D}^\alpha_R h_n(t)=2\operatorname{Im} Q_n^\alpha(e^{it})
\end{equation}
\tag{2.20}
by (2.11) and (2.12).
§ 3. Auxiliary resultsA function g is said to be completely monotone on the half-axis (0,\infty) if it is infinitely differentiable and (-1)^\nu g^{(\nu)}(x)\geqslant 0 for all \nu=0,1,2,3,\dots and all x>0. By the Hausdorff-Bernstein-Widder theorem a function g is completely monotone if and only if it can be expressed as where \mu is a nonnegative Borel measure such that the integral (3.1) converges for all x>0; here the measure \mu is finite if and only if g(0)<\infty. For the proof of this theorem, see, for example, [21], § 5.5. An example of a completely monotone function is given by g(x)=1/{x^\beta}, \beta>0; this function can be written as
\begin{equation*}
\frac{1}{x^\beta} =\frac{1}{\Gamma(\beta)}\int_0^\infty e^{-tx} t^{\beta-1}\,dt;
\end{equation*}
\notag
this representation can be derived from the formula We require the following important result, which we proved in [12]. Lemma 1 (see [12], Lemma 1). Let the polynomial of degree n\in\mathbb N have real coefficients satisfying, for some a\in\mathbb N, the m conditions
\begin{equation}
\sum_{k=1}^n (-1)^k c_k g^{(a\nu)}(k)=0, \qquad \nu=0,1,2,\dots,m-1,
\end{equation}
\tag{3.2}
1\leqslant m\leqslant n-1, where g is a completely monotone function equal to the Laplace transform of a measure with support on a set of cardinality at least n. Then Q_n has at least m sign changes on the interval (-1,0). For n\geqslant 1 consider the function of \alpha\geqslant 0. The following result holds for this function. and Proof. First we prove (3.3). Consider the polynomial
\begin{equation*}
h_n(t)=4^n \cos^{2n}\frac{t}{2}=\sum_{k=-n}^n C_{2n}^{n+k} \cos kt.
\end{equation*}
\notag
For even r\in \mathbb{N}, we have The polynomial h_n has a zero of multiplicity 2n at t=\pi. Hence h_n^{(2r)}(\pi)=0 for r=1,2,3,\dots,n-1. This proves (3.3).
Now let us verify (3.4). Consider the function
\begin{equation*}
\varphi_n(x)=\frac{C_{2n}^n}{2}+\sum_{k=1}^n (-1)^k C_{2n}^{n+k} e^{-k^2 x}.
\end{equation*}
\notag
We claim that For fixed x>0 consider the function \psi_x(y)=e^{-y^2/(4x)}/\sqrt{\pi x}, y\in(-\infty,\infty), whose Fourier transform is the Gauss-Weierstrass kernel
\begin{equation*}
\widehat{\psi_x}(\omega)=\int_{-\infty}^\infty \psi_x(y)e^{-iy\omega}\,dy= e^{-x\omega^2}.
\end{equation*}
\notag
Now (3.5) follows since
\begin{equation*}
\begin{aligned} \, \varphi_n(x) &=\frac{1}{2}\sum_{k=-n}^n (-1)^k C_{2n}^{n+k} \widehat{\psi_x}(k) =\frac{1}{2}\sum_{k=-n}^n (-1)^k C_{2n}^{n+k}\int_{-\infty}^\infty \psi_x(y)e^{-iky}\,dy \\ &=\frac{1}{2}\int_{-\infty}^\infty \psi_x(y) \biggl(\sum_{k=-n}^n (-1)^k C_{2n}^{n+k} e^{-iky}\biggr)\,dy =\frac{1}{2}\int_{-\infty}^\infty \psi_x(y)h_n(\pi-y)\,dy>0. \end{aligned}
\end{equation*}
\notag
In what follows we require the Mellin transform of \varphi_n,
\begin{equation}
g_n(s)=(\mathcal{M}\varphi_n)(s)=\int_0^\infty x^{s-1}\varphi_n(x)\,dx.
\end{equation}
\tag{3.6}
Let us find a domain where the integral in (3.6) defines an analytic function. To this end we find how \varphi_n(x) behaves as x\to 0 and x\to\infty. We have
\begin{equation*}
\varphi_n(0)=\frac{C_{2n}^n}{2}+\sum_{k=1}^n (-1)^k C_{2n}^{n+k} =\frac{1}{2}\sum_{k=-n}^n (-1)^k C_{2n}^{n+k}=\frac{1}{2}(1-1)^{2n}=0.
\end{equation*}
\notag
Next, an application of (3.3) shows that \varphi_n'(0)=\varphi_n''(0)=\dots=\varphi_n^{(n-1)}(0)= 0. In addition, \lim_{x\to\infty}\varphi_n(x)=C_{2n}^n/2=\mathrm{const}>0. This implies that, for -n<\sigma<0, the function x^{\sigma-1}\varphi_n(x) lies in L(0,\infty). Consequently, the function (3.6) is analytic in the strip -n<\operatorname{Re} s <0 (see, for example, Theorem 1 in [22]).
From the definition (3.6) of g_n and the property (3.5) we obtain the following important fact, which is required below: Having the function (3.6) at our disposal, we consider the functions g_n^r(s)=(\mathcal{M}\varphi_n^{(r)})(s). We claim that for each r=1,2,\dots,n the function g_n^r(s) is analytic in the half-plane \operatorname{Re} s>-n+r and, in addition, can be expressed in terms of the function (3.6) in the strip -n+r<\operatorname{Re} s <r. First consider g_n^1. We have
\begin{equation}
\begin{aligned} \, g_n^1(s) &=\int_0^\infty x^{s-1}\varphi_n'(x)\,dx =\int_0^\infty x^{s-1} \,d\varphi_n(x) \nonumber \\ &=x^{s-1}\varphi_n(x)\big|_0^\infty -(s-1)\int_0^\infty x^{s-2}\varphi_n(x)\,dx. \end{aligned}
\end{equation}
\tag{3.8}
The function \varphi_n'(x) behaves as O(x^{n-1}) as x \!\to\! 0 and decays exponentially as {x\!\to\!\infty}; hence the left-hand integral in (3.8) exists and is analytic for -n+ 1<\operatorname{Re} s <\infty. In the strip -n+1<\operatorname{Re} s<1 the term outside the integral vanishes, and the right-hand integral in (3.8) defines an analytic function. So
\begin{equation*}
g_n^1(s)=-(s-1) g_n(s-1), \qquad -n+1<\operatorname{Re} s<1.
\end{equation*}
\notag
A similar analysis shows that the functions g_n^r(s), r=2,3,\dots,n, are analytic in the half-plane \operatorname{Re} s>-n+r, and
\begin{equation}
g_n^r(s)=(-1)^r(s-r)(s-r+1)\cdots(s-1) g_n(s-r), \qquad -n+r<\operatorname{Re} s <r.
\end{equation}
\tag{3.9}
In the half-plane \operatorname{Re} s>0,
\begin{equation}
\begin{aligned} \, \notag g_n^r(s) &=\int_0^\infty x^{s-1} \varphi_n^{(r)}(x)\,dx =\int_0^\infty x^{s-1} \biggl(\sum_{k=1}^n (-1)^k C_{2n}^{n+k} k^{2r} (-1)^r e^{-k^2 x}\biggr)dx \\ &=(-1)^r\Gamma(s) \sum_{k=1}^n (-1)^k C_{2n}^{n+k} k^{2r-2s}. \end{aligned}
\end{equation}
\tag{3.10}
For s\in(0,1) we have
\begin{equation*}
\begin{gathered} \, g_n^r(s)=(-1)^r\Gamma(s) \sum_{k=1}^n (-1)^k C_{2n}^{n+k} k^\alpha=(-1)^r\Gamma(s) S_n(\alpha), \\ \alpha=2r-2s=2r-\beta, \qquad \beta=2s\in(0,2). \end{gathered}
\end{equation*}
\notag
Thus, for 0<\operatorname{Re} s<r the function g_n^r(s) can be defined by (3.9) and (3.10) alike. Let us find the sign of g_n^r(s) for 1\leqslant r\leqslant n and s\in(0,1). The sign of the product (s-r)(s-r+ 1)\dotsb(s-1) is (-1)^r. For such s the function g_n(s-r) is positive by (3.7). This establishes (3.4) for 1\leqslant r\leqslant n and completes the proof of Lemma 2. Lemma 3. For n\geqslant 2 and 2n - 4<\alpha<2n - 2 let Q_n^\alpha be the polynomial defined by (2.10). Then 1) all the zeros of Q_n^\alpha, save one, lie in the open unit disc; 2) in addition, Q_n^\alpha has a real zero x_0=x_0(n,\alpha)<-1. Proof. To prove 1) we apply Lemma 1 to Q_n^\alpha. Let c_k be the coefficients of the polynomial Q_n^\alpha, that is,
\begin{equation*}
c_k =C_{2n}^{n+k}k^\alpha=C_{2n}^{n+k}k^{2n-4+\beta}, \qquad 0<\beta<2, \quad k=1,2,\dots,n.
\end{equation*}
\notag
We take a=2 in this lemma and consider the completely monotone function g(t)=1/t^\beta. We claim that
\begin{equation*}
S_\nu=\sum_{k=1}^n (-1)^k c_k g^{(2\nu)}(k)=0, \qquad \nu=0,1,\dots,n-3.
\end{equation*}
\notag
Indeed, using (3.3) (see Lemma 2) we obtain
\begin{equation*}
\begin{aligned} \, \frac{\Gamma(\beta)}{\Gamma(\beta+2\nu)}S_\nu &=\sum_{k=1}^n (-1)^k C_{2n}^{n+k}\frac{k^{2n-4+\beta}}{k^{2\nu+\beta}} \\ &=\sum_{k=1}^n (-1)^k C_{2n}^{n+k} k^{2(n-2-\nu)}=0, \qquad \nu=0,1,\dots,n-3. \end{aligned}
\end{equation*}
\notag
So the assumptions of Lemma 1 are met. Hence n-2 zeros of Q_n^\alpha lie on (-1,0). Another zero of Q_n^\alpha is at 0. Let us estimate the remaining zero.
To prove assertion 2) of the lemma it suffices to see that Q_n^\alpha(-1) has the correct sign, that is, It is easily checked that
\begin{equation*}
Q_n^\alpha(-1)=\sum_{k=1}^n (-1)^k C_{2n}^{n+k} k^\alpha=S_n(\alpha).
\end{equation*}
\notag
Now the required fact follows from (3.4) in Lemma 2. Lemma 3 is proved. Then Proof. Since
\begin{equation*}
u_n(t)=2\operatorname{Re} Q_n^\alpha(e^{it})=2\sum_{k=1}^n C_{2n}^{n+k} k^\alpha \cos kt,
\end{equation*}
\notag
we have
\begin{equation*}
u_n(0)=2\sum_{k=1}^n C_{2n}^{n+k} k^\alpha>0\quad\text{and} \quad u_n(\pi)=2\sum_{k=1}^n (-1)^k C_{2n}^{n+k} k^\alpha=2S_n(\alpha).
\end{equation*}
\notag
Hence \operatorname{sign} u_n(\pi)=(-1)^{n-1} by (3.4) in Lemma 2.
Next we have and another appeal to (3.4) shows that \operatorname{sign} u_n''(\pi)=(-1)^{n-1}, which proves Lemma 4.§ 4. Proofs of the main results4.1. Proof of Theorem 1Let us show that, for 2n-4<\alpha<2n- 2, for the Riesz derivative we have B_n(\alpha,R)_0>n^\alpha. By (2.20), D^\alpha_R h_n(t)=2\operatorname{Re} Q_n^\alpha(e^{it}). We denote this polynomial by u_n(t)= u_n^\alpha(t). We claim that not all of its zeros lie on the period. By Lemma 3 the polynomial Q=Q_n^\alpha has precisely n-1 zeros in the unit disc, and it has no zeros on the unit circle. As t ranges from -\pi to \pi, the function Q(e^{it}) describes a curve Z(t). By the argument principle this curve makes n-1 circuits about the origin. Hence u_n has at least 2n-2 zeros on the period. By Lemma 4
\begin{equation}
u_n(0)>0\quad\text{and} \quad \operatorname{sign} u_n(\pi)=(-1)^{n-1}.
\end{equation}
\tag{4.1}
Hence the zeros of u_n lie on the intervals (-\pi,0) and (0,\pi). Since the polynomial u_n is even, each of these intervals contains the same number m\in\{n-1,n\} of zeros. Assume that the polynomial u_n has \ell distinct zeros 0<t_1<\dots<t_\ell<\pi of multiplicity \kappa_1,\dots,\kappa_\ell, 1\leqslant\ell\leqslant m, on (0,\pi), where \kappa_1+\dots+\kappa_\ell=m. If t_1 has even multiplicity \kappa_1, then on the interval (t_1,t_2) the polynomial u_n(t) has the same sign as u_n(0). If \kappa_1 is odd, then the sign of u_n(t) on (t_1,t_2) is opposite to that of u_n(0). So \operatorname{sign} u_n(t)=(-1)^{\kappa_1} \operatorname{sign} u_n(0) for t\in(t_1,t_2). A similar analysis shows that
\begin{equation*}
\operatorname{sign} u_n(t)=(-1)^{\kappa_1}\dotsb(-1)^{\kappa_j}\operatorname{sign} u_n(0), \qquad t\in(t_j,t_{j+1}), \quad j=1,\dots,\ell-1.
\end{equation*}
\notag
For t\in(t_\ell,\pi] we have
\begin{equation}
\operatorname{sign} u_n(t)=(-1)^{\kappa_1} \dotsb(-1)^{\kappa_\ell}\operatorname{sign} u_n(0)=(-1)^m \operatorname{sign} u_n(0), \qquad t\in(t_\ell,\pi].
\end{equation}
\tag{4.2}
But m is equal to either n-1 or n. Hence m=n-1 by (4.1) and (4.2). As a result, u_n has precisely 2n-2 zeros on the period. Now let \alpha\in(2r,2r+2), r=0,1,\dots,n-3. We claim that the polynomial D^\alpha_R h_n cannot have 2n zeros on the period. If D^\alpha_R h_n had 2n zeros, then (D^\alpha_R h_n)^{(2n-4-2r)}=(-1)^{n-2-r} D^{2n-4+\alpha-2r}_R h_n would have 2n zeros on the period, but this is not so by the above since 2n-4<2n-4+\alpha-2r<2n-2. Now assume that \alpha=0. The constant term of the polynomial D^0 h_n is zero, and so D^0 h_n has at least two zeros on the period. But it has at most two such zeros, because its derivative (D^0 h_n(t))'=h_n'(t) is positive for t\in(-\pi,0) and negative for t\in(0,\pi). Thus, for \alpha\in[0,2)\cup(2,4)\cup(4,6)\cup\dots\cup(2n-4,2n-2) the polynomial D^\alpha_R h_n has at most 2n-2 zeros on the period. For the remaining we have Indeed, for even \alpha\in\mathbb N, the Riesz derivative coincides, up to the sign of (-1)^{\alpha/2}, with the classical derivative of order \alpha, and now (4.3) follows from the sharp Bernstein inequality (1.14), which was proved by Arestov. For \alpha\geqslant 2n-2 equality (4.3) also follows from this author’s result in [12].Thus, we have shown that, for p=0 for n\in\mathbb{N} and \alpha\geqslant 0, the best constant (2.18) in inequality (1.11) is equal to B_n(\alpha,R)_0=n^\alpha if and only if \alpha satisfies (1.17). Now the conclusion of Theorem 1 is secured by (1.8). 4.2. Proof of Theorem 2Let \alpha\in[0,1)\cup(1,3)\cup(3,5)\cup\dots\cup (2n- 5,2n-3). We show that the polynomial \widetilde{D}^\alpha_R h_n cannot have 2n zeros on the period. If \widetilde{D}^\alpha_R h_n had 2n zeros on the period, then (\widetilde{D}^\alpha_R h_n)'=D^{\alpha+1}_R h_n would have 2n zeros on the period, but this is impossible by what was proved in § 4.1, because \alpha +1 < 2n - 2 and is not a natural number. So, for \alpha\in[0,1)\cup(1,3)\cup(3,5)\cup\dots\cup(2n-5,2n-3) the polynomial \widetilde{D}^\alpha_R h_n has at most 2n-2 zeros on the period. Let us show that, for we haveFor odd \alpha\in\mathbb N, the Riesz derivative coincides, up to the sign of (-1)^{(\alpha-1)/2}, with the classical derivative of order \alpha, and so (4.5) follows from (1.14). For {\alpha\geqslant 2n-2}, (4.5) is secured by [12]. It remains to consider the case when \alpha\in(2n-3,2n-2). Let us show that for such \alpha all 2n zeros of the polynomial v_n=v_n^\alpha=\widetilde{D}^\alpha_R h_n lie on the period. It is clear that v_n^\alpha=(u_n^{\alpha-1})', \alpha - 1\in(2n - 4,2n - 3). The polynomial u_n^{\alpha-1} has 2n-2 zeros -\pi< \tau_1<\dots< \tau_{2n-2}<\pi. So the polynomial v_n has 2n-3 zeros on the interval [\tau_1, \tau_{2n-2}], and it also vanishes at \pi. In addition, v_n has two more zeros on the intervals (-\pi,\tau_1) and (\tau_{2n-2},\pi) because by (3.12) u_n^{\alpha-1}(\pi) and (u_n^{\alpha-1})''(\pi) are positive or negative simultaneously. So, all the 2n zeros of the polynomial v_n lie on the period. This proves equality (4.5) for \alpha as in (4.4). Thus, for p=0, n\in\mathbb{N} and \alpha\geqslant 0 the best constant (2.19) in inequality (1.12) is \widetilde{B}_n(\alpha,R)_0=n^\alpha if and only if \alpha satisfies condition (1.18). Now the conclusion of Theorem 2 follows from (1.8). § 5. Extremal polynomialsIn this section we describe the sets of extremal polynomials in inequalities (1.9) and (1.10) for \alpha given by (1.17) and (1.18), respectively, and 0\leqslant p<\infty. To do this we invoke some results due to Arestov [6], which provide necessary and sufficient conditions for a polynomial to be extremal in the inequality for the Szegő composition operator under some conditions on the operator. Following [6], we denote by \mathscr{P}_m^0, \mathscr{P}_m^\infty and \mathscr{P}_m^1 the subsets of the set of polynomials \mathscr{P}_m such that all of their m zeros lie in the disc |z|\leqslant 1, in the set |z|\geqslant 1, or on the circle |z|=1, respectively. Theorem B (Arestov; see Theorems 1, 2 and 5 in [6]). Let m\in\mathbb N, m\geqslant 2, and let \Lambda_m(z)=\sum_{k=0}^m C_m^k \gamma_k z^k\in\mathscr{P}_m^1. In addition, let the polynomial
\begin{equation*}
\Lambda_m^\backprime(z)=\sum_{k=0}^{m-2} C_{m-2}^k \gamma_{k+1} z^k
\end{equation*}
\notag
of degree m-2 also lie in \mathscr{P}_{m-2}^1. Then for all 0\leqslant p\leqslant\infty,
\begin{equation}
\|\Lambda_m P_m\|_p\leqslant |\gamma_m|\cdot\|P_m\|_p, \qquad P_m \in\mathscr{P}_m.
\end{equation}
\tag{5.1}
For p=0 this inequality turns to equality precisely for the polynomials P_m\in\mathscr{P}_m^0\cup\mathscr{P}_m^\infty, while for 0<p<\infty it turns to equality precisely for the polynomials az^m+b, a,b\in\mathbb C. For \Lambda_m\in\mathscr{P}_m^1, in view of (2.3) we have \|\Lambda_m\|_0=|\gamma_m|, where \gamma_m is its leading coefficient, and so inequality (5.1) is contained in Theorem A. Theorem 3. The following results hold. 1. For p=0 the extremal polynomials for inequalities with \alpha given by (1.17) and (1.18), respectively, are precisely the polynomials for which the polynomials P_{2n}, as defined by (2.1), lie in \mathscr{P}^0_{2n}\cup\mathscr{P}^\infty_{2n}. 2. For 0<p<\infty the extremal polynomials for inequalities (1.9) and (1.10) with \alpha given by (1.17) and (1.18), respectively, are precisely the polynomials c_{-n}e^{-int}+c_n e^{int} and c_{-n},c_n\in\mathbb C, respectively. Proof. For the proof we use Theorem B. We will show that the polynomials
\begin{equation*}
\Lambda_{2n}^\alpha(z)=\sum_{k=-n}^n C_{2n}^{n+k}\lambda_k z^{n+k}, \qquad \lambda_k=|k|^\alpha,
\end{equation*}
\notag
and
\begin{equation*}
\widetilde{\Lambda}_{2n}^\alpha(z)=\sum_{k=-n}^n C_{2n}^{n+k}\widetilde{\lambda}_k z^{n+k}, \qquad \widetilde{\lambda}_k=|k|^\alpha (\operatorname{sign} k)
\end{equation*}
\notag
(see (2.11) and (2.12)), for \alpha given by (1.17) and (1.18), respectively, satisfy and
It is clear that (\Lambda_{2n}^\alpha)^\backprime=\Lambda_{2(n-1)}^\alpha and \bigl(\widetilde{\Lambda}_{2n}^\alpha\bigr)^\backprime=\widetilde{\Lambda}_{2(n-1)}^\alpha. If \alpha\in\mathbb N is even, then the Riesz derivative of order \alpha coincides, up to sign, with the classical derivative: D^\alpha_R f_n=\pm f_n^{(\alpha)}, f_n\in\mathscr{T}_n. Hence, for such \alpha, for any m\in\mathbb N the polynomial \Lambda_{2m}^\alpha lies in \mathscr{P}_{2m}^1 (see [6] and [17]). In a similar way, for odd \alpha\in\mathbb N the polynomial \widetilde{\Lambda}_{2m}^\alpha lies in \mathscr{P}_{2m}^1. Next, Lemma 3 in [12] asserts that for any \alpha\geqslant 2n-2 all zeros of the polynomials D^\alpha_R h_n and \widetilde{D}^\alpha_R h_n (and, a fortiori, of D^\alpha_R h_{n-1} and \widetilde{D}^\alpha_R h_{n-1}) lie on the period. Hence, by (2.1) the polynomials \Lambda_{2n}^\alpha and \widetilde{\Lambda}_{2n}^\alpha, and also \Lambda_{2n-2}^\alpha and \widetilde{\Lambda}_{2n-2}^\alpha, lie in \mathscr{P}_{2m}^1. This proves (5.2) and (5.3). Now Theorem 3 follows from Theorem B. 
Citation in format AMSBIB
\Bibitem{Leo23}
This publication is cited in the following 2 articles:
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