Constants in Markov’s and Bernstein inequality on a finite interval in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}

In this paper we demonstrate the constants in the pointwise Bernstein inequality |P(α)(x)|≤2n(x-a)(b-x)α||P||[a,b],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |P^{(\alpha )}(x)|\le \left( \frac{2n}{\sqrt{(x-a)(b-x)}}\right) ^{\alpha }||P||_{[a,b]}, \end{aligned}$$\end{document}for the α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha -$$\end{document}th derivative of an algebraic polynomial in L∞-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }-$$\end{document}norms on an interval in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}$$\end{document}, where α≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 3$$\end{document}. This result was obtained using the tools of theory of pluripotential and we apply it to get the main result which is a new generalization of V. A. Markov’s type inequalities ||P(α)||p≤C1/p2b-aα||Tn(α)||[-1,1]n2/p||P||p,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} ||P^{(\alpha )}||_p\le C^{1/{p}}\left( \frac{2}{b-a}\right) ^{\alpha }||T^{(\alpha )}_{n}||_{[-1,1]}n^{2/p}||P||_{p}, \end{aligned}$$\end{document}for the α-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha -$$\end{document}th derivative of an algebraic polynomial in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}$$\end{document} norms, where p≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ge 1$$\end{document}. In particular, we show that for any α≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \ge 3$$\end{document} the constant C in the V. A. Markov inequality satisfies the condition C≤832·3,94741·πMα2331/p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\le 8\left( \frac{32\cdot 3,94741\cdot \pi M\alpha ^2}{3\sqrt{3}}\right) ^{1/p}$$\end{document}.

The Bernstein inequality (cf. [25]) holds for the interior points of the interval [a, b] and is of the following form The inequality is exact in the sense that for any fixed point x ∈ (a, b) as n → ∞ the formula (cf. [25]) is satisfied. For sets on the real line the general form of the Bernstein's inequality is given in (cf. [7]) and (cf. [36][37][38]). Let E ⊂ R be a compact set, then we get for algebraic polynomials P of degree at most n = 1, 2, . . . Let E ⊂ R and ω E (x) be the density of μ E with respect to Lebesgue measure wherever it exists. The measure μ E is called the equilibrium measure of E (cf. [18,38]). In a special case when E = [−1, 1] the inequality (1) becomes the original Bernstein inequality: because ω [−1,1] (x) = 1 π √ 1−x 2 . Let us recall some well known results on Markov's inequality. By the Markov theorem (cf. [22]) we obtain that for any polynomial P ||P || [a,b] For the Chebyshev polynomial of the form we get an equality. V. A. Markov (cf. [24], p.141.) generalized A. A. Markov's inequality to derivitives of an arbitrary order P (α) (x). He proved that for any P and any α = 1, . . . , n Futhermore, in this case, the extremal polynomial is the Chebyshev polynomial T n (x). Inequality (4) was generalized by Schwab (cf. [23,29]) who proved that for a polynomial P of total degree n, on a finite interval, we have The exact constant in the univariate Markov inequality in L 2 −norm was investigated in (cf. [27]). The authors of this article proved that for any polynomial P of total degree n = 1, 2, 3, 4, : G. Sroka (cf. [35]) showed the following important result: where constants C α are bounded and T n (x) = cos(n arccos x) (x ∈ [−1, 1]) are the Chebyshev polynomials of the first kind. Futheremore, he proved that C α ≤ 12 3 √ 2 e 2 for α ≥ 3.
In (cf. [14]) P. Borvein and V. Totik extended Markov and Bernstein inequalities (for α = 1) to arbitrary subsets of [−1, 1] and [−π , π ], respectively. In (cf. [15]) P. Borwein generalized the inequality (1) and (3) on disjoint intervals. In addition, in papers (cf. [26,34]), an estimation of the constant in the Markov's inequality for a simplex using minimal polynomials wer introduced as a novel benchmark problem. Markov's-type polynomial inequalities (or inverse inequalities), as well as Bernstein inequalities, are often found in many areas of applied mathematics, including popular numerical solutions of differential equations. Proper estimates of optimal constants in both types of inequalities can help to improve the bounds of numerical errors. More information on this topic can be found in the papers by M. Oszust, G. Sroka(cf. [26]), M.Baran and L. Białas-Cież (cf. [3]) and references.

The crucial tools
First let us recall some well known theorems and examples.
Siciak's ekstremal function on a compact subset E of C is defined by: || · || E is the maximum norm on E. We refer to (cf. [4-6, 8, 9, 12, 19, 30-32]) for definitions and basic properties connected with this important tool in pluripotential theory and its applications to approximation theory. where Before we continue, we remind Cauchy's inequality.
LetD(x 0 , r ) be a closed circle centered at the point x 0 and radius r > 0.
. Then for any x 0 ∈ C and for 1 ≤ α ≤ n, r > 0 We will use the following well known Bernstein-Walsh's inequality.

Proposition 1
For any x 0 ∈ C, P ∈ R n [x], and r > 0

Theorem 1
For any x 0 ∈ (a, b) and r ∈ R + := (0, +∞), we have: Proof From (6) it follows that: We denote . Thus if u belongs to the domain of f , then It is easy to check that the values of the function f at the endpoints of the respectively and for ≤ r , then we have the equality in the above inequality, because the function reaches a maximum on the circle z ∞ ≤ r . It is also easy to see that the case −r ≤ x = . Then for every x 0 ∈ (a, b) and for every 1 ≤ α ≤ n, Proof Applying Cauchy's (7) and Bernstein-Walsh's (8) inequalities, we get Notice that puting Combining the last inequality with (11) for we obtain the assertion of the theorem.

Lemma 1 For any t
Then After solving the equation Since at x = α √ n 2 −α 2 f changes sign from negative to positive, f (t) has a local minimum, at this point, equal to Then for any t ∈ (0, 1], we obtain which is equivalent to the inequality t + √ 1 + t 2 ≤ e t , for t ∈ (0, 1]. Proof Let consider the following function f (t) = t + √ Observe that the interval (0, 1] is contained in the domain of the function f . We prove that the function f is decreasing on the interval (0, 1). To this end, we calculate f (t) = √ 1+t 2 +t . This implies that f (t) < 0 for t ∈ (0, 1), so the function f is decreasing on the interval (0, 1). On the other hand, using de l'Hospital's rule we easily see that lim t→0 + t + √ 1 + t 2 1/t = e. Therefore f (t) < 0 for t ∈ (0, 1), this concludes the proof of the inequality.

On Bernstein inequality for the line segment [a, b] ⊂ R
By (10) we are able to deduce the following result of this paper: Then for any x 0 ∈ (a, b) and for any α ≥ 3 we obtain Proof From Lemma 2 for α ≤ n we get Hence from Theorem 2 for t = α n we get the estimates Finally, using inequality α! < e α 2 α for any 3 ≥ α ∈ N and Lemma 3, we obtain and the proof is complete.

Remark 1
In P. Borwein's and T. Erdélyi book (cf. [13]) (p. 258-260), we can find a sketch of the proof that is a slightly weaker version of P. Bernstein's inequality on [−1, 1] for α−th derivatives based on the M. A. Lachance'a argument from 1984 (cf. [21]). His method is based on Bernstein-Szegö's inequality and Bernstein's inequality for trigonometric polynomials.
Such an inequality was also applied in the manuscript (cf. [35]) to show the main result concerning transferring the classic V.A. Markov's inequality into the case of integral norms. In the manuscript (cf. [28]) the authors use Bernstein's inequality (3) to prove that Gauss-Jacobi (-Lobatto) nodes of suitable order are L ∞ −norming meshes for algebraic polynomials, in a wide range of Jacobi parameters. In (cf. [39]) it is shown that finite-dimensional univariate function spaces satisfying a Bernstein-like inequality admit norming meshes.

Remark 3
Theorem 3 is a special case of the Bernstein's inequality for α−th derivatives for convex bodies ( compact sets, convex with nonempty interior) in R. For example, if we set a = −b in the Theorem 3 then we have a central-symmetric body convex (i.e. its interior contains 0). If we assume 0 < a < b, in the Theorem 3 we get a non central-symetric convex body. It is worth mention that A. Kroó and S. Révész (cf. [20]) etc. investigated Bernstein and Markov inequalities in uniform norms for convex bodies.

Auxiliary lemma
In this section we show a result which will be needed in the proof of the main result. A crucial idea is a factorization of operator of α-th derivative from (R n [x], || · || p ) to itself by the space (R n [x], || · || [a,b] ). We begin with the following inequality.

Example 1
The numerical computations carried out using Mathematica programs imply that the constant J in the inequality: should be between 1 and 3, 94741.

Lemma 4
For each p ≥ 1 such that pα > 1 and for an arbitrary P ∈ R n [x], we have the inequality where B(α, n, p) : Proof Let us recall that Applying a version of Bernstein's inequality from Theorem 3 and V. A. Markov's inequality (5), we get the following inequality (for x ∈ (a, b)) This reduces to Hence, by (12), we obtain Integrating both sides of the last inequality gives one Now, by using the substitution hence we have Now, by using the substitution y = t α/2 n 2 (n 2 − 1) · · · (n 2 − (α − 1) 2 ) we calculate as claimed Observation 2 For fixed n ∈ N and α ≥ 1 we have

Remark 4
The Bojanov conjecture (cf. [10]) asserts that for α−th derivative it should be true that Applying the method of proof of Lemma 4 we can derive (for α = 1) inequalities equivalent to the Bojanov's inequalities (13), which are true in this case.

G. Sroka
The next, a non-obvious, corollary follows from Corollary 1 and earlier results by M. Baran, P. Ozorka (cf. [1]), and M. Baran, A. Kowalska, P. Ozorka (cf. [3] and was discussed mainly in the case p = 2). This indicates that in the case of L p norm the line segment [a, b] ⊂ R possesses the V. A. Markov property in the sense considered nad M. Baran, L. Białas-Cież (cf. [2,5]

Conflicts of interest The authors declare that they have no competing interests
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