0 Journal of Inequalities and Applications

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Weighted analogues of Bernstein-type inequalities on several intervals

Mehmet Ali Akturk1,2 and Alexey Lukashov1,3*

"Correspondence: alukashov@fatih.edu.tr 1 Department of Mathematics, Fatih University, Istanbul, 34500,Turkey 3Department of Mechanics and Mathematics, Saratov State University, Saratov, 410012, Russia Full list of author information is available at the end of the article

Abstract

We give weighted analogues of Bernstein-type inequalities for trigonometric polynomials and rational functions on several intervals. MSC: 41A17; 42A05; 41A20

Keywords: weighted polynomial inequalities; inequalities for derivatives of rational functions

ft Spri

1 Introduction

Inequalities for polynomials have been a classical object of studies for more than one century. Modern expositions can be found in books and surveys [1-3] and [4]. Recently weighted analogues of classical polynomial inequalities were considered (see, for instance, [5,6] and [7]). Other ways of generalizations are in replacing the domain of polynomials by more complicated (disconnected) sets and (or) in considering polynomials in more general Chebyshev systems. The main goal of the paper is to give simple proofs of weighted analogues of Bernstein-type inequalities on several intervals. They are inspired by weighted Bernstein-type inequalities from Section 5.2, E.4 in [1]. It turns out that for disconnected sets similar ideas allow to write down weighted versions with an explicit constant.

Throughout the paper, we use the notations

PC = p: p(x) = £ akxk, ak e R(C) (1)

I k-o J

for the set of algebraic polynomials and

TC = 11: t(x) = y + (ak cos kx + bk sin kx), ak, bk e R(C) J (2)

for the set of trigonometric polynomials with real (complex) coefficients; as a weight w, we consider an arbitrary continuous positive function on a suitable set, || • || is the uniform norm on this set.

The first theorem is a weighted analogue of the Bernstein-type inequality on several intervals.

Theorem 1 Let E be a set consisting of a finite number l > 2 of disjoint intervals, E =

UL [aj, bj] c [0,1], a1 < b1 < a2 < ••• < b2l, then there exists no depending on w and E such

ringer

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p'n(x)w(x)

ni(x - aj)(x - bj)\

< n\\pnw\\E, x e E

for every polynomial pn e Pf, n > no.

Next result is a weighted version of the Bernstein-type inequality for trigonometric polynomials on several intervals.

Theorem 2 Let w be any function which is continuous and positive on

E = Utj 02j] £ 2n ], < 02 < ••• < hl < 02l+l = 01 + 2n j=1

5(0) = J~[ sin

0 - Qj

Then there exists n° depending on w and E such that \t'n(0)w(0|5(0)\\ < n\\tnw\\£, 0 e E

for every polynomial tn e Tf, n > n°. Inequality (6) is sharp in the sense that it is not possible to replace n in (6) by n(1 - e) for arbitrary e > 0.

Now let 0 < a < n, and let

K = {ei0 | 0 e [-a,a]}

be the circular arc on the unit circle of central angle 2a and with a midpoint at 1. Next result is a weighted version of the inequality from [8].

Theorem 3 With the above notations and for any continuous positive function w, there exists n° depending on w and a such that

\p'n(ei0) w{eie) I

< n\\pnw\\Ka, 0 e [-a,a]

for every polynomial pn e Pf, n > n°.

Next we recall the definition of the harmonic measure a)(z, G, D) of a set G c dD at a point z e D relative to the domain D,

w(z, G, D) =

if 9 , - 7-gD (Z, n Jg dn

z)|dZ |,

where gD (Z, z) is the Green function of the domain D and n is the exterior normal at Z (see, for example, [9]).

Our last result is an extension of Rusak's inequality [10, p.57] to the case of several intervals.

Theorem 4 Let rn be a complex-valued algebraic fraction

rn(x) = xn + bn, (0)

V IPv (x)|

where b1,...,bn e C, pv (x) = nj=1(x - Xj )vj is a real polynomial of degree v which is positive on E = Ujl=1 [a2;_i, a2j], a1 < a2 < • •• < an satisfying the condition

\rn(x) \ < 1, x e E, (1)

and y (x) is a differentiablefunction on E. Then the estimate

\(rn(x)y(x))'\ {tp'n(x))2y2(x) + y/2(x), x e int(E) (12)

is valid. Here

pn(x) = — ( (2n - v)mE(to,x) + ^^ Vj^E(xj,x) J, (13)

2 \ j=1 '

mE(z, x) = dx («(z,[a1, x] n E, C\E)). Ifx is not a multiple root of y (x), then the equality sign is valid only for algebraic fractions

rn(x) = e cos pn (x), | e | =1 (14)

at the points x satisfying

(y (x) sin pn(x))' = 0 (15)

in the case when

(2n - v)«(TO,[a2^-1,a2k],C\E) + ^ vj«(x;,[a2k-1,a2k],C\E) = qk, (16)

where qk e N, k = 1,..., l.

Remark 1 A Markov-type inequality, which is obtained by a similar method, was announced in the conference [11].

In the following we use several auxiliary results.

Lemma 1 [12] Consider any algebraic fraction

rn(x)=xn+b1x:^ ••+bn, ()

V pv (x)

where bi,...,bn e R, and pv (x) = nV=i(x - xj)vj is a real polynomial of degree v which is

positive onE = [a1, a2] U ••• U [a2l-1, a2l] c R, a1 < ••• < a2l. Then

+ ^ <........ (

where yn(x) is given by (13).

For further reference, it is convenient to give a particular case of a version of Lemma 1 from [13].

Lemma 2 The following inequality holds for any trigonometric polynomial tn e Tn and

0 e int(E), E is a real compact subset of [0,2n]:

+t2(01 <■«■'& • (

rn£(z,x) = d^fc n \ei0: inf E < 0 < x},, (20)

and T£ = {ei0: 0 e E}.

Lemma 3 [12,14] The following assertions are equivalent. 1. The trigonometric polynomial tn deviates least from zero on

E = [01,02] U ••• U [02l-1,02i], 01< 02 < ••• < 02i with respect to the sup-norm among all trigonometric polynomials of degree N/2 with leading coefficients cos ty and sin ty, i.e.,

max I tn (0 )l = inf max

0eE 1 ci,dieR 0eE

NN cos ty cos — 0 + sin ty sin — 0 r 2 r 2

^ N - 2j J N - 2j + y / c, cos —2— 0 + d;- sin —2— 0

has the maximal possible number ofextremum points on E.

2. For every j = 1,..., l, the equilibrium measures of the arcs rj- = {ei0: 0 e [02j-1,02j]} are positive rational numbers. More precisely,

Nw(<x, r,, C\ rE ) = qN, q(N) e N, j = 1,..., l. (22)

3. There is a real trigonometric polynomial aN- i of order N -2 such that for a constant An >0,

4(0)-S(0)al_ i (0)= AN, (3)

where S(0) is given by (5).

If any of those assertions is valid, then

(a) the numbers qjN) are equal to the number ofzeros ofrN(Q) on Ej = [ify-i, &2j], j = 1,..., l;

(b) the polynomial tn may also be written in terms of mE (z, x) as xn(Q)=Ans cos ( W Nm£ (to, Z) dZ), Q e E, (24)

J£n[Q1,t

where e e {-1,1}.

Lemma 4 [15] The density of the equilibrium measure from (20), E = [0^ 02] U ••• U [02H, 02l], 01 < 02 < ••• < 02l is given by

^ (to, 0 ) = ± , (5)

where Q(0) = fj sin( ^), and fy e [02j, 02j+1], j = 1, ...,l, 02j+1 := 01 + 2n , are uniquely determined by

f 02j+1 Q(0)

-j=== dZ =0, j =1,...,l. (6)

Proof We want to present here a different proof of the lemma which uses the representations of extremal polynomials in (24).

(1) Suppose firstly «(to, Vj, C\Ve) = , pj e N, j = 1,..., l. Then by Lemma 3 the function

tn(Q) = cos I tf/ 2Nrns (to, Z) dZ , Q e E (27)

is a real trigonometric polynomial of order N. If we take a derivative, we get

2N-l 0 _ „

Tn(0)=NQ(0)[] sin—^, (28)

where fy, j = 1, ...,2N -1, are zeros of aN _ (0) and there is a real trigonometric polynomial l of order N - l such that

4(0) - S(0)o2n_l (0) = 1. (29) Hence

ia, o' • 0 - Pj sin(n fen[01,0]2N(to, z) dZ)

on ; (0) = c sin —-— =- .-. (30)

N-j( ) Ji 2 v '

Moreover, tn (0) has a maximal number of deviation points, and inner zeros of its derivative coincide with zeros of aN_ (0), and tn has one zero fy at each gap (02j, 02j+1), j = 1,..., l.

sin(n /Vn[„ 0] 2nmE(to, Z) dZ) Tjn (0) = ±n jen[01,0 ] -—— q(0)

= - sin n

/ 2NmE(to,Z)dZ )n2NmE(to,0),

Je n[01,0 ]

so we have

me (to, 0) =

!Q(0)l

yiS(0)T'

Now equality (26) follows from the representation (24). Uniqueness of §j's follows from the uniqueness of extremal trigonometric polynomials in Lemma 3.

(2) Using density of the systems of l arcs satisfying «(to, r, C\rE) e Q, j = 1,..., l, among all systems of l arcs (see, for instance, [16, 17] and references therein), we obtain the lemma. □

2 Proofs

Proof of Theorem 2 First consider tn e Tn. By the Weierstrass approximation theorem, for any n >0, there is qk e Tk such that

w(0) <

qk (0)

n!=1! sin( ^)!

— < (1 + n)w(0), 0 e E,

where are given by (26) in Lemma 4. Hence

Itn(0)w(0)|S(0)l1/2| <

tn(0)qk(0) l |S(0^ nj=1! sin(^)!

|S(0)|1/2

(tnqk )'(0)

tn(0)qk (0)

nl=1! sin(0-j)! !S(0)!1/2

nl=1! sin^)!

and, using Lemmas 2 and 4, we have

|tn(0)w(0)|S(0)|1/2| < (n + k)"tnqk'' e + "tn "e k"qk "e < (n + k)(1 + n)"tnw"E

0 ( 0-

1 I smf

+ —(1 + n)"tnW'E "w"e m

.70- §

< n"tnw"E

k, , 1 ,

1 + n + -(1 + n) +-(1 + n)"w"e

n f 0-

| | sin f

where m := min{w(0) : 0 e E}. Now, for every tn e Tn and e > 0, provided n > 0 is suffi-

n \ m 1

ciently small, n > n0 such that e > n + + mm II w||f ), we get

4(0)w(0)|S(0)|1/2| < n(1 + e)||tnW|E

n (0 - %

I I sinf

and because of || ]~[|=i sin() IIf < 1, we obtain, for sufficiently small e >0,

|i«(0)w(0)^S(0)j| < «UnwUs.

Thecaseof tn e T^ is proved then similarly to the proof of [1, Corollary 5.1.5]. The theorem is sharp even for the case w = 1. Namely, we cannot replace the multiplier n by n(1 - e) with any e > 0 in the right-hand side of (6). Take E = [-a, a], 0 < a < n. Then we have S(0) = sin(0—a) sin(^), fy1 = n and

• ( 0 - % sin1

Consider

tn(0) =cos( 2narccosl —

Take 0 = 0n = 2 arcsin(sin a sin n), then

K (0n)| = n

cos f J\S(0n)\

|S(0n)|| = n cos 2 > n(1 — e)

for sufficiently large n such that

sin — <

4n sin2 2 '

(2) □

Proofs of Theorems 1 and 3 are quite analogous and they use related inequalities from [18,19].

Proof of Theorem 4 Firstly we consider the case when the numerator pn(x) has real coefficients. Put rn(x) = cos w = cos(arccos rn(x)); using Lemma 1, we obtain

^rn(x)y (x)) =

sin wr'n (x)y (x)

1 —r2(x)

+ cos WY '(x)

< sin2 w + cos2 w

(r!n (x)) Y (x) ,2

1 — r2(x) < V (<p'n (x))2Y 2(x) + Y '2(x).

+ Y ' (x)

The validity of the estimate for complex-valued algebraic fractions is proved by the same trick as in [1, Corollary 5.1.5]. Equality sign in the last inequality in (43) is valid only for the function rn(x) = ecos<n(x), !e! = 1 if (16) holds [13, 20]. Equality sign in the second inequality in (43) then holds only for the same function at those points where

- sin ^2n(x) = cos <?2n(x) (4)

v'i-(x)y (x) Y '(x) ' Equality (44) is equivalent to (y(x) sin<2n(x))' = 0. □

Competing interests

The authors declare that they have no competing Interests. Authors' contributions

Allauthors jointly worked on the results and they read and approved the finalmanuscript. Author details

1 Department of Mathematics, Fatih University, Istanbul, 34500, Turkey. 2Department of Engineering Sciences, Istanbul University, Istanbul, 34320, Turkey. 3 Department of Mechanics and Mathematics, Saratov State University, Saratov, 410012, Russia.

Acknowledgements

The first author would like to thank the Scientific and TechnologicalResearch Councilof Turkey (TUBlTAK) for the financial support. The authors are deeply gratefulto reviewers for theircareful reading of the manuscript and remarks which helped to improve the presentation.

Received: 7 February 2013 Accepted:11 October 2013 Published: 07 Nov 2013 References

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10.1186/1029-242X-2013-487

Cite this article as: Akturk and Lukashov: Weighted analogues of Bernstein-type inequalities on several intervals.

Journal of Inequalities and Applications 2013, 2013:487