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volume 5, issue 4, article 103, 2004.

Received 27 March, 2004;

accepted 26 September, 2004.

Communicated by:A. Lupa¸s

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Journal of Inequalities in Pure and Applied Mathematics

LUPA ¸S-DURRMEYER OPERATORS

NAOKANT DEO

Department of Applied Mathematics Delhi College of Engineering Bawana Road, Delhi - 110042, India.

EMail:dr_naokant_deo@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 066-04

Lupa ¸s-Durrmeyer Operators Naokant Deo

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Abstract

In the present paper, we obtain Stechkin-Marchaud-type inequalities for some approximation operators, more precisely for Lupa¸s-Durrmeyer operators de- fined as in (1.1).

2000 Mathematics Subject Classification:26D15.

Key words: Stechkin-Marchaud-type inequalities, Lupa¸s Operators, Durrmeyer Op- erators.

The author is grateful to Prof. Dr. Alexandru Lupa¸s, University of Sibiu, Romania for valuable suggestions that greatly improved this paper.

Contents

1 Introduction. . . 3 2 Auxiliary Results. . . 5 3 Main Results . . . 7

References

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1. Introduction

Lupa¸s proposed a family of linear positive operators mapping C[0,∞) into C[0,∞), the class of all bounded and continuous functions on[0,∞), namely,

V_n(f, x) =

∞

X

k=0

p_n,k(x)f k

n

, x∈[0,∞),

wherep_n,k(x) = ^n+k−1_k

x^k(1 +x)^−n−k.

Motivated by Derriennic [1], Sahai and Prasad [5] proposed modified Lupa¸s operators defined, for functions integrable on[0,∞), by

(1.1) B_n(f, x) = (n−1)

∞

X

k=0

p_n,k(x) Z ∞

0

p_n,k(t)f(t)dt.

Wicken discussed Stechkin-Marchaud-type inequalities in [2] for Bernstein poly- nomials and obtained the following results:

w²_φ

f, 1

√n

≤Cn⁻¹

n

X

k=1

φ^−α(B_kf −f) _∞.

The main object of this paper is to give Stechkin-Marchaud-type inequalities for Lupa¸s-Durrmeyer operators. In the end of this section we introduce some definitions and notations.

Definition 1.1. For0≤λ≤1, 0< α <2r,0≤β≤2r, 0≤α(1−λ) +β≤2r (1.2) kfk₀ =kfk_0,α,β,λ= sup

x∈I

φ^{α(λ−1)−β}(x)f(x) ,

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(1.3) C_α,β,λ⁰ ={f ∈CB(I),kfk₀ <∞},

(1.4) kfk_r =kfk_r,α,β,λ = sup

x∈I

φ2r+α(λ−1)−β

(x)f^(2r)(x)

and

(1.5) C_α,β,λ^r =

f ∈C_B(I), f^(2r−1) ∈AC_loc,kfk_r <∞ , whereφ(x) = p

x(1 +x)andr= 0,1,2, . . .. Definition 1.2. Peetre’sK-functional is defined as (1.6) w^2r_φλ(f, t)α,β = sup

0<h≤t

sup

x±rhφ^λ(x)∈I

φ^{α(λ−1)−β}(x)∆^2r_hφλf(x)

and

(1.7) K_φ^λ(f, t^2r)_α,β = inf

g^(2r−1)∈ACloc

kf −gk₀+t^2rkgk_r ,

where AC_loc is the space of real valued absolute continuous and integrable functions on[0,1].

In second section of the paper, we will give some basic results, which will be useful in proving the main theorems; while in Section3the main results are given.

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2. Auxiliary Results

Some basic results are given here.

Lemma 2.1. Suppose that for nonnegative sequences {σ_n},{τ_n}withσ₁ = 0 the inequalityσ ≤ ^k_np

σ_k+τ_k, (1 ≤ k ≤ n),is satisfied forn ∈ N, p >0.

Then one has

(2.1) σn ≤Bpn^−p

n

X

k=1

k^p−1τk.

Lemma 2.2. Forf^(2s) ∈C_α,β,λ⁰ , s∈N₀, the following inequalities hold

(2.2)

B_n^(2s)f

r ≤C₁n^r f^(2s)

0, and

(2.3)

B_n^(2s)f

r ≤C₂n^{r+α(1−λ)2 +β2} f^(2s)

_∞.

Lemma 2.3. Forf^(2s) ∈C_α,β,λ^r , s∈N0, the following inequality holds

(2.4)

B_n^(2s)f r ≤

f^(2s) r.

Lemma 2.4. Let us suppose thatf^(2s) ∈C_α,β,λ⁰ , s∈N₀,0≤α(1−λ) +β ≤2, then

(2.5)

B_n^(2s)f r≤C

n

X

k=1

k^r−1

(Bkf−f)^(2s) 0+

f^(2s) _∞

! .

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Lemma 2.5. Suppose thatr∈N, x±rt∈I, 0≤β ≤2r, 0≤t≤ _16r¹ , then

(2.6)

Z _2r^t

−^t

2r

· · · Z _2r^t

−^t

2r

φ^−β x+

2r

X

j=1

u_j

!

du₁· · ·du_2r ≤C(β)t^2rφ^−β(x).

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3. Main Results

We are now ready to prove the main results of this paper.

Theorem 3.1. For the modulus of smoothness andK-functional

(3.1) K_φ^λ

f^(2s), 1 n^r

α,β

≤Cn^−r

n

X

k=1

k^r−1

(Bkf −f)^(2s) 0+

f^(2s) _∞

! ,

(3.2) w_φ^2rλ

f^(2s), 1

√n

α,β

≤Cn^−2−λr











n^2−λ1

X

k=1

k^{−r−12−λ}

(B_kf−f)^(2s) 0 +

f^(2s) _∞









 ,

wherek·k_∞denotes the supremum norm.

Proof of (3.1). Takingⁿ₂≤m≤nsuch that

(B_mf−f)^(2s) 0≤

(B_kf−f)^(2s) 0,

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(ⁿ₂ < k≤n),we have K_φλ

f^(2s), 1 n^r

α,β

≤

(B_mf−f)^(2s)

₀ +n^−r f_m^(2s)

r

≤ 2^r n^r

n

X

k=ⁿ₂

k^r−1

(B_kf −f)^(2s) 0

+Cn^−r

m

X

k=1

k^r−1

(B_kf −f)^(2s) 0+

f^(2s) ∞

!

≤Cn^−r

n

X

k=1

k^r−1

(B_kf −f)^(2s) 0+

f^(2s) _∞

! .

Proof of (3.2). By definition ofK-functional there existsg ∈C^r

α,β,λ such that

(3.3)

f^(2s)−g

0+n^−2−λr kgk_r ≤K_φ^λ

f, n^−2−λr

α,β

and (3.4)

∆^2r_hφλ(x)f^(2s)(x)

≤Cφ^{α(1−λ)+β}(x) f^(2s)

0

by Lemma2.5for aboveg, 0< hφ^λ(x)< _16r¹ , x±rhφ^λ(x)∈I, (3.5)

∆^2r_hφλ(x)g(x)

≤Ch^2rφ(−2r+α)(1−λ)+β

(x)kgk_r.

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Using (3.4) and (3.5), again for0< hφ^λ(x)< _16r¹ , x±rhφ^λ(x)∈I, we get (3.6)

∆^2r_hφλ(x)f^(2s)(x)

≤Cφ^{α(1−λ)+β}(x)

f^(2s)−g

₀+h^2rφ^2r(λ−1)(x)kgk_r . Forx±rhφ^λ(x)∈I , we obtain

(3.7) h²φ^2(λ−1)(x)≤

1 2n^2−λ1

−1

. From (3.6) and (3.7) we have

∆^2r_hφλ(x)f^(2s)(x) (3.8)

≤Cφ^{α(1−λ)+β}(x)K_φ^λ f^(2s), 1

2n^{2−λ1 −1}!

α,β

≤Cφ^{α(1−λ)+β}(x)n^−2−λr

×







h n^2−λr i

X

k=1

k^{−2−λr−1}

(Bkf −f)^(2s) 0+

f^(2s) _∞





.

Corollary 3.2. If0< α <2, f ∈C_B(I),then

|(Bnf)(x)−f(x)|=O (n^−1/2φ^1−λ(x))^α

⇒w²_φλ(f, t) = O(t^α),

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where

w²_φλ(f, t) = sup

0<h≤t

sup

x±hφ^λ(x)∈I

∆²_hφλf(x) .

This is the inverse part in [3].

In (1.4) and (1.5), forδ_n(x) =φ(x) +^√1_n,φ(x)replaced byδ_n(x), (3.1) also holds.

Corollary 3.3. If0< α <2r, f ∈C_B(I),then

|(Mnf)(x)−f(x)|=O (n^−1/2φ^1−λ(x))^α

⇒w^2r_φλ(f, t) =O(t^α),

where(M_nf)(x)is linear combination of(B_nf)(x).

This is the inverse parts in [4].

Remark 3.1. We also propose some other modifications of Lupa¸s operators as

Mn(f, x) = n

∞

X

k=0

pn,k(x) Z ∞

0

sn,k(t)f(t)dt

wheres_n,k(t) = e^−nt(nt)_k!^k andp_n,k(x)is defined in (1.1) for these operatorsM_n.

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References

[1] M.M. DERRIENNIC, Sur l’approximation de fonctions integrables sur [0,1] par des polynomes de Bernstein modifies, J. Approx, Theory, 31 (1981), 325–343.

[2] E. VAN WICKEN, Stechkin-Marchaud type inequalities in connection with Bernstein polynomials, Constructive Approximation, 2 (1986), 331–337.

[3] M. FELTEN, Local and global approximation theorems for positive linear operators, J. Approx. Theory, 94 (1998), 396–419.

[4] S. GUO, C. LIANDY. SUN, Pointwise estimate for Szasz-type, J. Approx.

Theory, 94 (1998), 160–171.

[5] A. SAHAIANDG. PRASAD, On simultaneous approximation by modified Lupa¸s operators, J. Approx., Theory, 45 (1985), 122–128.