Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 614810, 16 pages
doi:10.1155/2012/614810
Research Article
Approximation of the Summation-Integral-Type
q-Szász-Mirakjan Operators
Mei-Ying Ren1 and Xiao-Ming Zeng2
1
2
Department of Mathematics and Computer Science, Wuyi University, Wuyishan 354300, China
Department of Mathematics, Xiamen University, Xiamen 361005, China
Correspondence should be addressed to Mei-Ying Ren, npmeiyingr@163.com and Xiao-Ming
Zeng, xmzeng@xmu.edu.cn
Received 26 July 2012; Revised 20 September 2012; Accepted 24 October 2012
Academic Editor: Behnam Hashemi
Copyright q 2012 M.-Y. Ren and X.-M. Zeng. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We introduce summation-integral-type q-Szász-Mirakjan operators and study approximation
properties of these operators. We establish local approximation theorem. We give weighted
approximation theorem. Also we estimate the rate of convergence of these operators for functions
of polynomial growth on the interval 0, ∞.
1. Introduction
First of all, we mention some concepts and notations from q-calculus. All of the results can be
found in 1–5. In what follows, q is a real number satisfying 0 < q < 1.
For each nonnegative integer k, the q-integer kq and the q-factorial kq ! are defined
as
kq :
1,
1 − qk / 1 − q , q /
q 1,
k,
kq k − 1q · · · 1q ,
kq ! :
1,
k≥1
1.1
k 0.
Then for integers n, k, n ≥ k ≥ 0, we have
k 1q 1 qkq ,
kq qk n − kq nq .
1.2
2
Abstract and Applied Analysis
For the integers n, k, n ≥ k ≥ 0, the q-binomial coefficient is defined as
nq !
n
.
:
k q
kq !n − kq !
1.3
The two q-analogues of the exponential function are defined as
eq x :
∞
xn
1
∞ ,
!
n
1
−
1
−q x q
q
n0
|x| <
1
,
1−q
∞
∞
xn
Eq x :
1 1−q x q ,
qnn−1/2
!
n
q
n0
1.4
∞
i
where 1 x∞
q :
i0 1 q x.
It is easily observed that
eq xEq −x eq −xEq x 1.
1.5
The q-Jackson integral and the q-improper integral are defined as
a
∞
ftdq t : 1 − q a f aqn qn ,
0
∞/A
0
a > 0,
n0
n n
∞
q q
,
ftdq t : 1 − q
f
A A
n−∞
1.6
A > 0,
provided the sums converge absolutely.
For t > 0, the q-Gamma function is given by
Γq s KA, s
∞/A1−q
ts−1 eq −tdq t,
1.7
0
where
As
1 s
KA, s 1
1 A1−s
q ,
1A
A q
A > 0.
1.8
It was observed in 5 that Kx, s is a q-constant, that is, Kqx, s Kx, s. In
particular, for s ∈ N, KA, s qss−1/2 and KA, 0 1. Γq s 1 sq Γq s, Γq 1 1.
In 1997, Phillips 6 firstly introduced and studied q analogue of Bernstein
polynomials. After this, the applications of q-calculus in the approximation theory become
one of the main areas of research; many authors studied new classes of q-generalized
Abstract and Applied Analysis
3
operators for instance, see 7–12. In order to approximate integrable functions, in 2008
Gupta and Heping 13 introduced and studied the q-Durrmeyer-type operators as follows:
n
Mn,q f; x n 1q q1−k pn,k q; x
1
pn,k−1 q; qt ftdq t f0pn,0 q; x ,
1.9
0
k1
where f ∈ C0, 1, x ∈ 0, 1, n 1, 2, . . ., 0 < q < 1, and
n−k−1
n
pn,k q; x :
xk
1 − qs x .
k q
s0
1.10
In 2010, Mahmudov and Kaffaoğlu 14 defined and studied q-Szász-Durrmeyer
operators as follows:
Dn,q f; x nq
∞
k
q sn,k q; x
∞/1−q
sn,k q; t ftdq t,
1.11
0
k0
where f ∈ R0,∞ , 0 < q < 1, n ∈ N, and
sn,k q; x 1
q
Eq nq x
kk−1/2
nkq xk
kq !
.
1.12
In 2011, Aral and Gupta 15 introduced and studied q-generalized Szász-Durrmeyer
operators by means of the q-integral as follows:
qbn /1−qn ∞
nq q
q
q
Dn ft, x sn,k x
sn,k tftdq t,
bn k0
0
1.13
where f ∈ C0, ∞, 0 < q < 1, n ∈ N, {bn } is a sequence of positive numbers such that
limn → ∞ bn ∞, 0 ≤ x < αq n, αq n : bn /1 − qnq and
q
sn,k x k
x
E
−n
.
q
q
bn
qk1/2 kq !bn k
nq x
1.14
In the present paper, we will introduce a kind of q-analogue of the summation-integraltype Szász-Mirakjan operators as follows.
4
Abstract and Applied Analysis
Let f ∈ C0, ∞. For every n ∈ N, q ∈ 0, 1, x ∈ 0, ∞, and A > 0, the summationintegral-type q-Szász Mirakjan operators are defined as
∞
Sn,q f; x nq sn,k x, q
k0
∞/A1−q
2
qk ftsn,k t, q dq t,
1.15
0
where
sn,k x, q nq x
k
kq !
eq −nq x .
1.16
We will study the approximation properties of the operators 1.15. We point out
that the operators 1.15 considered in this paper are more general than the operators 1.11
considered in 14. It is easy to verify when A 1, the operators 1.15 reduce to the operators
1.11. Thus, the operators 1.11 are the special case of the operators 1.15. The literature 14
studied the local approximation and the Voronovskaja-type theorem for the operators 1.11.
In the present paper, besides the local approximation of the operators 1.15, our research
work is different from the literature in 14; we study the rate of convergence, the weighted
Korovkin-type theorem, and the weighted approximation error of the operators 1.15 and
obtain some new results. As regards the operators 1.13 considered in 15, it is obvious that
these operators are quite different from the operators 1.15. In 15, for the operators 1.13, a
direct approximation result in weighted function space with the help of a weighted Korovkin
type theorem on the finite interval was obtained. The weighted approximation error of these
operators in terms of weighted modulus of continuity was given. Also, an asymptotic formula
was established. In the present paper, we introduce a new weighted modulus of continuity
which is different from that in 15. We obtain some results of the weighted approximation
with the help of the new weighted modulus of continuity on the infinite interval.
2. Two Lemmas
For the operators Sn,q f; x defined by 1.15, we give the following two lemmas.
Lemma 2.1. For Sn,q ti ; x, i 0, 1, 2, 3, 4, we have
i Sn,q 1; x 1;
ii Sn,q t; x x/q2 1/qnq ;
iii Sn,q t2 ; x x2 /q6 3q qx/q5 nq 2q /q3 n2q ;
iv Sn,q t3 ; x x3 /q12 x2 /q11 nq 5q q3q q2 x/q9 n2q 6q 2q4q 2q2 2q 2q 3q /q6 n3q ;
v Sn,q t4 ; x x4 /q20 x3 /q19 nq 7q q5q q2 3q q3 x2 /q17 n2q 10q 2q8q 4q2 6q 4q3 4q 3q4 2q x/q14 n3q 10q 3q8q 5q2 6q 6q3 4q 4q4 2q 2q 3q 4q /q10 n4q .
Abstract and Applied Analysis
5
Proof. In view of the concepts of the q-improper integral and q-exponential function, for
nonnegative integer i, we have
∞/A1−q
ti sn,k t, q dq t 0
∞/A1−q
nq t
ti
kq !
0
k
eq −nq t dq t
∞/A/nq 1−q
1
ni1
q kq !
2.1
uki eq −udq u
0
k iq !
Γq k i 1
.
ki1ki/2
ni1
ni1
q kq !q
q kq !K A/nq , k i 1
i When i 0, by the formulas 1.5 and 2.1, we can get
Sn,q 1; x k
∞ nq x
k0
kq !
eq −nq x qkk−1/2 eq −nq x Eq nq x 1.
2.2
ii When i 1, using k 1q kq qk , by the formulas 1.5 and 2.1, we can get
Sn,q t; x k
∞ nq x
k0
kq !
2
kq qk
eq −nq x qk −3k−2/2
nq
k
∞ nq x
2
1 1
eq −nq x qk −3k−2/2 qnq
nq k1 k − 1q !
2.3
x
1
.
q2 qnq
iii When i 2, using k 2q k 1q k2q qk 1 2q kq q2k 2q , by the formulas
1.5 and 2.1, we can get
Sn,q t2 ; x k
∞ nq x
k0
kq !
eq −nq x qk
2
−5k−6/2
k2q qk 1 2q kq q2k 2q
n2q
6
Abstract and Applied Analysis
k
∞ nq x
2
1
e
−n
x
qk −5k−6/2
q
q
2
!
−
2
k
nq k2
q
k
∞ nq x
3q q 2
2q
−n
e
x
qk −3k−8/2 q
q
2
q3 n2q
nq k1 k − 1q !
3q q x
2q
x2
6 .
5
q
q nq
q3 n2q
2.4
iv When i 3, by the formulas 1.5 and 2.1, we can get
Sn,q t3 ; x k
∞ nq x
k0
kq !
2
k 3q k 2q k 1q
eq −nq x qk −7k−12/2
.
n3q
2.5
Using k3q k2q k1q k3q qk 12q 3q k2q q2k 2q 3q 2q 3q kq q 2q 3q , we have
3k
Sn,q t3 ; x
k
∞ nq x
2
1 −n
e
x
qk −7k−12/2
q
q
n3q k3 k − 3q !
k
∞ nq x
2
1
k −5k−16/2
2
e
−n
1
2q
q
1
x
q
2
3
q
q
q
q
n3q k2 k − 2q !
1
∞
n3q k1
nq x
k
k − 1q !
2.6
2
eq −nq x qk −3k−16/2
2q 3q
× 1 q 1 2q 3q q2 2q 3q 2q 3q q6 n3q
2q 3q
x x3
x2 11
.
5q q3q q2 6q 2q4q 2q2 2q 2
12
q
q nq
q9 nq
q6 n3q
v When i 4, similar to the case of i 3, by simple calculation we can get the desired
result.
Abstract and Applied Analysis
7
Lemma 2.2. Let q ∈ 0, 1, x ∈ 0, ∞, we have
i Sn,q t − x; x 1 − q2 /q2 x 1/qnq ;
ii Sn,q t − x2 ; x ≤ 2/q6 1 − q4 2/nq x1 x 2q /q3 n2q .
Proof. By Lemma 2.1, we have
Sn,q t − x; x Sn,q t; x − x 1 − q2
1
x
.
2
qnq
q
2.7
By Lemma 2.1, we have
Sn,q t − x2 ; x Sn,q t2 ; x − 2xSn,q t; x x2
2q
3q q
2
2
1
x
− 2 1 x
−
6
5
qnq
q
q
q nq
q3 n2q
2q
3q q
2
2
1
x1 x ≤
− 2 1 5
−
6
qn
q
q
q nq
q3 n2q
q
2q
2
2
4
x1 x ≤ 6 1−q .
nq
q
q3 n2q
2
2.8
3. Local Approximation
Let CB 0, ∞ denote the class of all real valued continuous bounded functions f on 0, ∞
endowed with the norm f sup{|fx| : x ∈ 0, ∞}. The K-functional is defined as
K2 f, δ inf f − g δg ,
g∈W 2
3.1
where δ > 0 and W 2 {g ∈ CB 0, ∞ : g , g ∈ CB 0, ∞}. By 16, page 177, Theorem 2.4
there exists an absolute constant C > 0 such that
K2 f, δ ≤ Cω2 f, δ ,
3.2
where
ω2 f, δ sup
√
sup fx 2h − 2fx h fx
0<h≤ δ x∈0,∞
3.3
8
Abstract and Applied Analysis
is the second order modulus of smoothness of f ∈ CB 0, ∞. By
ω f, δ sup sup fx h − fx,
3.4
0<h≤δ x∈0,∞
we denote the usual modulus of continuity of f ∈ CB 0, ∞.
Theorem 3.1. Let f ∈ CB 0, ∞. For every x ∈ 0, ∞, q ∈ 0, 1, we have
Sn,q f; x − fx ≤ Cω2 f, δn q, x ω f, δn q, x ,
where C is an absolute constant, δn q, x 2/q6 1 − q4 2/nq x1 x 2q /q3 n2q 3.5
1/2
.
Proof. For f ∈ CB 0, ∞, x ∈ 0, ∞, we define
Sn,q f; x Sn,q f; x − f
1
x
q2 qnq
fx.
3.6
By Lemma 2.1, we get Sn,q 1; x 1, Sn,q t; x x. Let g ∈ W 2 , x, t ∈ 0, ∞, by Taylor’s
formula
gt gx t − xg x t
t − ug udu,
3.7
x
we obtain
Sn,q g; x gx Sn,q
t
t − ug udu; x .
3.8
x
By the definition given by 3.6, for x ∈ 0, ∞, we have
t
t − ug udu; x Sn,q g; x − gx ≤ Sn,q
x
2
x/q 1/qnq x
1
− u g udu
2
x
qnq
q
t
≤ Sn,q |t − u| g u du; x
x
x/q2 1/qnq 1
x
− ug udu
2
q
qnq
x
⎡
2 ⎤
x
1
≤ ⎣Sn,q t − x2 ; x − x ⎦g .
2
qnq
q
3.9
Abstract and Applied Analysis
9
Since x/q2 1/qnq − x2 ≤ δn2 q, x, so, by Lemma 2.2, we have
Sn,q g; x − gx ≤ 2δn2 q, x g .
3.10
By the definition given by 1.15 and Lemma 2.1, we have
Sn,q f; x ≤ Sn,q 1; xf f .
3.11
So, by the definition given by 3.6, we obtain
Sn,q f; x ≤ Sn,q f; x 2f ≤ 3f .
3.12
Thus, for x ∈ 0, ∞, we have
Sn,q f; x − fx ≤
Sn,q f − g; x Sn,q g; x − gx
x
1
−
fx
gx − fx f
q2 qnq
≤ 4f − g 2δn2 q, x g ω f, δn q, x .
3.13
Hence, taking infimum on the right hand side over all g ∈ W 2 , we can get
Sn,q f; q; x − fx ≤ 4K2 f, δn2 q, x ω f, δn q, x .
3.14
By inequality 3.2, for every q ∈ 0, 1, we have
Sn,q f; x − fx ≤ Cω2 f, δn q, x ω f, δn q, x .
3.15
4. Rate of Convergence
Let Hxm 0, ∞ be the set of all functions f defined on 0, ∞ satisfying the condition |fx| ≤
Mf 1 xm , where m > 0, Mf is a constant depending only on f. Let Cxm 0, ∞ denote the
subspace of all continuous functions in Hxm 0, ∞. Also let Cx∗ m 0, ∞ be the subspace of all
functions f ∈ Cxm 0, ∞, for which lim|x| → ∞ fx/1 xm is finite. The norm on Cx∗ m 0, ∞
is fxm supx∈0,∞ |fx|/1 xm . We denote the modulus of continuity of f on the closed
interval 0, a, a > 0 by x ∈ 0, ∞ as
ωa f, δ sup
sup ft − fx,
δ > 0.
|t−x|≤δ x,t∈0,a
We observe that for f ∈ Cx∗ m 0, ∞, the modulus of continuity ωa f, δ → 0 as δ → 0 .
4.1
10
Abstract and Applied Analysis
Theorem 4.1. Let f ∈ Cx2 0, ∞, q ∈ 0, 1, and ωa1 f, δ be the modulus of continuity of f on the
finite interval 0, a 1 ⊂ 0, ∞, where a > 0. Then we have
Sn,q f; · − f ≤ 5Mf 1 a2 ηn2 q 2ωa1 f, ηn q ,
C0,a
where ηn q 2/q6 1 − q4 2/nq a1 a 2q /q3 n2q 0, a}.
1/2
4.2
, fC0,a max{|fx|, x ∈
Proof. For x ∈ 0, a and t > a 1, since t − x > 1, we have |ft − fx| ≤ Mf 2 x2 t2 ≤
Mf 2 3x2 2t − x2 ≤ 5Mf 1 a2 t − x2 .
For x ∈ 0, a and 0 ≤ t ≤ a 1, we have |ft − fx| ≤ ωa1 f, |t − x| ≤ 1 |t − x|/δωa1 f, δ with δ > 0.
So, for x ∈ 0, a and t ≥ 0, we may write
ft − fx ≤ 5Mf 1 a2 t − x2 1 |t − x| ωa1 f, δ .
δ
4.3
Thus, by Cauchy-Schwartz inequality, we obtain
Sn,q f; x − fx ≤ Sn,q ft − fx; x ≤ 5Mf 1 a2 Sn,q t − x2 ; x
ωa1 f, δ
1/2 1
2
Sn,q t − x ; x
.
1
δ
4.4
By Lemma 2.2, for x ∈ 0, a and ηn q 2/q6 1−q4 2/nq a1a2q /q3 n2q we have Sn,q t − x2 ; x ≤ ηn2 q. Hence, for every x ∈ 0, a, q ∈ 0, 1, we obtain
η q
Sn,q f; x − fx ≤ 5Mf 1 a2 ηn2 q ωa1 f, δ 1 n
.
δ
1/2
,
4.5
By taking δ ηn q, we immediately get the desired result.
Corollary 4.2. Assume that f ∈ Cx2 0, ∞, q ∈ 0, 1. Let M > 0, α ∈ 0, 1, f ∈ LipαM on
0, a 1, where a > 0. Then we have
2
Sn,q f; · − f 1
a
ηn2 q Mηnα q ,
≤
5M
f
C0,a
4.6
where ηn q and fC0,a are given in Theorem 4.1.
Proof. Let q ∈ 0, 1, M > 0, α ∈ 0, 1, f ∈ LipαM on 0, a 1. Then for any x ∈ 0, a, t ∈
0, a 1, we have |ft − fx| ≤ M|t − x|α . So, according to the proof of Theorem 4.1, for
x ∈ 0, a, t ≥ 0, we have
ft − fx ≤ 5Mf 1 a2 Sn,q t − x2 ; x M|t − x|α .
4.7
Abstract and Applied Analysis
11
Using the Hölder inequality with m 2/α, n 2/2 − α, for any x ∈ 0, a, t ≥ 0, we
get
α/2
Sn,q M|t − x|α ; x ≤ M Sn,q t − x2 ; x
.
4.8
So, for any x ∈ 0, a, t ≥ 0, we have
α/2
Sn,q f; x − fx ≤ 5Mf 1 a2 Sn,q t − x2 ; x M Sn,q t − x2 ; x
≤ 5Mf 1 a2 ηn2 q Mηnα q .
4.9
The desired result follows immediately.
5. Weighted Approximation
Now we give the weighted approximation result for the operators Sn,q f; x.
Theorem 5.1. Let the sequence q {qn } satisfies qn ∈ 0, 1 and qn → 1 as n → ∞. For f ∈
Cx∗ 2 0, ∞, we have
lim Sn,qn f; · − f x2 0.
n→∞
5.1
Proof. Using the Theorem in 17, we see that it is sufficient to verify the following three
conditions:
lim Sn,qn ei ; · − ei x2 0,
n→∞
5.2
where ei x xi , i 0, 1, 2.
Since Sn,qn e0 ; x 1, it is clear that limn → ∞ Sn,qn e0 ; · − e0 x2 0.
By Lemma 2.2, we have
Sn,q t; x − x
n
Sn,q e1 ; · − e1 2 sup
n
x
1 x2
x∈0,∞
≤
1 − qn2
qn2
1 − qn2
x
1
1
sup
≤
.
2
2
qn nqn
qn nqn
qn
x∈0,∞ 1 x
5.3
Since qn ∈ 0, 1 and limn → ∞ qn 1, we have nqn → ∞ as n → ∞ see 18, so, we
can obtain limn → ∞ Sn,qn e1 ; · − e1 x2 0.
12
Abstract and Applied Analysis
For i 2, by Lemma 2.1, we have
Sn,q t2 ; x − x2 1 − q6
x2
n
n
Sn,q e2 ; · − e2 2 sup
≤
sup
n
x
1 x2
qn6 x∈0,∞ 1 x2
x∈0,∞
3qn qn
2qn
x
sup
qn5 nqn x∈0,∞ 1 x2 qn3 n2qn
6
q
3
n
2qn
qn
1 − qn
≤
,
qn6
qn5 nqn
qn3 n2qn
5.4
which implies that limn → ∞ Sn,qn e2 ; · − e2 x2 0. In a word, we complete the proof.
It is known that, if f is not uniformly continuous on the interval 0, ∞, then the usual
first modulus of continuity ωf, δ does not tend to zero as δ → 0 . For every f ∈ Cx∗ 2 0, ∞,
we would like to take a weighted modulus of continuity Ωf, δ which tends to zero as δ →
0 .
For every f ∈ Cx∗ 2 0, ∞, let
Ω f, δ sup
0<h≤δ,x≥0
fx h − fx
1 h2 1 x2 .
5.5
The weighed modulus of continuity Ωf, δ was defined by İspir in 19. It is known that
Ωf, δ has the following properties.
Lemma 5.2 see 19. Let f ∈ Cx∗ 2 0, ∞. Then
i Ωf, δ is a monotone increasing function of δ;
ii for each f ∈ Cx∗ 2 0, ∞, limδ → 0 Ωf, δ 0;
iii for each m ∈ N, Ωf, mδ ≤ mΩf, δ;
iv for each λ ∈ R , Ωf, λδ ≤ 1 λΩf, δ.
Theorem 5.3. Let f ∈ Cx∗ 2 0, ∞, q ∈ 0, 1. Then we have the inequality
⎛ 1/2 ⎞
3
4
Sn,q f; · − f 5 ≤ M q Ω⎝f, 1 − q ⎠,
x
nq
5.6
where Mq is a positive constant independent f and n.
Proof. By the definition of Ωf, δ and Lemma 5.2 iv, for every δ > 0, we have
ft − fx ≤ 1 t − x2 1 x2
|t − x|
Ω f, δ .
1
δ
5.7
Abstract and Applied Analysis
13
Then, we obtain
Sn,q f; x − fx ≤ 1 x2 Ω f, δ Sn,q 1 t − x2 1 |t − x| ; x
δ
!
≤ 1 x2 Ω f, δ Sn,q 1 t − x2 ; x
Sn,q
1 t − x2
|t − x|
δ
;x
5.8
"
.
Also by the Cauchy-Schwartz inequality, we have
Sn,q
1 t − x
2
|t − x|
δ
;x
#1/2
!
2 "1/2
|t − x|2
2
Sn,q
≤ Sn,q 1 t − x ; x
;x
.
δ2
5.9
Consequently,
Sn,q f; x − fx
≤ 1 x2 Ω f, δ
×
⎧
⎨
⎩
Sn,q 1 t − x2 ; x Sn,q
1 t − x2
2
1/2 Sn,q
;x
|t − x|2
;x
δ2
1/2 ⎫
⎬
.
⎭
5.10
In view of Lemmas 2.1 and 2.2, we get
Sn,q
2
1 t − x ; x ≤ 1 6
q
2
2
≤ 6
q
2
1−q nq
4
3
2−q nq
4
x1 x 1 x2
≤ M1 q 1 x2 .
Also,
Sn,q
1 t − x
2
2
;x
1 2Sn,q t − x2 ; x Sn,q t4 ; x
− 4xSn,q t3 ; x 6x2 Sn,q t2 ; x − 4x3 Sn,q t; x x4
2q
q3 n2q
5.11
14
Abstract and Applied Analysis
4
6
4
1
−
−
1
q20 q12 q6 q2
*
1 3
x
7q q5q q2 3q q3
19
q nq
x4
4 6 4
− 11
5q q3q q2 5
3q q −
qnq
q nq
q nq
⎡
x2 ⎣
+
2
4
1 2
3
4
−
2
2q8
4q
4q
3q
6
4
2
10
q
q
q
q
q
q6 q2
q17 n2q
⎤
62q
4 ⎦
−
6q 2q4q 2q2 2q q9 n2q
q3 n2q
⎡ 2 3q q
4
⎢
x⎣
−
qnq
q5 nq
1
q14 n3q
⎛
⎝1 22q
q3 n2q
10q 3q8q 5q2 6q 6q3 4q 4q4 2q
2q 3q 4q
q10 n4q
⎤
42q 3q ⎥
−
⎦
q6 n3q
⎞
⎠ ≤ 16 x4 80 x3 164 x2 84 x 1 28
q20
q19
q17
q14
q10
2
≤ M2 q 1 x 2 ,
Sn,q
|t − x|2
;x
δ2
#1/2
⎤1/2
⎡ 2
1⎣ 2
2
q
⎦
1 − q4 x1 x ≤
δ q6
nq
q3 n2q
1/2
1
3
4
≤ M3 q 1 − q 1 x.
δ
nq
5.12
Now from inequalities 5.10–5.12, we have
⎡
⎢
Sn,q f; x − fx ≤ Ω f, δ ⎢M1 q 1 x2 1 x2
⎣
⎤
1/2
.
2
⎥
1
3
1 − q4 1 x2 1 x⎥
M2 q M3 q
⎦
δ
nq
Abstract and Applied Analysis
15
⎡
⎢
2
2
≤ Ω f, δ ⎢
⎣M1 q 1 x 1 x
.
1
M2 q M3 q M4
δ
1 − q4 3
nq
1/2
⎤
⎥
1 x5 ⎥
⎦,
5.13
where M4 supx≥0 1 x2 2 1 x/1 x5 .
Taking δ 1 − q4 3/nq 1/2 , from the above inequality we can obtain the desired
result.
Acknowledgments
This work is supported by the National Natural Science Foundation of China Grant no.
61170324, the Class A Science and Technology Project of Education Department of Fujian
Province of China Grant no. JA12324, and the Natural Science Foundation of Fujian
Province of China Grant no. 2010J01012. The authors thank the associate editor and the
referees for several important comments and suggestions which improve the quality of the
paper.
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