Journal of Inequalities in Pure and
Applied Mathematics
ON A NEW TYPE OF MEYER-KONIG AND ZELLER OPERATORS
VIJAY GUPTA
School of Applied Sciences
Netaji Subhas Institute of Technology
Azad Hind Fauj Marg, Sector 3 Dwarka
New Delhi-110045 INDIA.
EMail: vijaygupta2001@hotmail.com
volume 3, issue 4, article 57,
2002.
Received 21 December, 2001;
accepted 6 June, 2002.
Communicated by: R.N. Mohapatra
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this present paper, we introduce a new and simple integral modification of
the Meyer-Konig and Zeller Bezier type operators and study the rate of convergence for functions of bounded variation. Our result improves and corrects the
results of Guo (J. Approx. Theory, 56 (1989), 245–255 ), Zeng (Comput. Math.
Appl., 39 (2000), 1–13; J. Math. Anal. Appl., 219 (1998), 364–376), etc.
2000 Mathematics Subject Classification: 41A25, 41A30.
Key words: Rate of convergence, Bounded variation functions, Total variation,
Meyer-Konig-Zeller-Bezier operators, Beta function.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
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1.
Introduction
For a function defined on [0, 1], the Meyer-Konig and Zeller operators Pn , n ∈
N [6] are defined by
∞
X
k
(1.1)
Pn (f, x) =
mn,k (x) f
, x ∈ [0, 1] ,
n
+
k
k=0
where
mn,k (x) =
n+k−1
k
xk (1 − x)n .
On a New Type of Meyer-Konig
and Zeller Operators
The rates of convergence of some integral modifications of the operators
(1.1) were discussed in [1] – [5] and [7]. Recently, Zeng [8] generalized the operators (1.1) and its integral modification and estimated the rate of convergence
for functions of bounded variation. We introduce a new generalization of the
Meyer-Konig-Zeller operators for functions defined on [0, 1] as
Z 1
∞
X
(α)
(1.2)
Bn,α (f, x) =
Qn,k (x)
bn,k (t) f (t) dt, x ∈ [0, 1] ,
k=0
0
Vijay Gupta
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where α ≥ 1,
(α)
Qn,k (x) =
∞
X
j=k
and
bn,k (t) =
!α
mn,j (x)
−
∞
X
!α
mn,j (x)
j=k+1
(n + k)! k
t (1 − t)n−1 .
k! (n − 1)!
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(α)
For further properties of Qn,k (x), we refer readers to [8]. Actually, the main
purpose in introducing these operators (1.2) is that some approximation formulae for Bn,α , n ∈ N become simpler than the corresponding results for the other
modifications considered in [8]. For α = 1, the operators (1.2) reduce to
Bn,1 (f, x) =
∞
X
k=0
Z
mn,k (x)
1
bn,k (t) f (t) dt.
0
In the present paper, we study the behaviour of Bn,α (f, x) for functions of
bounded variation and give an estimate on the rate of convergence for these new
integrated Meyer-Konig-Zeller-Bezier operators. In the last section, we give the
correct estimates for some other generalized Meyer-Konig-Zeller type operators
considered in [2], [7] and [8].
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
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2.
Auxiliary Results
In this section we give certain results, which are necessary to prove the main
result.
Lemma 2.1. [7]. For all k, n ∈ N, x ∈ (0, 1] , we have
mn,k (x) < √
where the constant
√1
2e
1
,
2enx
On a New Type of Meyer-Konig
and Zeller Operators
is the best possible.
Lemma 2.2. For r ∈ N0 ( the set of non negative integers ), if we define
r
Bn,1 ((t − x) , x) =
∞
X
k=0
then
Bn,1
Z
mn,k (x)
1
bn,k (t) (t − x)r dt,
0
2 (1 − x)2
4x
(t − x) , x ≤
+
.
n − 1 (n − 1) (n − 2)
2
Proof. Put er (x) = xr , r = 0, 1, 2, . . . . The moments of the operators Bn,1 f
are given by
∞
X
(n + k)!
mn,k (x)
Bn,1 (t , x) =
B (k + r + 1, n) ,
k! (n − 1)!
k=0
r
Vijay Gupta
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where B is the Beta function. Obviously Bn,1 e0 = e0 = 1.
∞ X
(k + 1)
n+k−1
Bn,1 (t, x) = (1 − x)
xk
k
(n + k + 1)
k=0
∞ X
(k + 1) (n + k − 1)
n+k−2
n
·
≥ (1 − x)
xk
k−1
k
(n + k + 1)
k=0
∞
X
2
2
n+k−1
n
k+1
≥ (1 − x)
= 1−
x.
x
1−
k
n
+
1
n
+
1
k=0
n
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
Next,
Bn,1 t2 , x
∞ X
k (k − 1) + 4k + 2
n+k−1
n
= (1 − x)
xk
k
(n + k + 1) (n + k + 2)
k=0
∞
∞
∞
X
X
(1 − x)n X (n + k − 3)! k
(n + k − 3)! k
(n + k − 3)! k
≤
x +4
x +2
x
(n − 1)! k=2 (k − 2)!
(k − 1)!
k!
k=1
k=0
(∞ ∞ X n+k−1 X
4
n+k−1
n
k+2
≤ (1 − x)
x
+
xk+1
k
k
(n
−
1)
k=0
k=0
2
n+k−1
+
xk
k
(n − 1) (n − 2)
4x (1 − x)
2 (1 − x)2
=x +
+
.
(n − 1)
(n − 1) (n − 2)
2
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Combining these, we get
Bn,1 (t − x)2 , x = Bn,1 t2 , x − 2xBn,1 (t, x) + x2
2 (1 − x)2
4x
≤
+
.
(n − 1) (n − 1) (n − 2)
In particular, given any λ > 4 and any x ∈ (0, 1), there is an integer N (λ, x)
such that for all n ≥ N (λ, x)
λx
.
Bn,1 (t − x)2 , x ≤
n
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
Lemma 2.3. For all x ∈ (0, 1] and k ∈ N, there holds
α
(α)
.
Qn,k (x) ≤ αmn,k (x) < √
2enx
Proof. It is easy to verify that |aα − bα | ≤ α |a − b| , (0 ≤ a, b ≤ 1, α ≥ 1) .
Then by Lemma 2.1, we obtain
α
(α)
Qn,k (x) ≤ αmn,k (x) < √
.
2enx
P
(α)
Lemma 2.4. Let Kn,α (x, t) = ∞
k=0 Qn,k (x) bn,k (t) , and λ > 4, n ≥ N (λ, x)
then
Z y
αλx
(2.1)
λn,α (x, y) =
Kn,α (x, t) dt ≤
,
0 ≤ y < x,
n (x − y)2
0
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Z
(2.2)
1
1 − λn,α (x, y) =
Kn,α (x, t) dt ≤
z
αλx
,
n (z − x)2
x < z ≤ 1.
Proof. We first prove (2.1), as follows
Z
y
Z
Kn,α (x, t) dt ≤
0
y
Kn,α (x, t)
0
(x − t)2
dt
(x − y)2
1
2
2 Bn,α (t − x) , x
(x − y)
αBn,1 (t − x)2 , x
αλx
≤
≤
,
2
(x − y)
n (x − y)2
≤
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
by Lemma 2.2. The proof of (2.2) is similar.
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Lemma 2.5. [8, p. 5]. For x ∈ (0, 1), we have
Contents
X
nx
<k
1−x
1
5
mn,k (x) −
≤ √ .
2
2 nx
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3.
Main Result
In this section we prove the following main theorem
Theorem 3.1. Let f be a function of bounded variation on [0, 1] , α ≥ 1. Then
for every x ∈ (0, 1) and λ > 4 and n ≥ max {N (λ, x) , 3}, we have
α
1
f (x+) +
f (x−)
Bn,α (f, x) −
α+1
α+1
1 α2 + α − 2
α
≤
+√
|f (x+) − f (x−)|
2
α+1
2enx
Vijay Gupta
1−x
√
+

f (t) − f (x−) , 0 ≤ t < x;





0,
t = x;
gx (t) =





f (t) − f (x+) , x < t ≤ 1
Wb
a
(gx ) is the total variation of gx on [a, b].
Proof. Clearly
(gx ) ,
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(3.1)
k=1
x− √x
k
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where
and
(2λα + x)
nx
n x+
X
_k
On a New Type of Meyer-Konig
and Zeller Operators
1
α
Bn,α (f, x) −
f (x+) +
f (x−)
α+1
α+1
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α−1
α+1
≤ Bn,α (sgn (t − x) , x) +
·
|f (x+) − f (x−)|
2
+ |Bn,α (gx , x)| .
First, we have
Bn,α (sgn (t − x) , x) =
=
∞
X
k=0
∞
X
(α)
Qn,k
Z
1
bn,k (t) dt −
(x)
x
(α)
Qn,k
Z
1
Z
bn,k (t) dt − 2
0
∞
X
(α)
bn,k (t) dt.
0
(α)
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Pk
j=0
∞
X
(α)
Qn,k (x)
mn,j (x)
∞
X
k=0
k
X
mn,k (x)
k
X
j=0
mn,j (x) =
R1
x
bn,k (t) dt,
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mn,j (x)
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j=0
k=0
≤ −1 + 2α
!
j=0
k=0
= −1 + 2
k
X
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
Bn,α (sgn (t − x) , x)
Qn,k (x) 1 −
bn,k (t) dt
x
Qn,k (x)
Using Lemma 2.1, Lemma 2.3 and the fact that
we have
=1−2
x
0
Z
k=0
∞
X
bn,k (t) dt
0
(x)
k=0
=1−2
x
Z
mn,j (x)
J. Ineq. Pure and Appl. Math. 3(4) Art. 57, 2002
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"
= −1 + α + α
∞
X
mn,k (x)
mn,j (x) −
j=0
k=0
≤ α − 1 + αmn,k (x)
k
X
∞
X
∞
X
#
(mn,k (x))2
k=0
mn,k (x) ≤ α − 1 + √
k=0
α
.
2enx
Thus
Bn,α (sgn (t − x) , x) +
(3.2)
α−1
α+1
≤
α2 + α − 2
α
.
+√
α+1
2enx
Next we estimate Bn,α (gx , x). By the Lebesgue-Stieltjes integral representation, we have
Z 1
Bn,α (gx , x) =
Kn,α (x, t) gx (t) dt
0
Z
Z
Z =
+
+
Kn,α (x, t) gx (t) dt
I1
I2
I3
= E1 + E2 + E3 ,
say,
where
x
x
1−x
1−x
I1 = 0, x − √ , I2 = x − √ , x + √
and I3 = x + √ , 1 .
n
n
n
n
We first estimate E1 . Writing y = x − √xn and using Lebesgue-Stieltjes integraRt
tion by parts with λn,α (x, t) = 0 Kn,α (x, u) du, we have
Z y
Z y
E1 =
gx (t) dt (λn,α (x, t)) = gx (y+) λn,α (x, y) −
λn,α (x, t) dt (gx (t)) .
0
0
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
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x
Since |gx (y+)| ≤ Vy+
(gx ), it follows that
|E1 | ≤
x
_
Z
y
λn,α (x, t) dt −
(gx ) λn,α (x, y) +
x
_
0
y+
!
(gx ) .
t
By using (2.1) of Lemma 2.4, we get
x
_
αλx
αλx
|E1 | ≤
(gx )
2 +
n
n (x − y)
y+
Z
y
0
!
x
_
1
dt − (gx ) .
(x − t)2
t
Integrating the last term by parts we have after simple computation
Wx
Z y Wx
(g
)
αλx
x
0
t (gx )
+2
|E1 | ≤
3 dt .
n
x2
0 (x − t)
Now replacing the variable y in the last integral by x − √xu , we obtain


x
n
x
n
x
_
X
_
X
_
αλ 
 ≤ 2λα
(gx ) +
(gx ) .
(3.3)
|E1 | ≤
nx 0
nx
x
x
√
√
k=1
k=1
x−
x−
k
k
Using the similar method of Lemma 2.4 and equation (2.2), we get
(3.4)
|E3 | ≤
2αλ
nx
k=1
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1−x
√
n x+
X
_k
On a New Type of Meyer-Konig
and Zeller Operators
(gx ) .
Page 12 of 22
x
J. Ineq. Pure and Appl. Math. 3(4) Art. 57, 2002
Finally we estimate E2 .
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h
For t ∈ x −
√x , x
n
+
1−x
√
n
i
, we have
√
x+ 1−x
|gx (t)| = |gx (t) − gx (x)| ≤
_k
(gx ) ,
x− √x
k
and therefore
√
x+ 1−x
_k
|E2 | ≤
Z
a
dt λn (x, t) dt ≤ 1, for (a, b) ⊂ [0, 1], therefore
|E2 | ≤
_
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1−x
√
k
(gx ) ≤
x− √x
k
1
n
n x+
X
_k
(gx ) .
k=1 x− √x
|Bn,α (gx , x)| ≤
(2λα + x)
nx
k=1 x− √x
k
Combining (3.1), (3.2) and (3.6), our theorem follows.
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1−x
√
n x+
X
_k
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k
Finally collecting the estimates (3.2) – (3.5), we obtain
(3.6)
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
√
x+ 1−x
(3.5)
dt (λn,α (x, t)) .
x− √x
k
k
Since
k
(gx )
x− √x
Rb
√
x+ 1−x
(gx ).
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4.
Some Examples
Recently Zeng [8] introduced some other generalized Meyer-Konig and Zeller
operators as
!Z
Z 1
(α)
∞
X
a
Qn,k (x)
R
(4.1) M n,α (f, x) =
f (t) dt =
f (t) Kn,α,2 (x, t) dt,
1dt
I
0
k
I
k
k=0
(α)
P
Q χk (t)
R n,k
where α ≥ 1, Kn,α,2 (x, t) = ∞
k=0 l χk (u)du and χk is the characteristic func
k
(α)
k+1
with respect to I = [0, 1] and Qn,k (x) is
tion of the interval Ik = n+k , n+k+1
as defined by (1.1). In particular, for α = 1, the operators (4.1) reduce to the
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
a
operators M n,1 (f, x) studied by Guo [2], as
a
M n,1 (f, x) =
(4.2)
where
a
mn,k (x) = (n + 1)
∞
X
a
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Z
mn,k (x)
f (t) dt,
k=0
Ik
n+k+1
k
xk (1 − x)n .
We have noticed that Theorem 2 in [8] and Theorem 4.2 in [7] are not correct,
there are some misprints. This motivated us to correct these estimates and in
this section we have been able to correct and achieve improved estimates over
the results of Zeng ([7, 8]), Guo [2] and Love et al. [5].
The misprinted estimate obtained by Zeng [8] is as follows:
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Theorem 4.1. Let f be a function of bounded variation on [0, 1]. Then for every
x ∈ (0, 1) and n sufficiently large, we have
a
1
1
M n,α (f, x) − α f (x+) − 1 − α f (x−)
2
2
√
x+ 1−x
n
k
5α
5α X _
≤√
|f (x+) − f (x−)| +
(gx ) ,
nx + 1 k=1 √x
nx + 1
x−
where gx (t) and
k
On a New Type of Meyer-Konig
and Zeller Operators
Wb
a (gx ) are as defined by Theorem 3.1.
Remark 4.1. It is remarked that there are misprints in Lemma 6 and Lemma 8
of Zeng [8]. Actually the author [8] has not verified his Lemmas 6 and 8 and he
had taken these results directly from the paper of Guo [2] (see Lemmas 5 and 6
of [2]). As the Lemmas 5 and 6 of [2] are not correct. Although these mistakes
were pointed out earlier by Love et al. [5] in 1994. Zeng also has not gone
through the paper of Love et al. [5] and in another paper he has obtained the
following misprinted exact bound for the operators (4.2):
Theorem 4.2. Let f be a function of bounded variation on [0, 1]. Then for every
x ∈ (0, 1) and n sufficiently large, we have
a
1
M n,1 (f, x) − [f (x+) − f (x−)]
2
1−x
√
n x+ k
1
1
5 X _
≤ 16 + √
|f (x+) − f (x−)| +
(gx ) ,
3√
nx k=1 √x
2e x 2 n
x−
k
Vijay Gupta
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where gx (t) and
Wb
a
(gx ) are as defined by Theorem 3.1.
5
Remark 4.2. It may be remarked here that the factor nx
in Theorem 4.2 and
5α
in Theorem 4.1 are not correct as the second moment estimate of Guo [2]
nx+1
was not obtained correctly.
We now give the correct and improved statement of Theorem 4.1 with an
outline of the proof. For α = 1 our result reduces to the corresponding improved
version of Theorem 4.2. Before giving the theorem we shall give a lemma which
is necessary in the proof of our Theorem 4.4.
1−δ
4
Lemma 4.3. Let n > 2, ωn = (n−1)δ
1 + 12(n−2)δ
where 0 < δ ≤ x ≤ 1−δ,
2
then
(i) for 0 < y < x there holds
Z y
αωn x3 (1 − x)
(4.3)
Kn,α,2 (x, t) dt ≤
;
(x − y)2
0
(ii) for x < y < 1 there holds
Z 1
αωn x (1 − x)3
(4.4)
Kn,α,2 (x, t) dt ≤
.
(z − x)2
z
Proof. Following [5, Lemma 7] for x ∈ (0, 1) , n > 2, we have
a
(4.5)
M n,1
4x (1 − x) 1
(1 − x)2
+ ·
.
(t − x) , x ≤
(n − 1)
3 (n − 1) (n − 2)
2
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
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Using (4.5), we have
Z y
Z
Kn,α,2 (x, t) dt ≤
0
0
y
(x − t)2
dt
(x − y)2
(t − x)2 , x
Kn,α,2 (x, t)
a
α
M
n,1
(x − y)2
#
"
α
(1 − x)2
4x (1 − x)
≤
+
(n − 1)
3 (n − 1) (n − 2)
(x − y)2
1
−1
4αx3 (1 − x)
δ
≤
1+
12 (n − 2)
δ 2 (n − 1) (x − y)2
3
αωn x (1 − x)
.
=
(x − y)2
≤
This completes the proof of (4.3). The proof of (4.4) is similar.
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
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Theorem 4.4. Let f be a function of bounded variation on [0, 1]. Then for every
x ∈ (0, 1) , n > 2 and 0 < δ ≤ x ≤ 1 − x, we have
a
1
1
M n,α (f, x) − α f (x+) − 1 − α f (x−)
2
2
1
5
α
√ |f (x+) − f (x−)|
≤
+√
2
nx
2e
1−x
√
X
n x+
_k
1
+ 2αωn x (1 − x) +
(gx ) ,
n − 1 k=1 √x
x−
k
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where gx (t) and
Lemma 4.3.
Wb
a
(gx ) are as defined by Theorem 3.1 and ωn is as given by
Proof. First
1
1
M n,α (f, x) − α f (x+) − 1 − α f (x−)
2
2
a
f (x+) − f (x−) a
≤ M n,α (gx , x) +
M n,α (sgn (t − x) , x)
2α
a
1
1
+ f (x+) − α f (x+) − 1 − α f (x−) M n,α (δx , x) .
2
2
a
a
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
Title Page
Following [8], we have M n,α (δx , x) = 0 and
Contents
a
M n,α (sgn (t − x) , x) ≤ α2α
X
nx
<k
1−x
mn,k (x) −
1
(α)
+ 2α Qn,k0 (x)
2
k0
k0 +1
where x ∈ n+k
0 , n+k 0 +1 .
Using Lemma 2.3 and Lemma 2.5, we have
a
α2α
5
1
√ .
M n,α (sgn (t − x) , x) ≤
+√
2
nx
2e
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a
Next, we estimate M n,α (gx , x), as follows
1
Z
a
M n,α (gx , x) =
Kn,α,2 (x, t) gx (t) dt
0
Z
x− √xn
=
Z
√
x+ 1−x
n
+
!
1
+
x− √xn
0
Z
√
x+ 1−x
n
Kn,α,2 (x, t) gx (t)dt
= R1 + R2 + R3 , say.
The evaluation of R1 , R2 , R3 are similar to work in [8]. We have
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
√
x+ 1−x
n
|R2 | ≤
_
x− √xn
√
x+ 1−x
k
n
1 X _
(gx ) ≤
(gx ) .
n − 1 k=1 √x
x−
k
Contents
Next suppose y = x − √xn . Using integration by parts with µn,α (x, t) =
Rt
Kn,α,2 (x, u) du, we have
0
Z y
R1 =
gx (t) dt (µn,α (x, t))
0
Z y
= gx (y+) µn,α (x, y) −
µn,α (x, t) dt (gx (t))
0
!
Z y
x
x
_
_
≤
(gx ) µn,α (x, y) +
µn,α (x, t) dt − (gx ) .
y+
0
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By (4.3) of Lemma 4.3, we have
x
_
αωn x3 (1 − x)
|R1 | ≤
(gx )
+ αωn x3 (1 − x)
2
(x − y)
y+
Z
y
0
!
x
_
1
dt − (gx ) .
(x − t)2
t
Integrating the last term by parts we get
Wx
Z y Wx
− t (gx )
3
0 (gx )
|R1 | ≤ αωn x (1 − x)
+2
dt .
x2
(x − t)3
0
Now replacing the variable y in the last integral by x −
√x ,
u
we have


x
n
x
_
X
_
1
 αωn x3 (1 − x)
|R1 | ≤ 2  (gx ) +
x
x
√
0
k=1
x−
≤ 2αωn x (1 − x)
n
X
k
x
_
Vijay Gupta
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(gx ) .
k=1 x− √x
k
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Finally using the similar methods, we have
|R3 | ≤ 2αωn x (1 − x)
On a New Type of Meyer-Konig
and Zeller Operators
n
x
X
_
k=1
Close
(gx ) .
x− √x
k
Combining the estimates of R1 , R2 , R3 , our theorem follows.
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Remark 4.3. In particular, for α = 1, by Theorem
4.4 it may be remarked
5
that the main theorem of Love et al. [5] i.e. x√nx |f (x+) − f (x−)| can be
improved to
√
5
1
+√
nx |f (x+) − f (x−)| .
2
2e
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and Zeller Operators
Vijay Gupta
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References
[1] S. GUO, Degree of approximation to functions of bounded variation by
certain operators, Approx. Theory and its Appl., 4(2) (1988), 9–18.
[2] S. GUO, On the rate of convergence of the integrated Meyer-Konig and
Zeller operators for functions of bounded variation, J. Approx. Theory, 56
(1989), 245–255.
[3] V. GUPTA, A sharp estimate on the degree of approximation to functions of
bounded variation by certain operators, Approx. Theory and its Appl., 11(3)
(1995), 106–107.
[4] V. GUPTA AND A. AHMAD, An improved estimate on the degree of approximation to functions of bounded variation by certain operators, Revista
Colombiana de Matematicas, 29 (1995), 119–126.
[5] E.R. LOVE, G. PRASAD AND A. SAHAI, An improved estimate of the
rate of convergence of the integrated Meyer-Konig and Zeller operators for
functions of bounded variation, J. Math. Anal. Appl., 187(1) (1994), 1–16.
On a New Type of Meyer-Konig
and Zeller Operators
Vijay Gupta
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[6] W. MEYER-KONIG AND K. ZELLER, Bernsteinsche Pootenzreihen, Studia Math., 19 (1960), 89–94.
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[7] X.M. ZENG, Bounds for Bernstein basis functions and Meyer-Konig and
Zeller basis functions, J. Math. Anal. Appl., 219 (1998), 364–376.
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[8] X.M. ZENG, Rates of convergence of bounded variation functions by two
generalized Meyer-Konig and Zeller type operators, Comput. Math. Appl.,
39 (2000), 1–13.
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