Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 4, Article 57, 2002
ON A NEW TYPE OF MEYER-KONIG AND ZELLER OPERATORS
VIJAY GUPTA
S CHOOL OF A PPLIED S CIENCES
N ETAJI S UBHAS I NSTITUTE OF T ECHNOLOGY
A ZAD H IND FAUJ M ARG , S ECTOR 3 DWARKA
N EW D ELHI -110045 INDIA.
vijaygupta2001@hotmail.com
Received 21 December, 2001; accepted 6 June, 2002
Communicated by R.N. Mohapatra
A BSTRACT. 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.
Key words and phrases: Rate of convergence, Bounded variation functions, Total variation, Meyer-Konig-Zeller-Bezier operators, Beta function.
2000 Mathematics Subject Classification. 41A25, 41A30.
1. I NTRODUCTION
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
n+k−1
mn,k (x) =
xk (1 − x)n .
k
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
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
091-01
0
2
V IJAY G UPTA
where α ≥ 1,
∞
X
(α)
Qn,k (x) =
!α
mn,j (x)
∞
X
−
j=k
!α
mn,j (x)
j=k+1
and
bn,k (t) =
(n + k)! k
t (1 − t)n−1 .
k! (n − 1)!
(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
∞
X
Bn,1 (f, x) =
1
Z
mn,k (x)
bn,k (t) f (t) dt.
0
k=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-ZellerBezier 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].
2. AUXILIARY R ESULTS
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
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
Z
mn,k (x)
k=0
1
bn,k (t) (t − x)r dt,
0
then
Bn,1
4x
2 (1 − x)2
(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
r
Bn,1 (t , x) =
∞
X
k=0
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
mn,k (x)
(n + k)!
B (k + r + 1, n) ,
k! (n − 1)!
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N EW TYPE M EYER -KONIG Z ELLER OPERATORS
3
where B is the Beta function. Obviously Bn,1 e0 = e0 = 1.
∞ X
(k + 1)
n+k−1
n
Bn,1 (t, x) = (1 − x)
xk
k
(n + k + 1)
k=0
∞
X
(k + 1) (n + k − 1)
n+k−2
·
≥ (1 − x)n
xk
k−1
k
(n + k + 1)
k=0
∞ X
2
n+k−1
n
k+1
≥ (1 − x)
x
1−
k
n+1
k=0
2
= 1−
x.
n+1
Next,
∞ X
k (k − 1) + 4k + 2
n+k−1
n
2
Bn,1 t , x = (1 − x)
xk
k
(n + k + 1) (n + k + 2)
k=0
∞
∞
X
(n + k − 3)! k
(1 − x)n X (n + k − 3)! k
x +4
x
≤
(n − 1)! k=2 (k − 2)!
(k − 1)!
k=1
+2
∞
X
(n + k − 3)!
k=0
k!
xk
(∞ ∞ X n+k−1 X
4
n+k−1
n
k+2
xk+1
≤ (1 − x)
x
+
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)
Combining these, we get
Bn,1 (t − x)2 , x = Bn,1 t2 , x − 2xBn,1 (t, x) + x2
2
≤
4x
2 (1 − x)2
+
.
(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
Lemma 2.3. For all x ∈ (0, 1] and k ∈ N, there holds
(α)
Qn,k (x) ≤ αmn,k (x) < √
α
.
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
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
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4
V IJAY G UPTA
Lemma 2.4. Let Kn,α (x, t) =
P∞
(α)
k=0
Z
Qn,k (x) bn,k (t) , and λ > 4, n ≥ N (λ, x) then
y
αλx
,
n (x − y)2
0
Z 1
αλx
1 − λn,α (x, y) =
Kn,α (x, t) dt ≤
,
n (z − x)2
z
Kn,α (x, t) dt ≤
λn,α (x, y) =
(2.1)
(2.2)
Proof. We first prove (2.1), as follows
Z y
Z
Kn,α (x, t) dt ≤
0 ≤ y < x,
x < z ≤ 1.
(x − t)2
Kn,α (x, t)
dt
(x − y)2
0
1
2
≤
2 Bn,α (t − x) , x
(x − y)
αBn,1 (t − x)2 , x
≤
(x − y)2
αλx
≤
,
n (x − y)2
0
y
by Lemma 2.2. The proof of (2.2) is similar.
Lemma 2.5. [8, p. 5]. For x ∈ (0, 1), we have
X
mn,k (x) −
nx <k
1−x
1 5
≤ √ .
2 2 nx
3. M AIN R ESULT
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
1
α
Bn,α (f, x) −
f (x+) +
f (x−) α+1
α+1
1−x
√
2
n x+
k
X
_
1 α +α−2
(2λα + x)
α
≤
+√
|f (x+) − f (x−)| +
(gx ) ,
2
α+1
nx
2enx
x
√
k=1 x−
k
where


f (t) − f (x−) , 0 ≤ t < x;




0,
t = x;
gx (t) =




 f (t) − f (x+) , x < t ≤ 1
and
Wb
a
(gx ) is the total variation of gx on [a, b].
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
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N EW TYPE M EYER -KONIG Z ELLER OPERATORS
5
Proof. Clearly
1
α
(3.1) Bn,α (f, x) −
f (x+) +
f (x−) α+1
α+1
α − 1 |f (x+) − f (x−)|
≤ Bn,α (sgn (t − x) , x) +
·
+ |Bn,α (gx , x)| .
α+1 2
First, we have
Z 1
Z x
∞
X
(α)
Bn,α (sgn (t − x) , x) =
Qn,k (x)
bn,k (t) dt −
bn,k (t) dt
=
x
k=0
∞
X
(α)
Qn,k
0
1
Z
Z
0
=1−2
(α)
Qn,k
0
(x)
bn,k (t) dt.
0
Using Lemma 2.1, Lemma 2.3 and the fact that
Bn,α (sgn (t − x) , x) = 1 − 2
(α)
Qn,k
Pk
j=0 mn,j (x) =
(x) 1 −
∞
X
k
X
(α)
Qn,k (x)
bn,k (t) dt, we have
!
x
mn,j (x)
∞
X
mn,k (x)
k
X
mn,j (x)
"
mn,j (x)
j=0
k=0
= −1 + α + α
!
j=0
k=0
≤ −1 + 2α
k
X
R1
j=0
k=0
= −1 + 2
x
Z
k=0
∞
X
bn,k (t) dt
bn,k (t) dt − 2
(x)
k=0
∞
X
x
∞
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)
k=0
α
≤α−1+ √
.
2enx
Thus
α2 + α − 2
α
−
1
α
(3.2)
+√
.
Bn,α (sgn (t − x) , x) + α + 1 ≤
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
1−x
1−x
x
I1 = 0, x − √ , I2 = x − √ , x + √
and I3 = x + √ , 1 .
n
n
n
n
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
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V IJAY G UPTA
We first estimate E1 . Writing y = x − √xn and using Lebesgue-Stieltjes integration by parts
Rt
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
Since |gx (y+)| ≤
x
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+
y
Z
0
!
x
_
1
dt − (gx ) .
(x − t)2
t
Integrating the last term by parts we have after simple computation
Wx
Z y Wx
αλx
0 (gx )
t (gx )
|E1 | ≤
+2
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
_
αλ 
2λα X _

(gx ) +
≤
(gx ) .
(3.3)
|E1 | ≤
nx 0
nx k=1 √x
x
√
k=1
x−
x−
k
k
Using the similar method of Lemma 2.4 and equation (2.2), we get
√
x+ 1−x
n
k
2αλ X _
|E3 | ≤
(gx ) .
nx k=1 x
(3.4)
Finally wehestimate E2 .
For t ∈ x − √xn , x +
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
Since
a
dt (λn,α (x, t)) .
x− √x
k
dt λn (x, t) dt ≤ 1, for (a, b) ⊂ [0, 1], therefore
√
x+ 1−x
(3.5)
k
(gx )
x− √x
k
Rb
√
x+ 1−x
|E2 | ≤
_k
x− √x
k
√
x+ 1−x
n
k
1X _
(gx ) ≤
(gx ) .
n k=1 √x
x−
k
Finally collecting the estimates (3.2) – (3.5), we obtain
√
x+ 1−x
(3.6)
|Bn,α (gx , x)| ≤
n
k
(2λα + x) X _
nx
(gx ).
k=1 x− √x
k
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
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N EW TYPE M EYER -KONIG Z ELLER OPERATORS
7
Combining (3.1), (3.2) and (3.6), our theorem follows.
4. S OME E XAMPLES
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
Ik
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 function of the interval
k
(α)
k+1
Ik = n+k , n+k+1
with respect to I = [0, 1] and Qn,k (x) is as defined by (1.1). In particular,
a
for α = 1, the operators (4.1) reduce to the operators M n,1 (f, x) studied by Guo [2], as
Z
∞
X
a
a
(4.2)
M n,1 (f, x) =
mn,k (x)
f (t) dt,
k=0
where
Ik
n+k+1
xk (1 − x)n .
mn,k (x) = (n + 1)
k
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:
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) − f (x+) − 1 −
f
(x−)
2α
2α
a
√
x+ 1−x
n
k
5α
5α X _
≤√
|f (x+) − f (x−)| +
(gx ) ,
nx + 1 k=1 √x
nx + 1
x−
k
Wb
where gx (t) and a (gx ) are as defined by Theorem 3.1.
Remark 4.2. 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 4.3. Let f be a function of bounded variation on [0, 1]. Then for every x ∈ (0, 1) and
n sufficiently large, we have
a
M n,1 (f, x) − 1 [f (x+) − f (x−)]
2
1−x
√
n x+
k
X
_
1
5
1
|f (x+) − f (x−)| +
≤ 16 + √
(gx ) ,
3√
nx k=1 √x
2e x 2 n
x−
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
k
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8
V IJAY G UPTA
W
where gx (t) and ba (gx ) are as defined by Theorem 3.1.
5
5α
Remark 4.4. It may be remarked here that the factor nx
in Theorem 4.3 and nx+1
in Theorem
4.1 are not correct as the second moment estimate of Guo [2] 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 4.3. Before
giving the theorem we shall give a lemma which is necessary in the proof of our Theorem 4.6.
4
1−δ
Lemma 4.5. Let n > 2, ωn = (n−1)δ2 1 + 12(n−2)δ where 0 < δ ≤ x ≤ 1 − δ, 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
M n,1
(4.5)
(1 − x)2
4x (1 − x) 1
+ ·
.
(t − x) , x ≤
(n − 1)
3 (n − 1) (n − 2)
2
Using (4.5), we have
Z y
Z
Kn,α,2 (x, t) dt ≤
0
≤
≤
≤
=
(x − t)2
Kn,α,2 (x, t)
dt
(x − y)2
0
a
α
2
2 M n,1 (t − x) , x
(x − y)
"
#
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
αωn x3 (1 − x)
.
(x − y)2
y
This completes the proof of (4.3). The proof of (4.4) is similar.
Theorem 4.6. 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
M n,α (f, x) − 1 f (x+) − 1 − 1 f (x−)
α
α
2
2
α
5
1
√ |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−
where gx (t) and
Wb
a
k
(gx ) are as defined by Theorem 3.1 and ωn is as given by Lemma 4.5.
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
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N EW TYPE M EYER -KONIG Z ELLER OPERATORS
9
Proof. First
a
M n,α (f, x) − 1 f (x+) − 1 − 1 f (x−)
α
α
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
Following [8], we have M n,α (δx , x) = 0 and
X
a
1 (α)
α
M n,α (sgn (t − x) , x) ≤ α2 mn,k (x) − + 2α Qn,k0 (x)
2
nx <k
1−x
k0
k0 +1
where x ∈ n+k
0 , n+k 0 +1 .
Using Lemma 2.3 and Lemma 2.5, we have
a
5
1
α2α
M n,α (sgn (t − x) , x) ≤
√
√
+
.
2
nx
2e
a
Next, we estimate M n,α (gx , x), as follows
Z 1
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+ 1−x
n
x− √xn
0
Z
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
√
x+ 1−x
n
|R2 | ≤
_
x− √xn
√
x+ 1−x
n
k
1 X _
(gx ) ≤
(gx ) .
n − 1 k=1 √x
x−
k
Next suppose y = x − √xn . Using integration by parts with µn,α (x, t) =
have
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))
Rt
0
Kn,α,2 (x, u) du, we
0
≤
x
_
Z
y
µn,α (x, t) dt −
(gx ) µn,α (x, y) +
0
y+
x
_
!
(gx ) .
t
By (4.3) of Lemma 4.5, we have
|R1 | ≤
x
_
y+
3
(gx )
αωn x (1 − x)
+ αωn x3 (1 − x)
2
(x − y)
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
Z
0
y
1
dt −
(x − t)2
x
_
!
(gx ) .
t
http://jipam.vu.edu.au/
10
V IJAY G UPTA
Integrating the last term by parts we get
Wx
W
− xt (gx )
|R1 | ≤ αωn x (1 − x)
dt .
(x − t)3
0
Now replacing the variable y in the last integral by x − √xu , we have


x
n
x
X _
1 _
 αωn x3 (1 − x)
|R1 | ≤ 2  (gx ) +
x
x
√
0
k=1
3
(gx )
+2
x2
0
x−
≤ 2αωn x (1 − x)
Z
y
k
n
X
x
_
k=1
x− √x
k
(gx ) .
Finally using the similar methods, we have
|R3 | ≤ 2αωn x (1 − x)
x
n
X
_
k=1
(gx ) .
x− √x
k
Combining the estimates of R1 , R2 , R3 , our theorem follows.
Remark 4.7. In particular,
for
α = 1, by Theorem 4.6 it may be remarked that the main theorem
5
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
R EFERENCES
[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.
[6] W. MEYER-KONIG AND K. ZELLER, Bernsteinsche Pootenzreihen, Studia Math., 19 (1960), 89–
94.
[7] X.M. ZENG, Bounds for Bernstein basis functions and Meyer-Konig and Zeller basis functions, J.
Math. Anal. Appl., 219 (1998), 364–376.
[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.
J. Inequal. Pure and Appl. Math., 3(4) Art. 57, 2002
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