# Onq-Bleimann, Butzer, and Hahn-Type Operators

Authors: Dilek Söylemez
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

Sequence ofq-Bleimann, Butzer, and Hahn operators which is based on a continuously differentiable function τ on R
+, with τ
0
=
0, inf
⁡
τ
'
x
≥
1, has been considered. Uniform approximation by such a sequence has been studied and degree of approximation by the operators has been obtained. Moreover, shape preserving properties of the sequence of operators have been investigated.


---

## Body

## 1. Introduction

In 1987, Lupaş [1] introduced the first q-analogue of Bernstein operators and investigated its approximation and shape preserving properties. Another q-generalization of the classical Bernstein polynomials is due to Phillips [2]. After that many generalizations of positive linear operators based on q-integers were introduced and studied by several authors. Some are in [3–13].Bleimann et al. [14] proposed a sequence of positive linear operators L
n defined by(1)
L
n
f
;
x
=
1
1
+
x
n
∑
k
=
0
n
f
k
n
-
k
+
1
n
k
x
k
,
x
≥
0
,
n
∈
Nfor f
∈
C
[
0
,
∞
), where C
[
0
,
∞
) denote the space of all continuous and real valued functions defined on [
0
,
∞
)
. Also the authors proved that L
n
f
;
x
→
f
x as n
→
∞ pointwise on 0
,
∞ for any f
∈
C
B
[
0
,
∞
), whereC
B
[
0
,
∞
) denote the space of all bounded functions from C
[
0
,
∞
). It is well known that (2)
f
C
B
=
sup
x
≥
0
⁡
f
xdefines a norm on C
B
[
0
,
∞
). Moreover, they showed that the convergence is uniform on each compact subset of 0
,
∞. In [15], using test functions t
/
(
1
+
t
)
v for v
=
0,1
,
2, Gadjiev and Çakar stated a Korovkin-type theorem for the uniform approximation of functions belonging to some suitable function spaces by some linear positive operators. As an application of this result, they proved uniform approximation by Bleimann, Butzer, and Hahn operators. Further results concerning such a sequence of operators and its generalizations may be found in [16–19].Now, we recall some notations fromq-analysis [20, 21].Theq-integer n and the q-factorial n
! are defined by(3)
n
≔
n
q
=
1
-
q
n
1
-
q
,
q
≠
1
n
,
q
=
1
,
for
n
∈
N
0
=
0, and(4)
n
!
≔
1
q
2
q
⋯
n
q
,
n
=
1,2
,
…
1
,
n
=
0
,
for
n
∈
N
,
0
!
=
1
,respectively, where q
>
0.For integers n
≥
r
≥
0 the q-binomial coefficient is defined as(5)
n
r
q
=
n
q
!
r
q
!
n
-
r
q
!
.Moreover, Euler identity is given by(6)
∏
s
=
0
n
-
1
1
+
q
s
x
=
∑
k
=
0
n
q
k
k
-
1
/
2
n
k
q
x
k
.Aral and Doğru [22] constructed the q-Bleimann, Butzer, and Hahn operators as(7)
L
n
,
q
f
;
x
=
1
l
n
x
∑
k
=
0
n
f
k
n
-
k
+
1
q
k
q
k
k
-
1
/
2
n
k
x
k
,where(8)
l
n
x
=
∏
s
=
0
n
-
1
1
+
q
s
xand f is defined on the semiaxis 0
,
∞
. The authors studied Korovkin-type approximation properties by using the test functions t
/
(
1
+
t
)
v for v
=
0,1
,
2. Moreover, they obtained rate of convergence of the operators and proved that rate of the q-Bleimann, Butzer, and Hahn operators is better than the classical one. A generalization of the q-Bleimann, Butzer, and Hahn operators was introduced by Agratini and Nowak in [23]. In this paper, the authors gave representation of the operators in terms of q-differences and investigated some approximation properties.A Voronovskaja-type result and monotonicity properties of these operators are investigated in [24].In [25], the authors introduced a new generalization of Bernstein polynomials denoted by B
n
τ and defined as(9)
B
n
τ
f
;
x
≔
B
n
f
∘
τ
-
1
;
τ
x
=
∑
k
=
0
n
n
k
τ
k
x
1
-
τ
x
n
-
k
f
∘
τ
-
1
k
n
,where B
nis the nth Bernstein polynomial, f
∈
C
0,1, x
∈
[
0,1
], and τ is a function that is continuously differentiable of infinite order on 0,1 such that τ
0
=
0, τ
1
=
1, and τ
′
x
>
0 for x
∈
0,1. Also, the authors studied some shape preserving and convergence properties concerning the generalized Bernstein operators B
n
τ
f
;
x.In [26], Aral et al. constructed sequences of Szasz-Mirakyan operators which are based on a function ρ. They studied weighted approximation properties and Voronovskaja-type results for these operators. They also showed that the sequence of the generalized Szász-Mirakyan operators is monotonically nonincreasing under the ρ-convexity of the original function. A similar generalization for Bleimann, Butzer, and Hahn operators is studied by Söylemez [27]. Also the class H
ω
τ was defined, a Korovkin-type theorem was given for the functions in this class, and uniform convergence of the generalized Bleimann, Butzer, and Hahn operators was obtained [27]. Moreover, the monotonicity properties of the operators were investigated.Now we recall the definition ofH
ω
τ that is a subspace of C
B
[
0
,
∞
) [27].Letω be a general modulus of continuity, satisfying the following properties:(a)
ωis continuous, nonnegative, and increasing function on 0
,
∞,
(b)
ω
δ
1
+
δ
2
≤
ω
δ
1
+
ω
δ
2,
(c)
l
i
m
δ
→
0
ω
δ
=
0
.The space of all real valued functionsf defined on 0
,
∞ satisfying(10)
f
x
-
f
y
≤
ω
τ
x
1
+
τ
x
-
τ
y
1
+
τ
yfor all x
,
y
∈
0
,
∞ is denoted by H
ω
τ
.It is clear from condition (b) that we have(11)
ω
n
δ
≤
n
ω
δ
,
n
∈
Nand one can get from the condition (a) that for any λ
>
0
(12)
ω
λ
δ
≤
ω
1
+
λ
λ
≤
1
+
λ
ω
δ
,where λ denotes the greatest integer that is not greater than λ.Now we define a new generalization ofq-Bleimann, Butzer, and Hahn operators for f
∈
C
[
0
,
∞
) by(13)
L
n
,
q
τ
f
;
x
=
1
l
n
τ
x
∑
k
=
0
n
f
∘
τ
-
1
k
n
-
k
+
1
q
k
q
k
k
-
1
/
2
n
k
τ
x
k
,where(14)
l
n
τ
x
=
∏
s
=
0
n
-
1
1
+
q
s
τ
x
,and τ is a continuously differentiable function on [
0
,
∞
) such that (15)
τ
0
=
0
,
inf
x
∈
0
,
∞
⁡
τ
′
x
≥
1
.An example of such a function τ is given in [26]. Note that, in the setting of the operators (13), we have (16)
L
n
,
q
τ
f
≔
L
n
,
q
f
∘
τ
-
1
∘
τ
,where the operators L
n
,
q are defined by (7). If τ
=
e
1, then L
n
,
q
τ
=
L
n
,
q
. Obviously, we have(17)
L
n
,
q
τ
1
;
x
=
1
,
L
n
,
q
τ
τ
1
+
τ
;
x
=
n
n
+
1
τ
x
1
+
τ
x
,
L
n
,
q
τ
τ
1
+
τ
2
;
x
=
n
n
-
1
n
+
1
2
q
2
τ
2
x
1
+
τ
x
1
+
q
τ
x
+
n
n
+
1
2
τ
x
1
+
τ
x
.In this study, we consider a generalization ofq-Bleimann, Butzer, and Hahn operators in the sense of [26], we investigate uniform convergence of L
n
,
q
τ
f
;
x
n
∈
N to f
x on 0
,
∞ for f
∈
H
ω
τ, and we obtain the degree of approximation. Moreover, we study shape preserving properties under τ-convexity of the function. Our results show that the new operators are sensitive to the rate of convergence to f, depending on the selection of τ. For the particular case τ
(
x
)
=
x, the previous results for q-Bleimann Butzer and Hahn operators are obtained.In order to ensure that the convergence properties holds, the author will assumeq
=
q
n is a sequence such that q
n
→
1 as n
→
∞ for 0
<
q
n
<
1, as in [22].Definition 1.
Letx
0
,
x
1
,
…
,
x
n be distinct points in the domain of f. Denote (18)
f
x
0
,
x
1
,
…
,
x
n
=
∑
r
=
0
n
f
x
r
∏
j
≠
r
n
x
r
-
x
j
,where r remains fixed and j takes all values from 0 to n, excluding r.Definition 2.
A continuous, real valued functionf is said to be convex in D
⊆
0
,
∞, if (19)
f
∑
i
=
1
m
α
i
x
i
≤
∑
i
=
1
m
α
i
f
x
ifor every x
1
,
x
2
,
…
,
x
m
∈
D and for every nonnegative number of α
1
,
α
2
,
…
,
α
m such that α
1
+
α
2
+
⋯
+
α
m
=
1.In [25] Cárdenas-Morales et al. introduced the following definition of τ-convexity of a continuous function.Definition 3.
A continuous, real valued functionf is said to be τ-convex in D, if f
∘
τ
-
1is convex in the sense of Definition 2.


## 2. Approximation Properties

In this section we deal with the promised approximation properties of the sequence ofq-Bleimann, Butzer, and Hahn operators. In [27], the following Korovkin-type theorem was given.Theorem 4.
LetT
n
τ
f
n
∈
N be a sequence of linear positive operators from H
ω
τ to C
B
0
,
∞
. If(20)
lim
n
→
∞
⁡
T
n
τ
τ
t
1
+
τ
t
v
;
x
-
τ
x
1
+
τ
x
v
C
B
=
0is satisfied for v
=
0,1
,
2, then for f
∈
H
ω
τ one has (21)
lim
n
→
∞
⁡
T
n
τ
f
-
f
C
B
=
0
.Now we are ready to give the following theorem.Theorem 5.
Suppose thatq
=
q
n, 0
<
q
n
<
1, and let q
n
→
1 as n
→
∞
. If L
n
,
q
τ is the operator defined by (13), then for any f
∈
H
ω
τ one has (22)
lim
n
→
∞
⁡
L
n
,
q
τ
f
-
f
C
B
=
0
.Proof.
According to Theorem4 we will show that (20) holds for L
n
,
q
τ. Obviously, from (17) we easily obtain that (23)
L
n
,
q
τ
1
;
x
=
1
.
L
n
,
q
τ
τ
t
1
+
τ
t
;
x
-
τ
x
1
+
τ
x
C
B
=
n
n
+
1
-
1
≤
1
q
n
-
1
q
n
n
+
1
-
1
,
L
n
,
q
τ
τ
t
1
+
τ
t
2
;
x
-
τ
x
1
+
τ
x
2
C
B
=
sup
x
≥
0
⁡
τ
x
1
+
τ
x
2
n
n
-
1
n
+
1
2
q
n
2
1
+
τ
x
1
+
q
n
τ
x
-
1
+
n
n
+
1
2
τ
x
1
+
τ
x
≤
1
q
n
2
1
-
q
n
2
-
2
n
+
1
+
1
n
+
1
2
.Therefore, the conditions (20) are satisfied. By Theorem 4, the proof is completed.Theorem 6.
Letq
=
q
n, 0
<
q
n
<
1, and let q
n
→
1 as n
→
∞
. Then one has(24)
L
n
,
q
τ
f
;
x
-
f
x
≤
2
ω
μ
n
τ
x
,for all f
∈
H
ω
τ. Here, (25)
μ
n
τ
x
=
τ
x
1
+
τ
x
2
1
-
2
n
n
+
1
+
n
n
-
1
n
+
1
2
q
n
2
1
+
τ
x
1
+
q
n
τ
x
+
n
n
+
1
2
τ
x
1
+
τ
x
.Proof.
Firstly, from (12) and (10) we can write(26)
f
t
-
f
x
≤
ω
τ
t
1
+
τ
t
-
τ
x
1
+
τ
x
≤
1
+
τ
t
/
1
+
τ
t
-
τ
x
/
1
+
τ
x
δ
ω
δand considering (17), we get(27)
L
n
,
q
τ
f
;
x
-
f
x
≤
L
n
,
q
τ
f
t
-
f
x
;
x
.
Equations (27) and (26) together imply that (28)
L
n
,
q
τ
f
;
x
-
f
x
≤
ω
δ
1
+
1
δ
L
n
,
q
τ
τ
t
1
+
τ
t
-
τ
x
1
+
τ
x
;
x
.Using the Cauchy-Schwarz inequality, we obtain (29)
L
n
,
q
τ
f
;
x
-
f
x
≤
ω
δ
1
+
1
δ
L
n
,
q
τ
τ
t
1
+
τ
t
-
τ
x
1
+
τ
x
2
;
x
1
/
2
.By choosing δ
=
μ
n
τ
x
=
L
n
,
q
τ
τ
t
/
(
1
+
τ
t
)
-
τ
x
/
(
1
+
τ
x
)
2
;
x
1
/
2, we get (30)
L
n
,
q
τ
f
;
x
-
f
x
≤
2
ω
μ
n
τ
x
,where (31)
sup
x
≥
0
⁡
μ
n
τ
x
≤
1
n
+
1
2which concludes the proof.


## 3. Shape Preserving Properties

Theorem 7.
Letf be a τ-convex function that is nonincreasing on 0
,
∞; then one has (32)
L
n
,
q
τ
f
;
x
≥
L
n
+
1
,
q
τ
f
;
xfor n
∈
N.Proof.
From (13), one can write(33)
L
n
,
q
τ
f
;
x
-
L
n
+
1
,
q
τ
f
;
x
=
1
∏
s
=
0
n
1
+
q
s
τ
x
∑
k
=
0
n
f
∘
τ
-
1
k
n
-
k
+
1
q
k
q
k
k
-
1
/
2
n
k
τ
x
k
1
+
q
n
τ
x
-
1
∏
s
=
0
n
1
+
q
s
τ
x
∑
k
=
0
n
+
1
f
∘
τ
-
1
k
n
-
k
+
2
q
k
q
k
k
-
1
/
2
n
+
1
k
τ
x
k
=
τ
x
n
+
1
l
n
+
1
τ
x
q
n
n
+
1
/
2
f
∘
τ
-
1
n
q
n
-
f
∘
τ
-
1
n
+
1
q
n
+
1
+
1
l
n
+
1
τ
x
∑
k
=
1
n
f
∘
τ
-
1
k
n
-
k
+
1
q
k
q
k
k
-
1
/
2
n
k
τ
x
k
+
1
l
n
+
1
τ
x
∑
k
=
0
n
-
1
f
∘
τ
-
1
k
n
-
k
+
1
q
k
q
k
k
-
1
/
2
q
n
n
k
τ
x
k
+
1
-
1
l
n
+
1
τ
x
∑
k
=
1
n
f
∘
τ
-
1
k
n
-
k
+
2
q
k
q
k
k
-
1
/
2
n
+
1
k
τ
x
k
=
τ
x
n
+
1
l
n
+
1
τ
x
q
n
n
+
1
/
2
f
∘
τ
-
1
n
q
n
-
f
∘
τ
-
1
n
+
1
q
n
+
1
+
1
l
n
+
1
τ
x
∑
k
=
0
n
-
1
f
∘
τ
-
1
k
+
1
n
-
k
q
k
+
1
q
k
k
-
1
/
2
q
k
n
k
+
1
τ
x
k
+
1
+
1
l
n
+
1
τ
x
∑
k
=
0
n
-
1
f
∘
τ
-
1
k
n
-
k
+
1
q
k
q
k
k
-
1
/
2
q
n
n
k
τ
x
k
+
1
-
1
l
n
+
1
τ
x
∑
k
=
0
n
-
1
f
∘
τ
-
1
k
+
1
n
-
k
+
1
q
k
+
1
q
k
k
-
1
/
2
q
k
n
+
1
k
+
1
τ
x
k
+
1
.Moreover, we have the following equalities that are proved in Lemma 3.1 of [24]: (34)
n
+
1
k
+
1
=
n
n
+
1
n
-
k
k
+
1
n
-
1
k
,
n
k
=
n
n
-
k
n
-
1
k
,
n
k
+
1
=
n
k
+
1
n
-
1
k
,which imply(35)
L
n
,
q
τ
f
;
x
-
L
n
+
1
,
q
τ
f
;
x
=
τ
x
n
+
1
l
n
+
1
τ
x
q
n
n
+
1
/
2
f
∘
τ
-
1
n
q
n
-
f
∘
τ
-
1
n
+
1
q
n
+
1
+
1
l
n
+
1
τ
x
∑
k
=
0
n
-
1
q
k
q
k
k
-
1
/
2
n
-
1
k
τ
x
k
+
1
f
∘
τ
-
1
k
+
1
n
-
k
q
k
+
1
n
k
+
1
+
f
∘
τ
-
1
k
n
-
k
+
1
q
k
q
n
-
k
n
n
-
k
-
f
∘
τ
-
1
k
+
1
n
-
k
+
1
q
k
+
1
n
n
+
1
n
-
k
k
+
1
=
1
l
n
+
1
τ
x
∑
k
=
0
n
-
1
q
k
q
k
k
-
1
/
2
n
n
+
1
n
-
k
k
+
1
n
-
1
k
τ
x
k
+
1
f
∘
τ
-
1
k
+
1
n
-
k
q
k
+
1
n
-
k
n
+
1
+
f
∘
τ
-
1
k
n
-
k
+
1
q
k
q
n
-
k
k
+
1
n
+
1
-
f
∘
τ
-
1
k
+
1
n
-
k
+
1
q
k
+
1
.From n
+
1
/
q
n
+
1
-
n
/
q
n
=
1
/
q
n
+
1
>
0 and by hypothesis, we have (36)
f
∘
τ
-
1
n
q
n
-
f
∘
τ
-
1
n
+
1
q
n
+
1
>
0
.Since f is τ-convex, by using Lemma  3.2 in [24] we obtain (37)
L
n
,
q
τ
f
;
x
-
L
n
+
1
,
q
τ
f
;
x
≥
0for x
∈
0
,
∞ and n
∈
N
. This proves the theorem.Theorem 8.
Suppose thatx
∈
0
,
∞
∖
k
/
n
-
k
+
1
q
k
:
k
=
0,1
,
…
,
n, and τ is linear. Then one has (38)
L
n
,
q
τ
f
;
x
-
f
x
q
=
-
τ
x
n
+
1
l
n
τ
x
τ
x
q
,
n
q
n
;
f
∘
τ
-
1
q
n
n
-
1
/
2
-
1
+
τ
x
l
n
τ
x
∑
k
=
0
n
-
1
τ
x
q
,
k
n
-
k
+
1
q
k
,
k
+
1
n
-
k
q
k
+
1
;
f
∘
τ
-
1
q
k
k
-
1
/
2
-
k
-
2
n
-
k
n
+
1
k
τ
x
k
.Proof.
From (17), we have (39)
L
n
,
q
τ
f
;
x
-
f
x
q
=
1
∏
s
=
0
n
-
1
1
+
q
s
τ
x
∑
k
=
0
n
f
∘
τ
-
1
k
n
-
k
+
1
q
k
-
f
x
q
q
k
k
-
1
/
2
n
k
τ
x
k
=
-
1
l
n
τ
x
∑
k
=
0
n
τ
x
q
-
k
n
-
k
+
1
q
k
τ
x
q
,
k
n
-
k
+
1
q
k
;
f
∘
τ
-
1
q
k
k
-
1
/
2
n
k
τ
x
k
.Using the equality (
k
/
n
-
k
+
1
)
n
k
=
n
k
-
1, we now get (40)
L
n
,
q
τ
f
;
x
-
f
x
q
=
-
1
l
n
τ
x
∑
k
=
0
n
τ
x
q
,
k
n
-
k
+
1
q
k
;
f
∘
τ
-
1
q
k
k
-
1
/
2
-
1
n
k
τ
x
k
+
1
+
1
l
n
τ
x
∑
k
=
1
n
τ
x
q
,
k
n
-
k
+
1
q
k
;
f
∘
τ
-
1
q
k
k
-
1
/
2
-
k
n
k
-
1
τ
x
k
=
-
τ
x
n
+
1
l
n
τ
x
τ
x
q
,
n
q
n
;
f
∘
τ
-
1
q
n
n
-
1
/
2
-
1
+
1
l
n
τ
x
∑
k
=
0
n
-
1
τ
x
q
,
k
+
1
n
-
k
q
k
+
1
;
f
∘
τ
-
1
-
τ
x
q
,
k
n
-
k
+
1
q
k
;
f
∘
τ
-
1
q
k
k
-
1
/
2
-
1
n
k
τ
x
k
+
1
.On the other hand we can write(41)
k
+
1
n
-
k
q
k
+
1
-
k
n
-
k
+
1
q
k
=
n
+
1
n
-
k
n
-
k
+
1
q
k
+
1
,which implies(42)
τ
x
q
,
k
+
1
n
-
k
q
k
+
1
;
f
∘
τ
-
1
-
τ
x
q
,
k
n
-
k
+
1
q
k
;
f
∘
τ
-
1
=
τ
x
q
,
k
n
-
k
+
1
q
k
,
k
+
1
n
-
k
q
k
+
1
;
f
∘
τ
-
1
n
+
1
n
-
k
n
-
k
+
1
q
k
+
1
.Now taking (42) in (40), we have (43)
L
n
,
q
τ
f
;
x
-
f
x
q
=
-
τ
x
n
+
1
l
n
τ
x
τ
x
q
,
n
q
n
;
f
∘
τ
-
1
q
n
n
-
1
/
2
-
1
+
τ
x
l
n
τ
x
∑
k
=
0
n
-
1
τ
x
q
,
k
n
-
k
+
1
q
k
,
k
+
1
n
-
k
q
k
+
1
;
f
∘
τ
-
1
n
+
1
n
-
k
n
-
k
+
1
q
k
+
1
q
k
k
-
1
/
2
-
1
n
k
τ
x
k
+
1
.Thus the proof is completed.Now, from Theorem8, we have the following corollary immediately.Corollary 9.
Letf be a τ-convex, nonincreasing function on 0
,
∞ and τ is linear. Then one has (44)
f
x
q
≤
L
n
,
q
τ
f
;
xfor any x
∈
0
,
∞
∖
k
/
n
-
k
+
1
q
k
:
k
=
0,1
,
…
,
n, n
∈
N
.



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*Source: 480925-2015-12-29.xml*