Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 816367, 15 pages
doi:10.1155/2008/816367
Research Article
q-Parametric Bleimann Butzer and Hahn Operators
N. I. Mahmudov and P. Sabancıgil
Eastern Mediterranean University, Gazimagusa, Turkish Republic of Northern Cyprus, Mersin 10, Turkey
Correspondence should be addressed to N. I. Mahmudov, nazim.mahmudov@emu.edu.tr
Received 4 June 2008; Accepted 20 August 2008
Recommended by Vijay Gupta
We introduce a new q-parametric generalization of Bleimann, Butzer, and Hahn operators in
∗
0, ∞. We study some properties of q-BBH operators and establish the rate of convergence
C1x
for q-BBH operators. We discuss Voronovskaja-type theorem and saturation of convergence for qBBH operators for arbitrary fixed 0 < q < 1. We give explicit formulas of Voronovskaja-type for the
q-BBH operators for 0 < q < 1. Also, we study convergence of the derivative of q-BBH operators.
Copyright q 2008 N. I. Mahmudov and P. Sabancıgil. 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.
1. Introduction
q-Bernstein polynomials
Bn,q fx :
n−k−1
n k
n k
f
1 − qs x
x
k
n
s0
k0
1.1
were introduced by Phillips in 1. q-Bernstein polynomials form an area of an intensive
research in the approximation theory, see survey paper 2 and references therein. Nowadays,
there are new studies on the q-parametric operators. Two parametric generalizations of qBernstein polynomials have been considered by Lewanowicz and Woźny cf. 3, an analog
of the Bernstein-Durrmeyer operator and Bernstein-Chlodowsky operator related to the qBernstein basis has been studied by Derriennic 4, Gupta 5 and Karsli and Gupta 6,
respectively, a q-version of the Szasz-Mirakjan operator has been investigated by Aral and
Gupta in 7. Also, some results on q-parametric Meyer-König and Zeller operators can be
found in 8–11.
In 12, Bleimann et al. introduced the following operators:
Hn fx n 1
k
n
f
xk ,
n
k
n−k1
1 x k0
x > 0, n ∈ N.
1.2
2
Journal of Inequalities and Applications
There are several studies related to approximation properties of Bleimann, Butzer, and
Hahn operators or, briefly, BBH, see, for example, 12–18. Recently, Aral and Doğru 19
introduced a q-analog of Bleimann, Butzer, and Hahn operators and they have established
some approximation properties of their q-Bleimann, Butzer, and Hahn operators in the
subspace of CB 0, ∞. Also, they showed that these operators are more flexible than classical
BBH operators, that is, depending on the selection of q, rate of convergence of the q-BBH
operators is better than the classical one. Voronovskaja-type asymptotic estimate and the
monotonicity properties for q-BBH operators are studied in 20.
In this paper, we propose a different q-analog of the Bleimann, Butzer, and Hahn
∗
0, ∞. We use the connection between classical BBH and Bernstein
operators in C1x
operators suggested in 16 to define new q-BBH operators as follows:
Hn,q fx : Φ−1 Bn1,q Φfx,
1.3
where Bn1,q is a q-Bernstein operator, Φ and Φ−1 will be defined later. Thanks to 1.3,
different properties of Bn1,q can be transferred to Hn,q with a little extra effort. Thus
the limiting behavior of Hn,q can be immediately derived from 1.3 and the well-known
properties of Bn1,q . It is natural that even in the classical case, when q 1, to define Hn in the
∗
0, ∞, the limit lf of fx/1 x as x→∞ has to appear in the definition of Hn .
space C1x
∗
0, ∞ the classical BBH operator has to be modified as follows:
Thus in C1x
n xn1
1
k
n
Hn fx f
, x > 0, n ∈ N.
1.4
xk lf
n
k
n−k1
1 x k0
1 xn
The paper is organized as follows. In Section 2, we give construction of q-BBH operators
and study some elementary properties. In Section 3, we investigate convergence properties
of q-BBH, Voronovskaja-type theorem and saturation of convergence for q-BBH operators for
arbitrary fixed 0 < q < 1, and also we study convergence of the derivative of q-BBH operators.
2. Construction and some properties of q-BBH operators
Before introducing the operators, we mention some basic definitions of q calculus.
Let q > 0. For any n ∈ N ∪ {0}, the q-integer n nq is defined by
n : 1 q · · · qn−1 ,
0 : 0;
2.1
and the q-factorial n! nq ! by
n! : 12 · · · n,
0! : 1.
2.2
For integers 0 ≤ k ≤ n, the q-binomial is defined by
n!
n
.
:
k
k!n − k!
2.3
Also, we use the following standard notations:
z; q0 : 1,
z; qn :
n−1
1 − qj z,
j0
n−k−1
n k
pn,k q; x :
x
1 − qs x,
k
s0
z; q∞ :
p∞k q; x :
xk
∞
1 − qj z,
j0
∞
1 − qk k! s0
2.4
1 − q x.
s
N. I. Mahmudov and P. Sabancıgil
3
It is agreed that an empty product denotes 1. It is clear that pnk q; x ≥ 0, p∞k q; x ≥ 0 ∀x ∈
0, 1 and
n
∞
pnk q; x p∞k q; x 1.
k0
2.5
k0
Introduce the following spaces.
Bρ 0, ∞ {f : 0, ∞→R | ∃Mf > 0 such that |fx| ≤ Mf ρx ∀x ∈ 0, ∞},
Cρ 0, ∞ {f ∈ Bρ 0, ∞ | f is continuous},
fx
Cρ∗ 0, ∞ f ∈ Cρ 0, ∞ | lim
lf exists and is finite ,
x→∞ ρx
fx
0 .
Cρ0 0, ∞ f ∈ Cρ 0, ∞ | lim
x→∞ ρx
2.6
It is clear that Cρ∗ 0, ∞ ⊂ Cρ 0, ∞ ⊂ Bρ 0, ∞. In each space, the norm is defined by
fρ sup
x≥0
|fx|
.
ρx
2.7
We introduce the following auxiliary operators. Firstly, let us denote
y
x
ψy , y ∈ 0, 1,
ψ −1 x , x ∈ 0, ∞.
1−y
1x
2.8
Secondly, let Φ : Cρ∗ 0, ∞→C0, 1 be defined by
⎧
fψy
⎪
⎪
,
if y ∈ 0, 1,
⎨
ρψy
Φfy :
fx
⎪
⎪
⎩lf lim
, if y 1.
x→∞ ρx
2.9
Then Φ is a positive linear isomorphism, with positive inverse Φ−1 : C0, 1→Cρ∗ 0, ∞ defined
by
Φ−1 gx ρxgψ −1 x,
g ∈ C0, 1, x ∈ 0, ∞.
2.10
For f ∈ C0, 1, t > 0, we define the modulus of continuity ωf; t as follows:
ωf; t : sup{|fx − fy| : |x − y| ≤ t, x, y ∈ 0, 1}.
2.11
We introduce new Bleimann-, Butzer-, and Hahn- BBH type operators based on q-integers
as follows.
Definition 2.1. For f ∈ Cρ∗ 0, ∞, the q-Bleimann, Butzer, and Hahn operators are given by
Hn,q fx : Φ−1 Bn1,q Φfx
n
fψk/n 1
n1
pn1,k q; ψ −1 x lf ρxψ −1 x ,
ρx
ρψk/n 1
k0
where
pn1,k q; ψ −1 x :
n−k
k
n1
ψ −1 x
1 − qs ψ −1 x,
k
s0
k 0, 1, . . . , n.
n ∈ N,
2.12
2.13
4
Journal of Inequalities and Applications
Note that for q 1, ρ 1 x and lf 0, we recover the classical Bleimann, Butzer, and
0, it is new Bleimann, Butzer, and Hahn operators
Hahn operators. If q 1, ρ 1 x but lf /
0
0, ∞ then
with additional term lf xn1 /1 xn . Thus if f ∈ C1x
n f
Hn,q fx :
k0
k
k
q n − k 1
n−k qx k n
s x
1−q
.
k
1x
1x
s1
2.14
To present an explicit form of the limit q-BBH operators, we consider
k ∞
ψ −1 x −1
p∞k q; ψ x :
1 − qk k! s0
1 − qs ψ −1 x.
2.15
Definition 2.2. Let 0 < q < 1. The linear operator defined on Cρ∗ 0, ∞ given by
H∞,q fx : ρx
∞
fψ1 − qk k
k0 ρψ1 − q p∞k q; ψ −1 x
2.16
is called the limit q-BBH operator.
Lemma 2.3. Hn,q , H∞,q : Cρ∗ 0, ∞→Cρ∗ 0, ∞ are linear positive operators and
Hn,q fρ ≤ fρ ,
H∞,q fρ ≤ fρ .
2.17
Proof. We prove the first inequality, since the second one can be done in a like manner. Thanks
to the definition, we have
|Hn,q fx| ≤ ρxfρ
n
n1
pn1,k q; ψ −1 x ρx|lf |ψ −1 x
k0
≤ ρxfρ
n
n1
pn1,k q; ψ −1 x ρxfρ ψ −1 x
2.18
k0
ρxfρ
n1
pn1,k q; ψ −1 x ρxfρ .
k0
Lemma 2.4. The following recurrence formula holds:
m j m − 1 x m−1
t
1
t
j
j
Hn,q ρt
n
H
x ρt
x.
q
n−1,q
j
1t
1t
n 1m−1 1 x j0
2.19
In particular, we have
Hn,q ρx ρx,
x
t
x ρx
,
Hn,q ρt
1t
1x
Hn,q 1x 1,
2 2
1
x
t
x
.
Hn,q ρt
ρx
x ρx
2
1t
1x
1 x n 1
2.20
N. I. Mahmudov and P. Sabancıgil
5
Proof. We prove only the recurrence formula, since the formulae 2.20 can easily be obtained
by standard computations. Since lf 1 for f ρtt/1 tm , we have
m t
Hn,q ρt
x
1t
m
n1
n x
k
ρx
pn1,k q; ψ −1 x ρx
n 1
1x
k0
ρx
ρx
m k n1
n−k n x
x
k
x
n1
ρx
1 − qs
k
n 1
1x
1x
1x
s0
k0
n
km−1
k0 n
1m−1
n
k−1
x
1x
k n−k 1 − qs
s0
x
1x
ρx
x
1x
n1
n m−1
m − 1 qj k − 1j
ρx
j
n 1m−1
k1 j0
×
k n1
n−k x
x
x
n
1 − qs
ρx
k−1
1x
1x
1x
s0
2.21
m − 1
x m−1
qj nj
m−1 1 x
j
n 1
j0
1
j n n1
x
t
x
ρx
× Hn−1,q ρt
x − ρx
1t
1x
1x
j m − 1
x m−1
t
j
j
n
H
ρt
x
q
n−1,q
j
1t
n 1m−1 1 x j0
1
x
ρx
1x
n1 1−
1
n 1m−1
m−1
j0
m−1 j j
q n
j
j m − 1
x m−1
t
j
j
q n Hn−1,q ρt
x.
j
1t
n 1m−1 1 x j0
1
Next theorem shows the monotonicity properties of q-BBH operators.
∗
Theorem 2.5. If f ∈ C1x
0, ∞ is convex and
n
n 1
−f
qn1 ≥ 0,
lf f
qn
qn1
2.22
then its q-BBH operators are nonincreasing, in the sense that
Hn,q fx ≥ Hn1,q fx,
n 1, 2, . . . , q ∈ 0, 1, x ∈ 0, ∞.
2.23
6
Journal of Inequalities and Applications
Proof. We begin by writing
Hn,q fx − Hn1,q fx
n−k n qx k x
k
n
f k
1 − qs
k
1x
1x
q n − k 1
s1
k0
n1
qx k n−k1
xn1
k
n1
s x
− f k
.
1−q
lf
k
1x
1x
q n − k 2
1 xn1
s1
k0
2.24
We now split the first of the above summations into two, writing
x
1x
k n−k s1
x
1−q
1x
s
ψk qn−k1 ψk1 ,
2.25
x
.
1x
2.26
where
ψk x
1x
k n−k1
1 − qs
s1
The resulting three summations may be combined to give
Hn,q fx − Hn1,q fx
n k
n k
f k
q ψk qn−k1 ψk1 k
q n − k 1
k0
n1
n1 k
x
n1 k
− f k
q ψk lf
k
1x
q n − k 2
k0
n
n1
k
k − 1
n k
n
f k
q ψk f k−1
qn1 ψk
k
k
−
1
q
n
−
k
1
q
n
−
k
2
k0
k1
n1
n1 k
x
n1 k
− f k
q ψk lf
k
1x
q n − k 2
k0
n1
n1
n
n
n 1
x
x
n1
n1
l
,
−
f
q
ak qk ψk f
f
k
qn
1x
1x
qn1
k1
2.27
where
ak qn−k1 k
k
k − 1
k
n − k 1
f k
f k−1
−f k
.
n 1
n 1
q n − k 1
q n − k 2
q n − k 2
2.28
By assumption, the sum of the last three terms of 2.27 is positive. Thus to show
monotonicity of Hn,q it suffices to show nonnegativity of ak , 0 ≤ k ≤ n. Let us write
α
n − k 1
,
n 1
x1 k
,
qk n − k 1
x2 k − 1
.
qk n − k 2
2.29
N. I. Mahmudov and P. Sabancıgil
7
Then it follows that
qn−k1 k
,
n 1
qn−k2 k − 1
k
αx1 1 − αx2 k
1
n − k 2
q n 1
1 − qn−k2 qn−k2 1 − qk−1 k
k
,
k
k
q n 1
1 − qn−k2
q n − k 2
1−α
2.30
and we see immediately that
ak αfx1 1 − αfx2 − fαx1 1 − αx2 ≥ 0,
2.31
and so Hn,q fx − Hn1,q fx ≥ 0.
Remark 2.6. It is easily seen that
n
n 1
lf f
−
f
qn1
qn
qn1
qn 1
n
n 1
1
Φf1 Φf
− Φf
.
n 2
n 2
n 2
n 1
n 2
2.32
The condition 2.22 follows from convexity of Φf. On the other hand, Φf is convex if f is
convex and nonincreasing, see 16.
3. Convergence properties
Theorem 3.1. Let q ∈ 0, 1, and let f ∈ Cρ∗ 0, ∞. Then
Hn,q f − H∞,q fρ ≤ CqωΦf, qn1 ,
3.1
where Cq 4/q1 − q ln1/1 − q 2.
Proof. For all x ∈ 0, ∞, by the definitions of Hn,q fx and H∞,q fx, we have that
n
fψk/n 1
pn1,k q; ψ −1 x
ρψk/n
1
k0
n1
∞
fψ1 − qk x
lf ρx
p q; ψ −1 x
− ρx
k ∞k
1x
ρψ1
−
q
k0
n1 k
Φf
ρx
− Φf1 − qk pn1,k q; ψ −1 x
n 1
k0
Hn,q f − H∞,q f ρx
ρx
− ρx
n1
Φf1 − qk − Φf1pn1,k q; ψ −1 x − p∞k q; ψ −1 x
k0
∞
Φf1 − qk − Φf1p∞k q; ψ −1 x
kn2
: I1 I2 I3 .
3.2
8
Journal of Inequalities and Applications
First, we estimate I1 , I 3 . By using the following inequalities:
0≤
1 − qk
qn1 1 − qk k
k
− 1 − qk −
1
−
q
≤ qn1 ,
n 1
1 − qn1
1 − qn1
0 ≤ 1 − 1 − qk qk ≤ qn1 ,
3.3
k ≥ n 2,
we get
|I1 | ≤ ρxωΦf, qn1 |I3 | ≤ ρx
∞
n1
pn1,k q; ψ −1 x ρxωΦf, qn1 ,
k0
3.4
−1
ωΦf, q p∞k q; ψ x ≤ ρxωΦf, q
k
n1
.
kn2
Next, we estimate I2 . Using the well-known property of modulus of continuity
ωg, λt ≤ 1 λωg, t,
λ > 0,
3.5
we get
|I2 | ≤ ρx
n1
ωΦf, qk |pn1,k q; ψ −1 x − p∞k q; ψ −1 x|
k0
≤ ρxωΦf, qn1 n1
1 qk−n−1 |pn1,k q; ψ −1 x − p∞k q; ψ −1 x|
3.6
k0
≤ 2ρxωΦf, qn1 : ρx
n1
1 qn1 k0
qk |pn1,k q; ψ −1 x − p∞k q; ψ −1 x|
2
ωΦf, qn1 Jn1 ψ −1 x,
qn1
where
Jn1 ψ −1 x n1
qk |pn1,k q; ψ −1 x − p∞k q; ψ −1 x|.
3.7
k0
Now, using the estimation 2.9 from 21, we have
Jn1 ψ −1 x ≤
n1
qn1
1 ln
pn1,k q; ψ −1 x p∞k q; ψ −1 x
q1 − q 1 − q k0
2qn1
1
ln
.
≤
q1 − q 1 − q
3.8
From 3.6 and 3.8, it follows that
|I2 | ≤ ρx
1
4
ln
ωΦf, qn1 .
q1 − q 1 − q
From 3.4, and 3.9, we obtain the desired estimation.
3.9
N. I. Mahmudov and P. Sabancıgil
9
∗
0, ∞. Then H∞,q fx fx ∀x ∈ 0, ∞
Theorem 3.2. Let 0 < q < 1 be fixed and let f ∈ C1x
if and only if f is linear.
Proof. By definition of H∞,q we have
H∞,q fx Φ−1 B∞,q Φfx.
3.10
Assume that H∞,q fx fx. Then B∞,q Φfx Φfx. From 22, we know that
B∞,q g g if and only if g is linear. So B∞,q Φfx Φfx if and only if Φfx 1 − xfx/1 − x Ax B. It follows that fx 1 xAx/1 x B A Bx B.
The converse can be shown in a similar way.
∗
Remark 3.3. Let 0 < q < 1 be fixed and let f ∈ C1x
0, ∞. Then the sequence {Hn,q fx}
does not approximate fx unless f is linear. It is completely in contrast to the classical case.
Theorem 3.4. Let q qn satisfies 0 < qn < 1 and let qn →1 as n→∞. For any x ∈ 0, ∞ and for any
f ∈ Cρ∗ 0, ∞, the following inequality holds:
1
|Hn,qn fx − fx| ≤ 2ω Φf, λn x ,
ρx
3.11
where λn x x/1 x2 1/n 1qn .
Proof. Positivity of Bn1,qn implies that for any g ∈ C0, 1
|Bn1,qn gx − gx| ≤ Bn1,qn |gt − gx|x.
3.12
On the other hand,
|Φft − Φfx| ≤ ωΦf, |t − x|
1
≤ ωΦf, δ 1 |t − x| ,
δ
δ > 0.
3.13
This inequality and 3.12 imply that
1
|Bn1,qn Φfx − Φfx| ≤ ωΦf, δ 1 Bn1,qn |t − x|x ,
δ
|Φ−1 Bn1,qn Φfx − Φ−1 Φfx|
1
≤ ωΦf, δ Φ−1 1 Φ−1 Bn1,qn |t − x|x
δ
1/2
1
2
−1
−1
≤ ρxωΦf, δ 1 Bn1,qn |t − ψ x| ψ x
δ
2
2 1/2 1
1
x
x
x
ρxωΦf, δ 1 −
δ
1x
1x
1 x2 n 1qn
1/2 1
1
x
,
ρxωΦf, δ 1 δ 1 x2 n 1qn
by choosing δ λn x, we obtain desired result.
3.14
10
Journal of Inequalities and Applications
Corollary 3.5. Let q qn satisfies 0 < qn < 1 and let qn →1 as n→∞. For any f ∈ Cρ∗ 0, ∞ it holds
that
lim Hn,qn fx − fxρ 0.
3.15
n→∞
Next, we study Voronovskaja-type formulas for the q-BBH operators. For the qBernstein operators, it is proved in 23 that for any f ∈ C1 0, 1,
lim
n→∞
n
Bn,q fx − B∞,q fx Lq f, x
qn
uniformly in x ∈ 0, 1, where
⎧∞
k
k−1 xk
⎪
⎨ k f 1 − qk − f1 − q − f1 − q x; q∞ ,
q; qk
1 − qk − 1 − qk−1 Lq f, x : k0
⎪
⎩
0,
3.16
0 ≤ x < 1,
x 1.
3.17
Similarly, we have the following Voronovskaja-type theorem for the q-BBH operators for fixed
q ∈ 0, 1. Before stating the theorem we introduce an analog of Lq f, x for q-BBH operators
∞
x
,q
Vq f, x : Φ−1 Lq Φfx k
1x
∞ k0
1 − qk 1
qk f1 − qk /qk − qk−1 f1 − qk−1 /qk−1 1 − qk
−
−
f
× f
qk
qk
qk
1 − qk − 1 − qk−1 1
xk
q, qk 1 xk−1
∞
k
k
k−1
/qk−1 1 − qk 1
x
k−1 f1 − q /q − f1 − q
;q
k f −
q
1x
qk
qk
qk−1 − qk
∞ k0
×
×
xk
1
.
q; qk 1 xk−1
Theorem 3.6. Let 0 < q < 1, f ∈
lim
n→∞
3.18
∗
C1x
0, ∞
∩ C1 0, ∞, and Φf is differentiable at x 1. Then
n 1
Hn,q fx − H∞,q fx Vq f, x,
qn1
3.19
∗
in C1x
0, ∞.
Proof. We estimate the difference
n 1
Δx : n1 Hn,q fx − H∞,q fx − Vq f, x
q
n 1
n1 Φ−1 Bn1,q Φfx − Φ−1 B∞,q Φfx − Φ−1 Lq Φfx
q
n 1
Φ−1
B
−
B
−
L
Φ
fx
n1,q
∞,q
q
n1
q
n 1
−1
.
1 x
Φfψ
B
−
B
−
L
x
n1,q
∞,q
q
n1
q
3.20
N. I. Mahmudov and P. Sabancıgil
11
Since Φf is well defined on whole 0, 1, from 23, Theorem 1, we get that
n 1
0.
B
−
B
−
L
Φfu
lim Δ1x ≤ lim sup n1,q
∞,q
q
n1
n→∞
n→∞ 0≤u≤1
q
3.21
Theorem is proved.
Remark 3.7. It is clear that Φf is differentiable in 0, 1 if f ∈ C1 0, ∞. If Φf is not
differentiable at x 1, then
lim
n→∞
n 1
Hn,q fx − H∞,q fx Vq f, x,
qn1
3.22
uniformly on any 0, A ⊂ 0, ∞.
Theorem 3.8. If f ∈ C2 0, ∞ and qn →1 as n→∞, then
lim n 1qn {Hn,qn fx − fx} n→∞
1 f x1 x2 x
2
3.23
uniformly on any 0, A ⊂ 0, ∞.
Proof. By definition of Hn,qn ,
Hn,qn fx − fx Φ−1 Bn1,qn Φfx − Φ−1 Φfx
Φ−1 Bn1,qn − IΦfx
3.24
−1
1 xBn1,qn − IΦfψ x,
and if L : 1/2f x1 − xx, then
1 f x1 x2 x Φ−1 LΦfx 1 xLΦfψ −1 x
2
1
1 xΦf ψ −1 xψ −1 x1 − ψ −1 x.
2
On the other hand, by 24, Corollary 5.2 we have that
1
lim sup n 1qn Bn1,qn − IΦfu − Φf uu1 − u 0.
n→∞ 0≤u≤1
2
3.25
3.26
Now, the result follows from the following inequality:
n 1 {Hn,q fx − fx} − 1 f x1 x2 x
qn
n
2
1
1 xn 1qn Bn1,qn − IΦfψ −1 x − 1 x Φf ψ −1 xψ −1 x1 − ψ −1 x
2
1
≤ 1 A sup n 1qn Bn1,qn − IΦfu − Φf uu1 − u.
2
0≤u≤A/1A
3.27
The theorem is proved.
12
Journal of Inequalities and Applications
From Theorem 3.6, we have the following saturation of convergence for the q-BBH
operators for fixed q ∈ 0, 1.
∗
0, ∞ ∩ C1 0, ∞. Then
Corollary 3.9. Let 0 < q < 1 and f ∈ C1x
Hn,q fx − H∞,q fx1x oqn1 3.28
if and only if Vq f, x ≡ 0, and this is equivalent to
f
1 − qk
qk
1
1
− k−1
k
q
q
1 − qk f
qk
1 − qk−1 −f
,
qk−1
k 1, 2, . . . .
3.29
∗
Theorem 3.10. Let 0 < q < 1 and f ∈ C1x
0, ∞ ∩ C1 0, ∞. If f is a convex function, then
n1
Hn,q fx − H∞,q fx1x oq if and only if f is a linear function.
Proof. If Hn,q f − H∞,q f1x oqn1 , then by Corollary 3.9
f
1 − qk
qk
1 − qk qk−1 − qk
1 − qk−1 f
−
f
,
q2k−1
qk
qk−1
k 1, 2, . . . .
3.30
Hence for k 1, 2, . . .
1−qk /qk
1 − qk
t
dt 0.
f
−
f
qk
1−qk−1 /qk−1
3.31
Since f is convex and f is continuous on 0, ∞, we get f t f 1 − qk /qk ∀t ∈ 1 −
qk−1 /qk−1 , 1 − qk /qk . Hence f t ≡ f 0, and therefore ft At B. Conversely, if f is
linear, then Hn,q fx − H∞,q fx1x 0.
One of the remarkable properties of the q-Bernstein approximation is that derivatives
of Bn f of any order converge to corresponding derivatives of f, see 25. Next theorem
shows the same property for Hnq for the first derivative.
∗
Theorem 3.11. Let f ∈ C1x
0, ∞ ∩ C1 0, ∞ and let {qn } be a sequence chosen so that the sequence
εn n
1 qn qn2
· · · qnn−1
−1
3.32
converges to zero from above faster than {1/3n }. Then
lim Hn,qn fx f x
3.33
x
Hn,qn fx 1 xBn1,qn Φf
.
1x
3.34
n→∞
uniformly on any 0, A ⊂ 0, ∞.
Proof. By definition
N. I. Mahmudov and P. Sabancıgil
13
Since Hn,qn fx is a composition of differentiable functions, it is differentiable at any x ∈
0, A and
d
d
x
Hn,qn fx 1 xBn1,qn Φf
dx
dx
1x
3.35
1 d
x
x
Bn1,qn Φf
Bn1,qn Φf
.
1x
1 x dx
1x
By 24, Theorem 4.1
2 x
x
x
Bn1,q Φf
≤ 2ω Φf, Bn1,q t − x
−
Φf
,
n
n
1x
1x 1x
1x
3.36
and by 25, Theorem 3
d
x
x
0.
Bn1,qn Φf
− Φf
lim sup n→∞
dx
1x
1x 3.37
0≤x≤A
Thus the desired limit follows from the following inequality:
d
Hn,q fx − d fx
n
dx
dx
d
x
d
Hn,qn fx −
1 xΦf
dx
dx
1x x
x
x
x
1 d Bn1,q Φf
≤ Bn1,qn Φf
− Φf
−
Φf
n
1x
1x
1 x dx
1x
1x 2 d
x
x
x
x
≤ 2ω Φf, Bn1,qn t −
Bn1,qn Φf
− Φf
1x
1x
dx
1x
1x d
x
1
x
x
Bn1,q Φf
2ω Φf,
−
Φf
n
dx
2 n 1
1x
1x 1 x
qn
d
A
x
x
.
− Φf
≤ 2ω Φf,
Bn1,qn Φf
n 1qn
dx
1x
1x 3.38
Remark 3.12. In 1, it is shown that
Bn1,q fx n1 n1
k0
where
k
Δk f0 xk ,
i
, Δ0 fi fi , Δk1 fi Δk fi1 − qk Δk fi ,
n 1
k
i k − j
j jj−1/2 k
k
.
Δ fi −1 q
f
j
n 1
j0
3.39
fi f
3.40
14
Journal of Inequalities and Applications
Immediately from the definition of Hn,q , we get an analog of 3.39 for Hn,q :
Hn,q fx Φ−1 Bn1,q Φfx
n1 n1 k
Φ−1
Δ Φf0 xk
k
k0
n1 xk
n1 k
.
Δ Φf0
k
1 xk−1
k0
3.41
Acknowledgment
The research is supported by the Research Advisory Board of Eastern Mediterranean
University under project BAP-A-08-04.
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