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Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 DOI 10.1186/s13660-015-0767-4
R E S E A R C H Open Access
Some approximation results onBernstein-Schurer operators defined by(p, q)-integersMohammad Mursaleen1,2*, Md Nasiruzzaman1 and Ashirbayev Nurgali3
*Correspondence:[email protected] of Mathematics,Aligarh Muslim University, Aligarh,202002, India2Operator Theory and ApplicationsResearch Group, Department ofMathematics, Faculty of Science,King Abdulaziz University, P.O. Box80203, Jeddah, 21589, Saudi ArabiaFull list of author information isavailable at the end of the article
AbstractRecently, Mursaleen et al. (On (p,q)-analogue of Bernstein operators, arXiv:1503.07404)introduced and studied the (p,q)-analog of Bernstein operators by using the idea of(p,q)-integers. In this paper, we generalize the q-Bernstein-Schurer operators using(p,q)-integers and obtain a Korovkin type approximation theorem. Furthermore, weobtain the convergence of the operators by using the modulus of continuity andprove some direct theorems.
MSC: 41A10; 41A25; 41A36
Keywords: q-integers; (p,q)-integers; Bernstein operator; (p,q)-Bernstein operator;q-Bernstein-Schurer operator; (p,q)-Bernstein-Schurer operator; modulus of continuity
1 Introduction and preliminariesIn , Bernstein [] introduced the following sequence of operators Bn : C[, ] → C[, ]defined for any n ∈N and f ∈ C[, ]:
Bn(f ; x) =n∑
k=
(nk
)xk( – x)n–kf
(kn
), x ∈ [, ]. (.)
By applying the idea of q-integers, the q-Bernstein operators were introduced by Lupaş[] and later by Philip []. Since then, many authors introduced q-generalization of variousoperators and investigated several approximation properties. For instance, the q-analog ofStancu-Beta operators in [] and []; the q-analog of Bernstein-Kantorovich operators in[]; the q-Baskakov-Kantorovich operators in []; the q-Szász-Mirakjan operators in [];the q-Bleimann-Butzer-Hahn operators in [] and in []; the q-analog of Baskakov andBaskakov-Kantorovich operators in []; the q-analog of Szász-Kantorovich operators in[]; and the q-analog of generalized Bernstein-Schurer operators in []. Besides this, wealso refer to some recent related work on this topic: e.g. [] and [].
First we give here some notations on the (p, q)-calculus.The (p, q)-integer was introduced in order to generalize or unify several forms of q-
oscillator algebras well known in the earlier physics literature related to the representation
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Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 2 of 12
theory of single parameter quantum algebras []. The (p, q)-integer [n]p,q is defined by
[n]p,q =pn – qn
p – q, n = , , , . . . , < q < p ≤ .
The (p, q)-binomial expansion is
(x + y)np,q := (x + y)(px + qy)
(px + qy
) · · · (pn–x + qn–y)
and the (p, q)-binomial coefficients are defined by
[nk
]
p,q
:=[n]p,q!
[k]p,q![n – k]p,q!.
Details on the (p, q)-calculus can be found in []. For p = , all the notions of the (p, q)-calculus are reduced to the q-calculus [].
In , Schurer [] introduced and studied the operators Sm,� : C[,� + ] → C[, ]defined by
Sm,�(f ; x) =m+�∑
k=
(m + �
k
)xk( – x)m+�–kf
(km
), x ∈ [, ], (.)
for any m ∈N and fixed � ∈N.The q-analog of the Bernstein-Schurer operators is defined as follows (cf. []):
Bm,�(f ; q; x) =m+�∑
k=
[m + �
k
]
q
xkm+�–k–∏
s=
( – qsx
)f(
[k]q
[m]q
), x ∈ [, ], (.)
for any m ∈N, f ∈ C[,� + ], and fixed �.Recently, Mursaleen et al. [] applied (p, q)-calculus in approximation theory and intro-
duced first (p, q)-analog of Bernstein operators. They have also introduced and studied theapproximation properties of the (p, q)-analog of the Bernstein-Stancu operators in [].
In this paper, we introduce the (p, q)-analog of these operators. We investigate someapproximation properties of these operators and obtain the rate of convergence by usingthe modulus of continuity. We also establish some direct theorems.
2 Construction of (p, q)-Bernstein-Schurer operatorsLet < q < p ≤ . We construct the class of generalized (p, q)-Bernstein-Schurer operatorsas follows:
Bp,qm,�(f ; x) =
m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)f(
[k]p,q
[m]p,q
), x ∈ [, ], (.)
for any m ∈N, f ∈ C[,� + ], and fixed �. Clearly, the operators defined by (.) are linearand positive. If we put p = in (.), then the (p, q)-Schurer operators are reduced to theq-Bernstein-Schurer operators.
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 3 of 12
Lemma . For Bp,qm,�(·; ·) given by (.), we have the following identities:
(i) Bp,qm,�(e; x) = ;
(ii) Bp,qm,�(e; x) = [m+�]p,q
[m]p,qx;
(iii) Bp,qm,�(e; x) = [m+�]p,q(px+(–x))m+�–
p,q[m]
p,qx + [m+�]p,q[m+�–]p,q
[m]p,q
qx;
(iv) Bp,qm,�(e; x) = [m+�]p,q
[m]p,q
(px + ( – x))m+�–p,q x + q(q + p) [m+�]p,q[m+�–]p,q
[m]p,q
(px +
( – x))m+�–p,q x + [m+�]p,q[m+�–]p,q[m+�–]p,q
[m]p,q
qx,
where ej(t) = tj, j = , , , . . . .
Proof(i) For < q < p ≤ we use the well-known identity from []
n∑
k=
[nk
]
p,q
xkn–k–∏
s=
(ps – qsx
)= , x ∈ [, ].
Suppose we choose n = m + �.Since
( – x)m+�–kp,q =
m+�–k–∏
s=
(ps – qsx
),
we get
m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)= .
Consequently, Bp,qm,�(e; x) = .
(ii) Clearly we have
Bp,qm,�(e; x) =
m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
) [k]p,q
[m]p,q
= xm+�–∑
k=
[m + �
k +
]
p,q
xkm+�–k–∏
s=
(ps – qsx
) [k + ]p,q
[m]p,q{as k → k + }
= x[m + �]p,q
[m]p,q
m+�–∑
k=
[m + � –
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)
= x[m + �]p,q
[m]p,q.
(iii)
Bp,qm,�(e; x) =
m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
) [k]p,q
[m]p,q
=[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 4 of 12
× xkm+�–k–∏
s=
(ps – qsx
)[k]p,q
{using [k]
p,q = [k]p,q.[k]p,q}
=[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
× xkm+�–k–∏
s=
(ps – qsx
)(pk– + q[k – ]p,q
),
by using [k]p,q = pk– + q[k – ]p,q.Therefore we have
Bp,qm,�(e; x) =
[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkpk–m+�–k–∏
s=
(ps – qsx
)
+[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkq[k – ]p,q
m+�–k–∏
s=
(ps – qsx
)
= x[m + �]p,q
[m]p,q
m+�–∑
k=
[m + � –
k
]
p,q
xkpkm+�–k–∏
s=
(ps – qsx
)
+ q[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xk[k – ]p,q
m+�–k–∏
s=
(ps – qsx
)
= x[m + �]p,q
[m]p,q
m+�–∑
k=
[m + � –
k
]
p,q
(px)k( – x)m+�–k–p,q
+ q[m + �]p,q[m + � – ]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)
=[m + �]p,q(px + ( – x))m+�–
p,q
[m]p,q
x
+ xq[m + �]p,q[m + � – ]p,q
[m]p,q
m+�–∑
k=
[m + � –
k –
]
p,q
× xkm+�–k–∏
s=
(ps – qsx
).
Hence the desired result is proved.(iv)
Bp,qm,�(e; x) =
m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
) [k]p,q
[m]p,q
=[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)[k]
p,q
=[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkp(k–)m+�–k–∏
s=
(ps – qsx
)
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 5 of 12
+[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkq[k – ]p,q
m+�–k–∏
s=
(ps – qsx
)
+ q[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkpk–[k – ]p,q
m+�–k–∏
s=
(ps – qsx
).
A small calculation shows that
[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkp(k–)m+�–k–∏
s=
(ps – qsx
)
= x[m + �]p,q
[m]p,q
(px + ( – x)
)m+�–p,q ,
[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkq[k – ]p,q
m+�–k–∏
s=
(ps – qsx
)
= xq [m + �]p,q[m + � – ]p,q
[m]p,q
(px + ( – x)
)m+�–p,q
+ xq [m + �]p,q[m + � – ]p,q[m + � – ]p,q
[m]p,q
.
Also
q[m + �]p,q
[m]p,q
m+�∑
k=
[m + � –
k –
]
p,q
xkpk–[k – ]p,q
m+�–k–∏
s=
(ps – qsx
)
= pqx [m + �]p,q[m + � – ]p,q
[m]p,q
(px + ( – x)
)m+�–p,q .
This completes the proof. �
Lemma . Let Bp,qm,�(·; ·) be given by (.). Then, for any x ∈ [, ] and < q < p ≤ , we
have the following identities:(i) Bp,q
m,�(e – ; x) = [m+�]p,q[m]p,q
x – ;
(ii) Bp,qm,�(e – x; x) = ( [m+�]p,q
[m]p,q– )x;
(iii)
Bp,qm,�
((e – x); x
)
=[m + �]p,q
[m]p,q
(px + ( – x)
)m+�–p,q x
+((
[m + �]p,q
[m]p,q–
)
+[m + �]p,q
[m]p,q
(q[m + � – ]p,q – [m + �]p,q
))x.
3 On the convergence of (p, q)-Bernstein-Schurer operatorsLet f ∈ C[,γ ]. The modulus of continuity of f , denoted by ω(f , δ), gives the maximumoscillation of f in any interval of length not exceeding δ > and it is given by the relation
ω(f , δ) = sup|y–x|≤δ
∣∣f (y) – f (x)∣∣, x, y ∈ [,γ ].
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 6 of 12
It is well known that limδ→+ ω(f , δ) = for f ∈ C[,γ ] and for any δ > one has
∣∣f (y) – f (x)∣∣ ≤
( |y – x|δ
+ )
ω(f , δ). (.)
For q ∈ (, ) and p ∈ (q, ] obviously we have limm→∞[m]p,q = p–q . In order to obtain the
convergence results of the operator Bp,qm,�, we take a sequence qm ∈ (, ) and pm ∈ (qm, ]
such that limm→∞ pm = and limm→∞ qm = , so we get limm→∞[m]pm ,qm = ∞.
Theorem . Let p = pm, q = qm satisfying < qm < pm ≤ such that limm→∞ pm = ,limm→∞ qm = . Then for each f ∈ C[,� + ],
limm→∞ Bpm ,qm
m,� (f ; x) = f , (.)
is uniformly on [, ].
Proof The proof is based on the well-known Korovkin theorem regarding the convergenceof a sequence of linear and positive operators, so it is enough to prove the conditions
Bpm ,qmm,� (ej; x) = xj, j = , , , {as m → ∞}
uniformly on [, ].Clearly we have
limm→∞ Bpm ,qm
m,� (e; x) = .
By making a simple calculation we get
limm→∞
[m + �]pm ,qm
[m]pm ,qm= , as < qm < pm ≤ .
Since < qm < pm ≤ , we get
limm→∞
[m + �]pm ,qm
[m]pm ,qm
= .
Hence we have
limm→∞ Bpm ,qm
m,� (e; x) = x,
limm→∞ Bpm ,qm
m,� (e; x) = x. �
Theorem . If f ∈ C[,� + ], then
∣∣Bp,qm,�(f ; x) – f (x)
∣∣ ≤ ωf (δm),
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 7 of 12
where
δm = x∣∣∣∣[m + �]p,q
[m]p,q–
∣∣∣∣
+
√[m + �]p,q
[m]p,q·√
x(q[m + � – ]p,q – [m + �]p,q) + x(px + ( – x))m+�–p,q
[m]p,q.
Proof
∣∣Bp,qm,�(f ; x) – f (x)
∣∣ ≤m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)∣∣∣∣f(
[k]p,q
[m]p,q
)– f (x)
∣∣∣∣
≤m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)( | [k]p,q[m]p,q
– x|δ
+ )
ω(f , δ).
By using the Cauchy inequality and Lemma ., we have
∣∣Bp,qm,�(f ; x) – f (x)
∣∣
≤(
δ
{m+�∑
k=
[m + �
k
]
p,q
xk(
[k]p,q
[m]p,q– x
) m+�–k–∏
s=
(ps – qsx
)}
+
)ω(f , δ)
={
δ
(Bp,q
m,�(e; x) – xBp,qm,�(e; x) + xBp,q
m,�(e; x))
+ }ω(f , δ)
={
δ
( [m + �]p,q(px + ( – x))m+�–p,q
[m]p,q
x
+ x(
[m + �]p,q[m + � – ]p,q
[m]p,q
q – [m + �]p,q
[m]p,q+
))
+ }ω(f , δ)
={
δ
((x(
[m + �]p,q
[m]p,q–
))
+(√
[m + �]p,q
[m]p,q
·√
x(q[m + � – ]p,q – [m + �]p,q) + x(px + ( – x))m+�–p,q
[m]p,q
))
+ }ω(f , δ)
≤{
δ
(x∣∣∣∣[m + �]p,q
[m]p,q–
∣∣∣∣
+
√[m + �]p,q
[m]p,q
·√
x(q[m + � – ]p,q – [m + �]p,q) + x(px + ( – x))m+�–p,q
[m]p,q
)+
}ω(f , δ),
by using (a + b) ≤ (|a| + |b|).
Hence we obtain the desired result by choosing δ = δm. �
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 8 of 12
4 Direct theorems on (p, q)-Bernstein-Schurer operatorsThe Peetre K-functional is defined by
K(f , δ) = inf{(‖f – g‖ + δ
∥∥g ′′∥∥): g ∈W},
where
W ={
g ∈ C[,� + ] : g ′, g ′′ ∈ C[,� + ]}
.
Then there exists a positive constant C > such that K(f , δ) ≤ Cω(f , δ ), δ > , where the
second order modulus of continuity is given by
ω(f , δ
)
= sup<h<δ
supx∈[,�+]
∣∣f (x + h) – f (x + h) + f (x)∣∣.
Theorem . Let f ∈ C[,� + ], g ′ ∈ C[,� + ] and satisfying < q < p ≤ . Then for alln ∈N there exists a constant C > such that
∣∣∣∣Bp,qm,�(f ; x) – f (x) – xg ′(x)
([m + � – ]p,q
[m]p,q–
)∣∣∣∣ ≤ Cω(f , δm(x)
),
where
δm(x) =
[m + � – ]p,q
[m]p,q
(px + ( – x)
)m+�–p,q x
+((
[m + �]p,q
[m]p,q–
)
+[m + �]p,q
[m]p,q
(q[m + � – ]p,q – [m + �]p,q
))x.
Proof Let g ∈W. Then from the Taylor expansion, we get
g(t) = g(x) + g ′(x)(t – x) +∫ t
x(t – u)g ′′(u) du, t ∈ [,A],A > .
Now by Lemma ., we have
Bp,qm,�(g; x) = g(x) + xg ′(x)
([m + �]p,q
[m]p,q–
)+ Bp,q
m,�
(∫ t
x(e – u)g ′′(u) du; p, q; x
),
∣∣∣∣Bp,qm,�(g; x) – g(x) – xg ′(x)
([m + �]p,q
[m]p,q–
)∣∣∣∣ ≤ Bp,qm,�
(∣∣∣∣∫ t
x
∣∣(e – u)∣∣∣∣g ′′(u)
∣∣du; p, q; x∣∣∣∣
)
≤ Bp,qm,�
((e – x); p, q; x
)∥∥g ′′∥∥.
Hence we get
∣∣∣∣Bp,qm,�(g; x) – g(x) – xg ′(x)
([m + �]p,q
[m]p,q–
)∣∣∣∣
≤ ∥∥g ′′∥∥(
[m + �]p,q
[m]p,q
(px + ( – x)
)m+�–p,q x
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 9 of 12
+((
[m + �]p,q
[m]p,q–
)
+[m + �]p,q
[m]p,q
(q[m + � – ]p,q – [m + �]p,q
))x
).
On the other hand we have∣∣∣∣B
p,qm,�(f ; x) – f (x) – xg ′(x)
([m + � – ]p,q
[m]p,q–
)∣∣∣∣
≤ ∣∣Bp,qm,�
((f – g); x
)– (f – g)(x)
∣∣
+∣∣∣∣B
p,qm,�(g; x) – g(x) – xg ′(x)
([m + � – ]p,q
[m]p,q–
)∣∣∣∣.
Since
∣∣Bp,qm,�(f ; x)
∣∣ ≤ ‖f ‖,
we have∣∣∣∣B
p,qm,�(f ; x) – f (x) – xg ′(x)
([m + � – ]p,q
[m]p,q–
)∣∣∣∣
≤ ‖f – g‖
+∥∥g ′′∥∥
([m + �]p,q
[m]p,q
(px + ( – x)
)m+�–p,q x
+((
[m + �]p,q
[m]p,q–
)
+[m + �]p,q
[m]p,q
(q[m + � – ]p,q – [m + �]p,q
))x
).
Now taking the infimum on the right hand side over all g ∈W, we get
∣∣∣∣Bp,qm,�(f ; x) – f (x) – xg ′(x)
([m + � – ]p,q
[m]p,q–
)∣∣∣∣ ≤ CK(f , δ
m(x)).
In view of the property of the K-functional, we get
∣∣∣∣Bp,qm,�(f ; x) – f (x) – xg ′(x)
([m + � – ]p,q
[m]p,q–
)∣∣∣∣ ≤ Cω(f , δm(x)
).
This completes the proof. �
Theorem . Let f ∈ C[,�+] be such that f ′, f ′′ ∈ C[,�+], and the sequence {pm}, {qm}satisfying < qm < pm ≤ such that pm → , qm → and pm
m → α, qmm → β as m → ∞,
where ≤ α,β < . Then
limm→∞[m]pm ,qm
(Bpm ,qm
m,� (f ; x) – f (x))
=x(λ – αx)
f ′′(x)
is uniform on [,� + ], where < λ ≤ .
Proof From the Taylor formula, we have
f (t) = f (x) + f ′(x)(t – x) +
f ′′(x)(t – x) + r(t, x)(e – x),
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 10 of 12
where r(t, x) is the remainder term and limt→x r(t, x) = , therefore we have
[m]pm ,qm
(Bpm ,qm
m,� (f ; x) – f (x))
= [m]pm ,qm
(f ′(x)Bpm ,qm
m,�((e – x); x
)+
f ′′(x)
Bpm ,qmm,�
((e – x); x
)
+ Bpm ,qmm,�
(r(t, x)(t – x); x
)).
Now by applying the Cauchy-Schwartz inequality, we have
Bpm ,qmm,�
(r(t, x)(t – x); x
) ≤√
Bpm ,qmm,�
(r(t, x); x
) ·√
Bpm ,qmm,�
((t – x); x
).
Since r(x, x) = , and r(t, x) ∈ C[,� + ], from Theorem ., we have
Bpm ,qmm,�
(r(t, x); x
)= r(x, x) = ,
which implies that
Bpm ,qmm,�
(r(t, x)(t – x); x
)= ,
limm→∞[m]pm ,qm
(Bpm ,qm
m,�((e – x); x
))= x lim
m→∞[m]pm ,qm
([m + �]pm ,qm
[m]pm ,qm–
)= ,
limm→∞[m]pm ,qm
(Bpm ,qm
m,�((e – x); x
))
= x limm→∞[m]pm ,qm
[m + �]pm ,qm
[m]pm ,qm
(pmx + ( – x)
)m+�–pm ,qm
+ x limm→∞[m]pm ,qm
(([m + �]pm ,qm
[m]pm ,qm–
)
+[m + �]pm ,qm
[m]pm ,qm
(qm[m + � – ]pm ,qm – [m + �]pm ,qm
)),
limm→∞[m]pm ,qm
(Bpm ,qm
m,�((e – x); x
))= λx – αx = x(λ – αx),
where λ ∈ (, ] depends on the sequence {pm}.Hence we have
limm→∞[m]pm ,qm
(Bpm ,qm
m,� (f ; x) – f (x))
=x(λ – αx)
f ′′(x).
This completes the proof. �
Now we give the rate of convergence of the operators Bp,qm,�(f ; x) in terms of the elements
of the usual Lipschitz class LipM(ν).Let f ∈ C[, m + �], M > and < ν ≤ . We recall that f belongs to the class LipM(ν) if
the inequality
∣∣f (t) – f (x)∣∣ ≤ M|t – x|ν (
t, x ∈ (, ])
is satisfied.
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 11 of 12
Theorem . Let < q < p ≤ . Then for each f ∈ LipM(ν) we have
∣∣Bp,qm,�(f ; x) – f (x)
∣∣ ≤ Mδνm(x),
where
δm(x) =
[m + �]p,q
[m]p,q
(px + ( – x)
)m+�–p,q x
+((
[m + �]p,q
[m]p,q–
)
+[m + �]p,q
[m]p,q
(q[m + � – ]p,q – [m + �]p,q
))x.
Proof By the monotonicity of the operators Bp,qm,�(f ; x), we can write
∣∣Bp,qm,�(f ; x) – f (x)
∣∣ ≤ Bp,qm,�
(∣∣f (t) – f (x)∣∣; p, q; x
)
≤m+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)∣∣∣∣f(
[k]p,q
[m]p,q
)– f (x)
∣∣∣∣
≤ Mm+�∑
k=
[m + �
k
]
p,q
xkm+�–k–∏
s=
(ps – qsx
)∣∣∣∣[k]p,q
[m]p,q– x
∣∣∣∣ν
= Mm+�∑
k=
(Pm,�,k(x)
([k]p,q
[m]p,q– x
)) νP
–ν
m,�,k(x),
where Pm,�,k(x) =[ m+�
k
]p,qxk ∏m+�–k–
s= (ps – qsx).Now applying the Hölder inequality, we have
∣∣Bp,qm,�(f ; x) – f (x)
∣∣ ≤ M
(m+�∑
k=
Pm,�,k(x)(
[k]p,q
[m]p,q– x
)) ν
(m+�∑
k=
Pm,�,k(x)
) –ν
= M(Bp,q
m,�((e – x); x
)) ν .
Choosing δ : δm(x) =√
Bp,qm,�((e – x); x), we obtain
∣∣Bp,qm,�(f ; x) – f (x)
∣∣ ≤ Mδνm(x).
Hence, the desired result is obtained. �
5 ConclusionBy using the notion of (p, q)-integers we introduced (p, q)-Bernstein-Schurer operatorsand investigated some approximation properties of these operators. We obtained the rateof convergence by using the modulus of continuity and also established some direct theo-rems. These results generalize the approximation results proved for q-Bernstein-Schureroperators, which are directly obtained by our results for p = .
Competing interestsThe authors declare that they have no competing interests.
Mursaleen et al. Journal of Inequalities and Applications (2015) 2015:249 Page 12 of 12
Authors’ contributionsThe authors contributed equally and significantly in writing this paper. All authors read and approved the finalmanuscript.
Author details1Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India. 2Operator Theory and ApplicationsResearch Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589,Saudi Arabia. 3Science-Pedagogical Faculty, M. Auezov South Kazakhstan State University, Shymkent, 160012, Kazakhstan.
AcknowledgementsThe second author (MN) acknowledges the financial support of UGC BSR Fellowship, and the third author (AN) gratefullyacknowledges the financial support from M. Auezov South Kazakhstan State University, Shymkent.
Received: 27 April 2015 Accepted: 23 July 2015
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