# A Kantorovich-Stancu Type Generalization of Szasz Operators including Brenke Type Polynomials

Authors: Rabia Aktaş; Bayram Çekim; Fatma Taşdelen
Journal: Journal of Function Spaces and Applications
 (2013)

---

## Abstract

We introduce a Kantorovich-Stancu type modification of a generalization of Szasz operators defined by means of the Brenke type polynomials and obtain approximation properties of these operators. Also, we give a Voronovskaya type theorem for Kantorovich-Stancu type operators including Gould-Hopper polynomials.


---

## Body

## 1. Introduction

For each positiven and f∈CB([0,∞)) or C([0,∞))∩E, the Szasz-Mirakyan operators defined by
(1)Sn(f;x)∶=e-nx∑k=0∞‍(nx)kk!f(kn)
have an important role in the approximation theory [1]. Their Korovkin type approximation properties and rates of convergence have been investigated by many researchers. Recently, there is a growing interest in defining linear positive operators via special functions (see [2–13]). In particular, many authors have studied various generalizations of Szasz operators via special functions. In [14], Jakimovski and Leviatan constructed a generalization of Szasz operators by means of the Appell polynomials. Then, Ismail [15] presented another generalization of Szasz operators by means of Sheffer polynomials, which involves the operators (1) defined by Jakimovski and Leviatan in [14]. In [11],Varma et al.considered the following generalization of Szasz operators by means of the Brenke type polynomials, which are motivated by the operators defined by Jakimovski and Leviatanand Ismail, for x≥0 and n∈ℕ:
(2)Ln(f;x)∶=1A(1)B(nx)∑k=0∞pk(nx)f(kn)
under the following assumptions:
(3)(i)A(1)≠0,ak-rbrA(1)≥0,0≤r≤k,k=0,1,2,…,(ii)B:[0,∞)⟶(0,∞),(iii)(4)and(5)convergefor|t|<R(R>1),
where
(4)A(t)=∑r=0∞‍artr,a0≠0,B(t)=∑r=0∞‍brtr,br≠0(r≥0)
are analytic functions and the Brenke type polynomials [16] have generating functions of the form
(5)A(t)B(xt)=∑k=0∞‍pk(x)tk,
where
(6)pk(x)=∑r=0k‍ak-rbrxr,k=0,1,2,….The Kantorovich type of Szasz-Mirakyan operators is defined by [17]
(7)Kn(f;x)∶=ne-nx∑k=0∞‍(nx)kk!∫k/n(k+1)/nf(t)dt.
The approximation properties of the Szasz-Mirakyan-Kantorovich operators and their various iterates were studied by many authors in [12, 18–23].Recently, in [8], the Kantorovich type of the operators given by (2) under the assumptions (3) has been defined as
(8)Kn(f;x)∶=nA(1)B(nx)∑k=0∞‍pk(nx)∫k/n(k+1)/nf(t)dt,
where n∈ℕ, x≥0 and f∈C[0,∞), and some of its properties have been investigated.The purpose of this study is to introduce a Kantorovich-Stancu type modification of the operators given by (8) and to examine the approximation properties of these operators. We also present a Kantorovich-Stancu type of the operators including Gould-Hopper polynomials and then we prove a Voronovskaya type theorem for these operators including Gould-Hopper polynomials.


## 2. Construction of the Operators

For each positive integern, x≥0 and f∈CB([0,∞)), or C([0,∞))∩E, let us consider the following operators:
(9)Kn(α,β)(f;x)∶=n+βA(1)B(nx)∑k=0∞‍pk(nx)∫(k+α)/(n+β)(k+α+1)/(n+β)f(t)dt,
where α and β parameters satisfy the condition 0≤α≤β. For the approximation properties of Stancu type operators, we refer to [24–27].It is clear that forα=β=0,Kn(α,β)(f;x) reduces to the operators defined by (8).In the case ofB(t)=et and A(t)=1,with the help of (5) it follows that pk(x)=xk/k!. So the operator Kn(α,β)(f;x) gives the Kantorovich-Stancu type of Szasz-Mirakyan operators as follows:
(10)Kn(α,β)(f;x)∶=(n+β)e-nx∑k=0∞‍(nx)kk!∫(k+α)/(n+β)(k+α+1)/(n+β)f(t)dt,
where α and β parameters satisfy the condition 0≤α≤β.In the case ofα=β=0, the operator (10) turns out to be the Szasz-Mirakyan-Kantorovich operators given by (7).ForB(t)=et,Kn(α,β)(f;x) gives the Kantorovich-Stancu type of the operators Pn(f;x) proposed by Jakimovski and Leviatan in  [14].Now, for the operatorsKn(α,β) given by (9), we give some results which are necessary to prove the main theorem.Lemma 1.
Kantorovich-Stancu type operators, defined by (9), are linear and positive.Lemma 2.
For eachx∈[0,∞), the Kantorovich-Stancu type operators (9) have the following properties:
(11)Kn(α,β)(1;x)=1,(12)Kn(α,β)(s;x)=nn+βB′(nx)B(nx)x+A′(1)(n+β)A(1)+2α+12(n+β),(13)Kn(α,β)(s2;x)=(nn+β)2B′′(nx)B(nx)x2+nB′(nx)[2A′(1)+(2α+2)A(1)](n+β)2A(1)B(nx)x+1(n+β)2A(1){13A′′(1)+(2α+2)A′(1)HHHHHHHHHH+(α2+α+13)A(1)}.Proof.
From the generating function of the Brenke type polynomials given by (5), a few calculations reveal that
(14)∑k=0∞‍pk(nx)=A(1)B(nx),∑k=0∞‍kpk(nx)=A′(1)B(nx)+nxA(1)B′(nx),∑k=0∞‍k2pk(nx)=n2x2A(1)B′′(nx)+nxB′(nx){2A′(1)+A(1)}+B(nx){A′′(1)+A′(1)}.
By using these equalities, we obtain the assertions of the lemma by simple calculation.Lemma 3.
For eachx∈[0,∞), one has
(15)Kn(α,β)((s-x)2;x)={(nn+β)2B′′(nx)B(nx)-2nB′(nx)(n+β)B(nx)+1}x2+{nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)+(2α+1)nB′(nx)(n+β)2B(nx)hhh-2A′(1)(n+β)A(1)-2α+1n+βnB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)}x+A′′(1)+A′(1)(n+β)2A(1)+(2α+1)A′(1)(n+β)2A(1)+α2+α+(1/3)(n+β)2.Theorem 4.
Let(16)E∶={f:x∈[0,∞),f(x)1+x2isconvergentasx⟶∞},limy→∞B′(y)B(y)=1,limy→∞B′′(y)B(y)=1.
If f∈C[0,∞)∩E, then
(17)limn→∞Kn(α,β)(f;x)=f(x),
and the operators Kn(α,β) converge uniformly in each compact subset of [0,∞).Proof.
According to Lemma2, by considering the equality (16), we get
(18)limn→∞Kn(α,β)(si;x)=xi,i=0,1,2.
This convergence is satisfied uniformly in each compact subset of [0,∞). Then, the proof follows from the universal Korovkin-type property (vi) of Theorem  4.1.4 in [28].


## 3. Rates of Convergence

In this section, we compute the rates of convergence of the operatorsKn(α,β)(f) to f  by means of a classical approach, the second modulus of continuity, and Peetre’s K-functional.Letf∈C~[0,∞). Then for δ>0, the modulus of continuity of f denoted by w(f;δ) is defined to be
(19)w(f;δ)∶=supx,y∈[0,∞)|x-y|≤δ|f(x)-f(y)|,
where C~[0,∞)denotes the space of uniformly continuous functions on [0,∞).Then, for any δ>0 and each x∈[0,∞), it is well known that one can write
(20)|f(x)-f(y)|≤w(f;δ)(|x-y|δ+1).The next result gives the rate of convergence of the sequenceKn(α,β)(f) to f by means of the modulus of continuity.Theorem 5.
Forf∈C~[0,∞)∩E, one has
(21)|Kn(α,β)(f;x)-f(x)|≤2w(f;λn(x)),
where
(22)λ=λn(x)=Kn(α,β)((s-x)2;x)={(nn+β)2B′′(nx)B(nx)-2nB′(nx)(n+β)B(nx)+1}x2+{nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)+(2α+1)nB′(nx)(n+β)2B(nx)hhhhhhhh-2A′(1)(n+β)A(1)-2α+1n+β}x+A′′(1)+A′(1)(n+β)2A(1)+(2α+1)A′(1)(n+β)2A(1)+α2+α+(1/3)(n+β)2.Proof.
Using linearity of the operatorsKn(α,β), (11) and (20), we get
(23)|Kn(α,β)(f;x)-f(x)|≤n+βA(1)B(nx)∑k=0∞‍Pk(nx)×∫(k+α)/(n+β)(k+α+1)/(n+β)|f(s)-f(x)|ds≤n+βA(1)B(nx)∑k=0∞‍Pk(nx)×∫(k+α)/(n+β)(k+α+1)/(n+β)(|s-x|δ+1)w(f;δ)ds≤{1+n+βA(1)B(nx)δ∑k=0∞‍Pk(nx)hhhhhhh×∫(k+α)/(n+β)(k+α+1)/(n+β)|s-x|ds}w(f;δ).
According to the Cauchy-Schwarz inequality for integration, we obtain that
(24)∫(k+α)/(n+β)(k+α+1)/(n+β)|s-x|ds≤1n+β(∫(k+α)/(n+β)(k+α+1)/(n+β)|s-x|2ds)1/2
from which, it follows that
(25)∑k=0∞‍Pk(nx)∫(k+α)/(n+β)(k+α+1)/(n+β)|s-x|ds≤1n+β∑k=0∞‍Pk(nx)(∫(k+α)/(n+β)(k+α+1)/(n+β)|s-x|2ds)1/2.
By using the Cauchy-Schwarz inequality for summation on the right hand side of (25), we may write
(26)∑k=0∞‍Pk(nx)∫(k+α)/(n+β)(k+α+1)/(n+β)|s-x|ds≤A(1)B(nx)n+β(A(1)B(nx)n+βKn(α,β)((s-x)2;x))1/2=A(1)B(nx)n+β(Kn(α,β)((s-x)2;x))1/2=A(1)B(nx)n+β(λn(x))1/2,
where λn(x) is given by (22). Considering this inequality in (23), we find that
(27)|Kn(α,β)(f;x)-f(x)|≤{1+1δλn(x)}w(f;δ).
If we set δ=λn(x), the proof is completed.Now, we will study the rates of convergence of the operatorsKn(α,β) to f  by means of the second modulus of continuity and Peetre’s K-functional.Recall that the second modulus of continuity off∈CB[0,∞) is defined by
(28)w2(f;δ)∶=sup0<t≤δ∥f(·+2t)-2f(·+t)+f(·)∥CB,
where CB[0,∞)is the class of real valued functions defined on [0,∞) which are bounded and uniformly continuous with the norm ∥f∥CB=supx∈[0,∞)|f(x)|.Peetre’sK-functional of the function f∈CB[0,∞) is defined by
(29)K(f;δ)∶=infg∈CB2[0,∞){∥f-g∥CB+δ∥g∥CB2},
where
(30)CB2[0,∞)∶={g∈CB[0,∞):g′,g′′∈CB[0,∞)}
and the norm ∥g∥CB2∶=∥g∥CB+∥g′∥CB+∥g′′∥CB (see [29]). It is clear that the following inequality:
(31)K(f;δ)≤M{w2(f;δ)+min(1,δ)∥f∥CB}
holds for all δ>0. The constant M is independent of f and δ.Theorem 6.
Letf∈CB2[0,∞). If Kn(α,β) is defined by (9), then one has
(32)|Kn(α,β)(f;x)-f(x)|≤ζ∥f∥CB2,
where
(33)ζ=ζn(x)={(nn+β)2B′′(nx)2B(nx)-nB′(nx)(n+β)B(nx)+12}x2+{nB′(nx)[2A′(1)+(2α+2)A(1)]2(n+β)2A(1)B(nx)hhh-2A′(1)+(2α+1)A(1)2(n+β)A(1)+nn+βB′(nx)B(nx)-1nB′(nx)[2A′(1)+(2α+2)A(1)]2(n+β)2A(1)B(nx)}x+A′′(1)+A′(1)2(n+β)2A(1)+(2α+1)A′(1)2(n+β)2A(1)+α2+α+(1/3)2(n+β)2+2A′(1)+(2α+1)A(1)2(n+β)A(1).Proof.
We can write from the Taylor expansion off, the linearity of the operators Kn(α,β), and (11)
(34)Kn(α,β)(f;x)-f(x)=f′(x)Kn(α,β)(s-x;x)+12f′′(η)Kn(α,β)((s-x)2;x),η∈(x,s).
From Lemma 2, it is obvious that
(35)Kn(α,β)(s-x;x)={nn+βB′(nx)B(nx)-1}x+A′(1)(n+β)A(1)+2α+12(n+β)≥0
for s≥x. Thus, by considering Lemmas 2 and 3 in (34), one can write
(36)|Kn(α,β)(f;x)-f(x)|≤{{nn+βB′(nx)B(nx)-1}xhhhhhhh+A′(1)(n+β)A(1)+2α+12(n+β)}∥f′∥CB+12[nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx){(nn+β)2B′′(nx)B(nx)hhhhhhhh-2nB′(nx)(n+β)B(nx)+1}x2hhhhhhhh+{nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)hhhhhhhhhhhhhh+(2α+1)nB′(nx)(n+β)2B(nx)hhhhhhhhhhhhhh-2A′(1)(n+β)A(1)-2α+1n+βnB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)}xhhhhhhhhhhhhhh+A′′(1)+A′(1)(n+β)2A(1)+(2α+1)A′(1)(n+β)2A(1)hhhhhhhhhhhhhh+α2+α+(1/3)(n+β)2nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)]∥f′′∥CB≤[nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx){(nn+β)2B′′(nx)2B(nx)-nB′(nx)(n+β)B(nx)+12}x2hhhhhhhhh+{nB′(nx)[2A′(1)+(2α+2)A(1)]2(n+β)2A(1)B(nx)hhhhhHHHHhh-2A′(1)+(2α+1)A(1)2(n+β)A(1)hhhhHHHHhhh+nn+βB′(nx)B(nx)-1nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)}xHHHHHHH+A′′(1)+A′(1)2(n+β)2A(1)+(2α+1)A′(1)2(n+β)2A(1)HHHHHHH+α2+α+(1/3)2(n+β)2hhhhhHhhhh+2A′(1)+(2α+1)A(1)2(n+β)A(1)nB′(nx)[2A′(1)+A(1)](n+β)2A(1)B(nx)]∥f∥CB2
which completes the proof.Theorem 7.
Iff∈CB[0,∞), then one has
(37)|Kn(α,β)(f;x)-f(x)|≤2M{w2(f;δ)+min(1,δ)∥f∥CB},
where
(38)δ∶=δn(x)=12ζn(x)
and M>0 is a constant which is independent of the function f and δ. Also, ζn(x) is the same as in Theorem 6.Proof.
Suppose thatg∈CB2[0,∞). From Theorem 6, we have
(39)|Kn(α,β)(f;x)-f(x)|≤|Kn(α,β)(f-g;x)|+|Kn(α,β)(g;x)-g(x)|+|g(x)-f(x)|≤2∥f-g∥CB+ζ∥g∥CB2=2[∥f-g∥CB+δ∥g∥CB2].
Since the left-hand side of inequality (39) does not depend on the function g∈CB2[0,∞), we get
(40)|Kn(α,β)(f;x)-f(x)|≤2K(f;δ),
where K(f;δ) is Peetre’s K-functional defined by (29). By using the relation (31) in (39), the inequality
(41)|Kn(α,β)(f;x)-f(x)|≤2M{w2(f;δ)+min(1,δ)∥f∥CB}
holds.Remark 8.
In Theorems5–7, λn,ζn,δn→0 when n→∞ under the assumption (16).


## 4. Special Cases of the OperatorsKn(α,β) and Further Properties

Gould-Hopper polynomialsgkd+1(x,h)are defined through the identity
(42)gkd+1(x,h)=∑m=0[k/(d+1)]‍k!m!(k-(d+1)m)!hmxk-(d+1)m
and satisfy the generating function
(43)ehtd+1exp(xt)=∑k=0∞‍gkd+1(x,h)tkk!,
where, as usual, [·] denotes the integer part [30].The Gould-Hopper polynomials are Brenke-type polynomials for the special case ofA(t)=ehtd+1 and B(t)=et in (5). From (2), the operators including the Gould-Hopper polynomials are as follows:
(44)Ln*(f;x)∶=e-nx-h∑k=0∞‍gkd+1(nx,h)k!f(kn),
where x∈[0,∞) and h≥0  (see [11]).Similarly, the special caseA(t)=ehtd+1 and B(t)=et of (9) gives the following Kantorovich-Stancu type operators Kn*(α,β)(f;x) including the Gould-Hopper polynomials:
(45)Kn*(α,β)(f;x)∶=(n+β)e-nx-h∑k=0∞‍gkd+1(nx,h)k!×∫(k+α)/(n+β)(k+α+1)/(n+β)f(t)dt
under the assumption h≥0.Remark 9.
Forh=0, we have gkd+1(nx,0)=(nx)k and the operators given by (45) reduce to the Kantorovich-Stancu type of Szasz-Mirakyan operators given by (10).Remark 10.
Forα=β=0, the operators (45) give the Kantorovich type operators including the Gould-Hopper polynomials given by
(46)Kn*(f;x)∶=ne-nx-h∑k=0∞‍gkd+1(nx,h)k!∫k/n(k+1)/nf(t)dt
in [8].Remark 11.
Forh=0 in Remark 10, we get gkd+1(nx,0)=(nx)k and then the operators given by (46) reduce to the Szasz-Mirakyan-Kantorovich operators given by (7).Now, in order to prove a Voronovskaya type theorem for the operators given by (45), let us prove the following lemmas.Lemma 12.
For the operatorsKn*(α,β), one has
(47)Kn*(α,β)(1;x)=1,Kn*(α,β)(s;x)=nxn+β+h(d+1)n+β+2α+12(n+β),Kn*(α,β)(s2;x)=n2x2(n+β)2+nx(n+β)2×{2h(d+1)+(2α+2)}+1(n+β)2×[13h(h+1)(d+1)2+(2α+1)h(d+1)hGGhh+(α2+α+13)],Kn*(α,β)(s3;x)=n3x3(n+β)3+3n2x22(n+β)3×{2α+3+2(d+1)h}+nx2(n+β)3×{6h2(d+1)2+6h(d+1)hgggh×(3+d+2α)+12α+6α2+7}+14(n+β)3{4h3(d+1)3+6h2(d+1)2hhhhhhhhhhhhhh×(2α+2d+3)+4α3+6α2hhhhhhhhhhhhhh+4α+1+2h(d+1)hhhhhhhhhhhhhh×[2d2+d(2α+7)hhhhhhhhhhhhhhhh+12α+6α2+7]},Kn*(α,β)(s4;x)=n4x4(n+β)4+4n3x3(n+β)4(h(d+1)+α+2)+3n2x2(n+β)4{2h2(d+1)2+2h(d+1)hhhhhhhhhh×(2α+d+4)+2α2+6α+5}+2nx(n+β)4{2h3(d+1)3+6h2(d+1)2(α+d+2)hhhhhhhhhhh+h(d+1)gggggggggggg×(2d2+10d+6(3+d)α+6α2+15)hhhhhhhhh+(1+α)(3+2α(2+α))2h3}+15(n+β)4×{5h(d+1)4[1+h(7+h(6+h))]hh+10h(d+1)3[1+h(3+h)](1+2α)hh+10h(h+1)(d+1)2(1+3α(1+α))hh+5h(d+1)(1+2α[2+α(3+2α)])hh+5α4+10α3+10α2+5α+1}.Proof.
The proof follows from the generating function (43) for the Gould-Hopper polynomials.Lemma 13.
For eachx∈[0,∞), one has
(48)Kn*(α,β)((s-x)2;x)=β2x2(n+β)2+x[n(n+β)2{2h(d+1)+(2α+2)}hhHHHh-2h(d+1)+2α+1n+βn(n+β)2]+1(n+β)2×[h13(h+1)(d+1)2hhhhh+(2α+1)h(d+1)+(α2+α+13)],Kn*(α,β)((s-x)4;x)=β4x4(n+β)4-x3{2β2(-3n+(2h(d+1)+2α+1)β)(n+β)4}+x2{1(n+β)4[3n2-2nβ(6h(d+1)+6α+5)hhhhhhhhhhJhhh+2{3h2(d+1)2+3h(d+1)hhhhhhhhhhJhhh×(2α+d+2)+3α(1+α)+1h2}β2]1(n+β)4}+x{2n(n+β)4{2h3(d+1)3+6h2(d+1)2hhhhhhhhLhhhhhhj×(α+d+2)+h(d+1)hhhhhhhhLhhhhhhj×(2d2+10d+6(3+d)αhhhhhhhhLhhhhhhj+6α2+15)+(1+α)hhhhhhhhLhhhhhhj×(3+2α(2+α))2h3}hhhhhhh-1(n+β)3{4h3(d+1)3+6h2(d+1)2hhhhhhhhhhhhhhhhhh×(2α+2d+3)+4α3+6α2hhhhhhhhhhhhhhhhhh+4α+1+2h(d+1)hhhhhhhhhhhhhhhhhh×[2d2+d(2α+7)+6α2hhhhhhhhhhhhhhhhhhh+12α+7d2]}2n(n+β)4}+15(n+β)4{5h(d+1)4[1+h(7+h(6+h))]hhhhHHHHHh+10h(d+1)3[1+h(3+h)](2α+1)hhhhHHHHHh+10(d+1)2h(1+h)(1+3α(1+α))hhhhHHHHHh+5h(d+1)(1+2α[2+α(3+2α)])hhhhHHHHHh+5α4+10α3+10α2+5α+1(d+1)4}.Proof.
From Lemma12, the proof is obvious.Theorem 14.
Letf∈C2[0,a]. Then one has
(49)limn→∞(n+β)[Kn*(α,β)(f;x)-f(x)]=f′(x){βx+h(d+1)+2α+12}+xf′′(x)2!.Proof.
By Taylor’s theorem forf, we have
(50)f(s)=f(x)+(s-x)f′(x)+(s-x)22!f′′(x)+(s-x)2η(s;x),
where η(s;x)∈C[0,a] and lims→xη(s;x)=0. By applying the operator Kn*(α,β) to the both sides of (50), we have
(51)Kn*(α,β)(f;x)=f(x)+f′(x)Kn*(α,β)(s-x;x)+f′′(x)2!Kn*(α,β)((s-x)2;x)+Kn*(α,β)((s-x)2η(s;x);x).
According to Lemmas 12 and 13, the equality (51) can be written as follows:
(52)(n+β)[Kn*(α,β)(f;x)-f(x)]=(n+β){βn+βx+h(d+1)n+β+2α+12(n+β)}f′(x)+(n+β){x2(βn+β)2hhhhhhhhhhhh+x[2h(d+1)+2α+1n+βn(n+β)2{2h(d+1)+(2α+2)}hhhhhhhhhhhhhhh-2h(d+1)+2α+1n+βn(n+β)2]hhhhGggg+1(n+β)2[1313h(h+1)(d+1)2+(2α+1)hhhhhhHHHHHhhhhhGGG×h(d+1)+(α2+α+13)](βn+β)2}×f′′(x)2!+(n+β)Kn*(α,β)((s-x)2η(s;x);x),
where
(53)Kn*(α,β)((s-x)2η(s;x);x)=(n+β)e-nx-h∑k=0∞gkd+1(nx,h)k!×∫(k+α)/(n+β)(k+α+1)/(n+β)(s-x)2η(s;x)ds.
By applying Cauchy-Schwarz inequality, we can write
(54)(n+β)Kn*(α,β)((s-x)2η(s;x);x)≤(n+β)2e-nx-h∑k=0∞gkd+1(nx,h)k!×(∫(k+α)/(n+β)(k+α+1)/(n+β)(s-x)4ds)1/2×(∫(k+α)/(n+β)(k+α+1)/(n+β)η2(s;x)ds)1/2.
If we consider Cauchy-Schwarz inequality again on the right-hand side of inequality above, then we arrive at
(55)(n+β)Kn*(α,β)((s-x)2η(s;x);x)≤((n+β)3e-nx-h∑k=0∞gkd+1(nx,h)k!h×∫(k+α)/(n+β)(k+α+1)/(n+β)(s-x)4ds)1/2×((n+β)e-nx-h∑k=0∞gkd+1(nx,h)k!hhhh×∫(k+α)/(n+β)(k+α+1)/(n+β)η2(s;x)ds)1/2=(n+β)2Kn*(α,β)((s-x)4;x)Kn*(α,β)(η2(s;x);x).
From Lemma13, we have
(56)limn→∞(n+β)2Kn*(α,β)((s-x)4;x)=3x2.
On the other hand, since η(s;x)∈C[0,a] and lims→xη(s;x)=0, then it follows from Theorem 4 that
(57)limn→∞Kn*(α,β)(η2(s;x);x)=η2(x;x)=0.
Therefore, we conclude from (55), (56), and (57) that
(58)limn→∞(n+β)Kn*(α,β)((s-x)2η(s;x);x)=0
and then, by taking limit as n→∞ in (52) and using (58), we find
(59)limn→∞[Kn*(α,β)(f;x)-f(x)]=f′(x){βx+h(d+1)+2α+12}+xf′′(x)2!
which completes the proof.Remark 15.
Forα=β=0, Theorem 14 represents the Voronovskaya type theorem for the operators given by (46) (see [8]).Remark 16.
Forh=0, it yields a Voronovskaya type theorem for the Kantorovich-Stancu type of Szasz-Mirakyan operators given by (10).Remark 17.
Gettingα=β=h=0 in Theorem 14 gives the Voronovskaya type result for the Szasz-Mirakyan-Kantorovich operators given by (7).



---

*Source: 935430-2013-12-22.xml*