# Approximation Theorems for Generalized Complex Kantorovich-Type Operators

Authors: N. I. Mahmudov; M. Kara
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The order of simultaneous approximation and Voronovskaja-type results with quantitative estimate for complexq-Kantorovich polynomials (q>0) attached to analytic functions on compact disks are obtained. In particular, it is proved that for functions analytic in {z∈ℂ:|z|<R}, R>q, the rate of approximation by the q-Kantorovich operators (q>1) is of order q−n versus 1/n for the classical Kantorovich operators.


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## Body

## 1. Introduction

For each integerk≥0, the q-integer [k]q and the q-factorial [k]q! are defined by
(1.1)[k]q:={1-qk1-q,ifq∈R+∖{1},k,ifq=1,fork∈N,[0]q=0,[k]q!:=[1]q[2]q⋯[k]qfork∈ℕ,[0]!=1.
For integers 0≤k≤n, the q-binomial coefficient is defined by
(1.2)[nk]q:=[n]q![k]q![n-k]q!.
For fixed 1≠q>0, we denote the q-derivative Dqf(z) of f by
(1.3)Dqf(z)={f(qz)-f(z)(q-1)z,z≠0,f′(0),z=0.Let𝔻R be a disc 𝔻R:={z∈ℂ:|z|<R} in the complex plane ℂ. Denote by H(𝔻R) the space of all analytic functions on 𝔻R. For f∈H(𝔻R) we assume that f(z)=∑m=0∞‍amzm.In several recent papers, convergence properties of complexq-Bernstein polynomials, proposed by Phillips [1], defined by
(1.4)Bn,q(f;z)=∑k=0n‍f([k]q[n]q)[nk]qxk∏j=0n-k-1(1-qjx)=∑k=0n‍f([k]q[n]q)pn,k(q;z)
and attached to an analytic function f in closed disks, were intensively studied by many authors; see [2] and references their in. It is known that the cases 0<q<1 and q>1 are not similar to each other. This difference is caused by the fact that, for 0<q<1, Bn,q are positive linear operators on C[0,1] while for q>1, the positivity fails. The lack of positivity makes the investigation of convergence in the case q>1 essentially more difficult than that for 0<q<1. There are few papers [3–6] studying systematically the convergence of the q-Berntsein polynomials in the case q>1. If q≥1 then qualitative Voronovskaja-type and saturation results for complex q-Bernstein polynomials were obtained in Wang and Wu [5]. Wu [6] studied saturation of convergence on the interval [0,1] for the q-Bernstein polynomials of a continuous function f for arbitrary fixed q>1. On the other hand, Gal [7, 8], Anastassiou and Gal [9, 10], Mahmudov [11–13], and Mahmudov and Gupta [14] obtained quantitative estimates of the convergence and of the Voronovskaja’s theorem in compact disks, for different complex Bernstein-Durrmeyer type operators.The goal of the present note is to extend these type of results to complex Kantorovich operators based on theq-integers, in the case q>0, defined as follows:
(1.5)Kn,q(f;z)=∑k=0n‍pn,k(q;z)∫01f(q[k]q+t[n+1]q)dt.
Notice that in the case q=1, these operators coincide with the classical Kantorovich operators. For 0<q≤1 the operator Kn,q:C[0,1]→C[0,1] is positive and for q>1, it is not positive. The problems studied in this paper in the case q=1 were investigated in [2, 9].We start with the following quantitative estimates of the convergence for complexq-Kantorovich-type operators attached to an analytic function in a disk of radius R>1 and center 0.Theorem 1.1.
Letf∈H(𝔻R).(i)
Let0<q≤1 and 1≤r<R. For all z∈𝔻r and n∈ℕ, one has
(1.6)|Kn,q(f;z)-f(z)|≤3+q-12[n]q∑m=1∞‍|am|m(m+1)rm.(ii)
Let1<q<R<∞ and 1≤r<R/q. For all z∈𝔻r and n∈ℕ, one has
(1.7)|Kn,q(f;z)-f(z)|≤2[n]q∑m=1∞‍|am|m(m+1)qmrm.Remark 1.2.
(i) Since[n]q→(1-q)-1 as n→∞ in the estimate in Theorem 1.1(i) we do not obtain convergence of Kn,q(f;z) to f(z). But this situation can be improved by choosing 0<q=qn<1 with qn↗1 as n→∞. Since in this case [n]qn→∞ as n→∞, from Theorem 1.1(i) we get uniform convergence in 𝔻r.
(ii) Theorem1.1(ii) says that for functions analytic in 𝔻R, R>q, the rate of approximation by the q-Kantorovich operators (q>1) is of order q-n versus 1/n for the classical Kantorovich operators.Letf∈H(𝔻R). Let us define
(1.8)Lq(f;z):={1-2z2f′(z)+(1-z)(Dqf(z)-f′(z))1-q-1,if|z|<Rq,R>q>1,1-2z2f′(z)+z(1-z)2f′′(z),if|z|<R,0<q≤1.
It is not difficult to show that
(1.9)Lq(f;z)=q(1-z)∑m=1∞‍am[m]q-mq-1zm-1+1-2z2∑m=1∞‍ammzm-1=q∑m=1∞‍am([1]q+⋯+[m-1]q)zm-1(1-z)+1-2z2∑m=1∞‍ammzm-1,q>1.
Here we used the identity
(1.10)[m]q-mq-1=[1]q+⋯+[m-1]q.
The next theorem gives Voronovskaja-type result in compact disks, for complex q-Kantorovich operators attached to an analytic function in 𝔻R, R>1 and center 0.Theorem 1.3.
Letf∈H(𝔻R).(i)
Let0<q≤1 and 1≤r<R. For all z∈𝔻r and n∈ℕ one has
(1.11)|Kn,q(f;z)-f(z)-1-2z2[n+1]qf′(z)-z(1-z)2[n+1]qf′′(z)|≤28+q-12[n]q2∑m=2∞‍|am|m2(m-1)2rm.(ii)
Let1<q<R<∞ and 1≤r<R/q2. For all z∈𝔻r and n∈ℕ, one has
(1.12)|Kn,q(f;z)-f(z)-1[n+1]qLq(f;z)|≤14[n]q2∑m=2∞‍|am|m2(m-1)2q2mrm.Remark 1.4.
(i) In the hypothesis onf in Theorem 1.3(i) choosing 0<qn<1 with qn↗1 as n→∞, it follows that
(1.13)limn→∞[n+1]qn[Kn,qn(f;z)-f(z)]=1-2z2f′(z)+z(1-z)2f′′(z)
uniformly in any compact disk included in the open disk 𝔻R.
(ii) Theorem1.3(ii) gives explicit formulas of Voronovskaja-type for the q-Kantorovich polynomials for q>1.
(iii) Obviously the best order of approximation that can be obtained from the estimate Theorem1.3(i) is O(1/[n]qn2) and O(1/n2) for q=1, while the order given by Theorem 1.3(ii) is O(1/q2n), q>1, which is essentially better.Next theorem shows thatLq(f;z),q≥1, is continuous about the parameter q for f∈H(𝔻R), R>1.Theorem 1.5.
LetR>1 and f∈H(𝔻R). Then for any r,  0<r<R,
(1.14)limq→1+Lq(f;z)=L1(f;z)
uniformly on 𝔻r.As an application of Theorem1.3, we present the order of approximation for complex q-Kantorovich operators.Theorem 1.6.
Let1<q<R,1≤r<R/q2 (or 0<q≤1,1≤r<R) and f∈H(𝔻R). If f is not a constant function then the estimate
(1.15)∥Kn,q(f)-f∥r≥1[n+1]qCr,q(f),n∈ℕ
holds, where the constant Cr,q(f) depends on f,q and r but is independent of n.


## 2. Auxiliary Results

Lemma 2.1.
Letq>0. For all n∈ℕ,m∈ℕ∪{0}, z∈ℂ one has
(2.1)Kn,q(em;z)=∑j=0m‍(mj)qj[n]qj[n+1]qm(m-j+1)Bn,q(ej;z),
where em(z)=zm.Proof.
The recurrence formula can be derived by direct computation.(2.2)Kn,q(em;z)=∑k=0n‍pn,k(z)∑j=0m‍∫01‍(mj)qj[k]qjtm-j[n+1]qmdt=∑k=0n‍pn,k(z)∑j=0m‍(mj)qj[k]qj[n+1]qm(m-j+1)=∑j=0m‍(mj)qj[n]qj[n+1]qm(m-j+1)∑k=0n‍[k]qj[n]qjpn,k(z)=∑j=0m‍(mj)qj[n]qj[n+1]qm(m-j+1)Bn,q(ej;z).Lemma 2.2.
For allz∈𝔻r,r≥1 one has
(2.3)|Kn,q(em;z)|≤rm,n,m∈ℕ.Proof.
Indeed, using the inequality|Bn,q(ej;z)|≤rj (see [3]), we get
(2.4)|Kn,q(em;z)|≤∑j=0m‍(mj)qj[n]qj[n+1]qm(m-j+1)|Bn,q(ej;z)|≤1[n+1]qm∑j=0m‍(mj)qj[n]qjrm=(1+q[n]q[n+1]q)mrm=rm.Lemma 2.3.
For alln, m∈ℕ, z∈ℂ, 1≠q>0 one has
(2.5)Kn,q(em+1;z)=z(1-z)[n]qDqKn,q(em;z)+zKn,q(em;z)+1[n+1]qm+1∑j=0m+1‍(m+1j)qj[n]qj1(m-j+2)(1-j[n+1]qq(m+1)[n]q)Bn,q(ej;z).Proof.
We know that (see [2])
(2.6)z(1-z)[n]qDqBn,q(ej;z)=Bn,q(ej+1;z)-zBn,q(ej;z).
Taking the derivative of the formula (2.1) and using the above formula we have
(2.7)z(1-z)[n]qDqKn,q(em;z)=∑j=0m‍(mj)qj[n]qj[n+1]qm(m-j+1)z(1-z)[n]qDqBn,q(ej;z)=∑j=0m‍(mj)qj[n]qj[n+1]qm(m-j+1)(Bn,q(ej+1;z)-zBn,q(ej;z))=∑j=1m+1‍(mj-1)qj-1[n]qj-1[n+1]qm(m-j+2)Bn,q(ej;z)-zKn,q(em;z).
It follows that
(2.8)Kn,q(em+1;z)=z(1-z)[n]qDqKn,q(em;z)+zKn,q(em;z)+∑j=0m+1‍(m+1j)qj[n]qj[n+1]qm+1(m-j+2)Bn,q(ej;z)-∑j=1m+1‍(mj-1)qj-1[n]qj-1[n+1]qm(m-j+2)Bn,q(ej;z)=z(1-z)[n]qDqKn,q(em;z)+zKn,q(em;z)+1[n+1]qm+1(m+2)+∑j=1m+1‍(m+1j)qj-1[n]qj-1[n+1]qm(m-j+2)(m+1)q[n]q-j[n+1]q(m+1)[n+1]qBn,q(ej;z)=z(1-z)[n]qDqKn,q(em;z)+zKn,q(em;z)+∑j=0m+1‍(m+1j)qj-1[n]qj-1[n+1]qm(m-j+2)(m+1)q[n]q-j[n+1]q(m+1)[n+1]qBn,q(ej;z).
Here we used the identity
(2.9)(mj-1)=(m+1j)j(m+1).Form∈ℕ∪{0} define
(2.10)En,m(z):={Kn,q(em;z)-em(z)-1-2z2[n+1]qmzm-1-z(1-z)2[n+1]qm(m-1)zm-2,if0<q≤1,Kn,q(em;z)-em(z)-1-2z2[n+1]qmzm-1-∑j=1m-1[j]qqzm-1(1-z)[n+1]q,ifq>1.
Here it is assumed that ∑j=10‍[j]q=0.Lemma 2.4.
Letn, m∈ℕ.(a)
If0<q<1, one has the following recurrence formula:
(2.11)En,m(z)=z(1-z)[n]qDq(Kn,q(em-1;z)-em-1(z))+zEn,m-1(z)+(m-1[n+1]q-[m-1]q[n]q)zm-1(1-z)-1-2z2[n+1]qzm-1+1[n+1]qm∑j=0m‍(mj)qj[n]qj(m-j+1)mq[n]q-j[n+1]qmq[n]qBn,q(ej;z).(b)
Ifq>1, one has
(2.12)En,m(z)=z(1-z)[n]qDq(Kn,q(em-1;z)-em-1(z))+zEn,m-1(z)+[m-1]q[n]q[n+1]qzm-1(1-z)-1-2z2[n+1]qzm-1+1[n+1]qm∑j=0m‍(mj)qj[n]qj(m-j+1)mq[n]q-j[n+1]qmq[n]qBn,q(ej;z).Proof.
We give the proof for the caseq>1. The case 0<q<1 is similar to that of q>1.
(b) It is immediate thatEn,m(z) is a polynomial of degree less than or equal to m and that En,0(z)=En,1(z)=0.
Using the formula (2.5), we get
(2.13)En,m(z)=z(1-z)[n]qDq(Kn,q(em-1;z)-em-1(z))+[m-1]q[n]qzm-1(1-z)+z(En,m-1(z)+1-2z2[n+1]q(m-1)zm-2+∑j=1m-2‍[j]qqzm-2(1-z)[n+1]q)-1-2z2[n+1]qmzm-1-∑j=1m-1‍[j]qqzm-1(1-z)[n+1]q+1[n+1]qm∑j=0m‍(mj)qj[n]qj(m-j+1)mq[n]q-j[n+1]qmq[n]qBn,q(ej;z).
A simple calculation leads us to the following relationship:
(2.14)En,m(z)=z(1-z)[n]qDq(Kn,q(em-1;z)-em-1(z))+zEn,m-1(z)+[m-1]q[n]q[n+1]qzm-1(1-z)-1-2z2[n+1]qzm-1+1[n+1]qm∑j=0m‍(mj)qj[n]qj(m-j+1)mq[n]q-j[n+1]qmq[n]qBn,q(ej;z),
which is the desired recurrence formula.Remark 2.5.
Lemmas2.3 and 2.4 are true in the case q=1. In the formulae, we have to replace q-derivative by the ordinary derivative.


## 3. Proofs of the Main Results

We give proofs for the caseq>1. The case 0<q<1 and q=1 are similar to that of q>1.Proof of Theorem1.1.
The use of the above recurrence we obtain the following relationship:(3.1)Kn,q(em;z)-em(z)=z(1-z)[n]qDqKn,q(em-1;z)+z(Kn,q(em-1;z)-em-1(z))+1[n+1]qm∑j=0m‍(mj)qj[n]qj(m-j+1)mq[n]q-j[n+1]qmq[n]qBn,q(ej;z).
We can easily estimate the sum in the above formula as follows:
(3.2)|1[n+1]qm∑j=0m‍(mj)qj[n]qj1(m-j+1)(1-jm-jmq[n]q)Bn,q(ej;z)|≤1[n+1]qm(∑j=0m-1‍(m-1j)mm-jqj[n]qjm-j+1|1-jm-jmq[n]q|)|Bn,q(ej;z)|+qm-1[n]qm-1[n+1]qmrm≤2m(q[n]q+1)m-1+qm-1[n]qm-1[n+1]qmrm≤2m+1[n+1]qrm.
It is known that by a linear transformation, the Bernstein inequality in the closed unit disk becomes
(3.3)|Pm′(z)|≤mqr∥Pm∥qr,∀|z|≤qr,r≥1,
(where ∥Pm∥qr=max{|Pm(z)|:|z|≤qr}) which combined with the mean value theorem in complex analysis implies
(3.4)|Dq(Pm;z)|=|Pm(qz)-Pm(z)qz-z|≤∥Pm′∥qr≤mqr∥Pm∥qr,
for all |z|≤r, where Pm(z) is a complex polynomial of degree ≤m. From the above recurrence formula (3.1), we get
(3.5)|Kn,q(em;z)-em(z)|≤|z||1-z|[n]q|DqKn,q(em-1;z)|+|z||Kn,q(em-1;z)-em-1(z)|+2m+1[n+1]qrm≤r(1+r)[n]qm-1qr∥Kn,q(em-1)∥qr+r|Kn,q(em-1;z)-em-1(z)|+2m+1[n+1]qrm≤r|Kn,q(em-1;z)-em-1(z)|+2(m-1)[n]qqm-1rm+2m+1[n+1]qrm≤r|Kn,q(em-1;z)-em-1(z)|+4m[n]qqmrm.
By writing the last inequality for m=1,2,…, we easily obtain, step by step, the following:
(3.6)|Kn(em;z)-em(z)|≤4m[n]qqmrm+r4(m-1)[n]qqm-1rm-1+r24(m-2)[n]qqm-2rm-2+⋯+rm-14[n]qqr=4[n]qqmrm(m+m-1+⋯+1)≤2m(m+1)[n]qqmrm.
Since Kn,q(f;z) is analytic in 𝔻R, we can write
(3.7)Kn,q(f;z)=∑m=0∞‍amKn,q(em;z),z∈𝔻R,
which together with (3.6) immediately implies for all |z|≤r(3.8)|Kn,q(f;z)-f(z)|≤∑m=0∞‍|am||Kn,q(em;z)-em(z)|≤2[n]q∑m=1∞‍|cm|m(m+1)(qr)m.Proof of Theorem1.3.
A simple calculation and the use of the recurrence formula (2.5) lead us to the following relationship:
(3.9)En,m(z)=z(1-z)[n]qDq(Kn,q(em-1;z)-em-1(z))+zEn,m-1(z)+[m-1]q[n]q[n+1]qzm-1(1-z)+1[n+1]q(zm-Bn,q(em;z))+1[n+1]q(1-qm-1[n]qm-1[n+1]qm-1)Bn,q(em;z)+12[n+1]q(qm-1[n]qm-1[n+1]qm-1-1)Bn,q(em-1;z)+12[n+1]q(Bn,q(em-1;z)-zm-1)-(m-1)qm-2[n]qm-22[n+1]qmBn,q(em-1;z)+1[n+1]qm∑j=0m-2‍(mj)qj[n]qj(m-j+1)mq[n]q-j[n+1]qmq[n]qBn,q(ej;z):=∑k=19‍Ik.
Firstly, we estimate I3, I8. It is clear that
(3.10)|I3|≤[m-1]q[n]q[n+1]qrm-1(1+r),|I8|≤(m-1)2[n+1]q2|Bn,q(em-1;z)|≤(m-1)2[n+1]q2rm-1.
Secondly, using the known inequality
(3.11)1-∏k=1mxk≤∑k=1m‍(1-xk),0≤xk≤1,
to estimate I5, I6, I9. 							(3.12)|I5|≤1[n+1]q(1-qm-1[n]qm-1[n+1]qm-1)|Bn,q(em;z)|≤m-1[n+1]q2rm,|I6|≤12[n+1]q(1-qm-1[n]qm-1[n+1]qm-1)|Bn,q(em-1;z)|≤m-12[n+1]q2rm-1,|I9|≤1[n+1]qm∑j=0m-2‍(m-2j)m(m-1)(m-j)(m-j-1)qj[n]qj(m-j+1)(1-jm-jmq[n]q)rj≤2m(m-1)[n+1]qm-2[n+1]qmrm=2m(m-1)[n+1]q2rm.
Finally, we estimate I4, I7. We use [2, Theorem 1.1.2]
(3.13)|I4|+|I7|≤1[n+1]q|zm-Bn,q(em;z)|+12[n+1]q|Bn(em-1;z)-zm-1|≤2[m-1]q(m-1)[n]q[n+1]qrm+[m-2]q(m-2)[n]q[n+1]qrm-1.
Using (3.6), (3.10), (3.12), and (3.13) in (3.9) finally we have (m≥3)
(3.14)|En,m(z)|≤r(1+r)[n]q|Dq(Kn,q(em-1;z)-em-1(z))|+r|En,m-1(z)|+[m-1]q[n]q[n+1]qrm-1(1+r)+2[m-1]q(m-1)[n]q[n+1]qrm+m-1[n+1]q2rm+m-12[n+1]q2rm-1+[m-2]q(m-2)[n]q[n+1]qrm-1+(m-1)2[n+1]q2rm-1+2m(m-1)[n+1]q2rm≤r(1+r)[n]qm-1qr∥Kn,q(em-1)-em-1∥qr+r|En,m-1(z)|+10m[m-1]q[n]q2rm≤(m-1)(1+r)[n]q2(m-1)m[n]qq2(m-1)rm-1+r|En,m-1(z)|+10m[m-1]q[n]q2rm≤r|En,m-1(z)|+4m(m-1)2[n]q2q2mrm+10m(m-1)[n]q2qmrm≤r|En,m-1(z)|+14m2(m-1)2[n]q2q2mrm.
As a consequence, we get
(3.15)|En,m(z)|≤14m2(m-1)2[n]q2q2mrm.
This inequality combined with
(3.16)|Kn,q(f;z)-f(z)-1[n+1]qLq(f,z)|≤∑m=1∞‍|am||En,m(z)|
immediately implies the required estimate in statement.
Note that sincef(4)=∑m=4∞‍amm(m-1)(m-2)(m-3)zm-4 and the series is absolutely convergent for all |z|<R, it easily follows the finiteness of the involved constants in the statement.Proof of Theorem1.6.
For allz∈𝔻R and n∈ℕ, we get
(3.17)Kn,q(f;z)-f(z)=1[n+1]q{Lq(f;z)+[n+1]q(Kn,q(f;z)-f(z)-1[n+1]qLq(f;z))}.
We apply
(3.18)∥F+G∥r≥|∥F∥r-∥G∥r|≥∥F∥r-∥G∥r
to get
(3.19)∥Kn,q(f)-f∥r≥1[n+1]q{∥Lq(f;z)∥r-[n+1]q∥Kn,q(f;z)-f(z)-1[n+1]qLq(f;z)∥r}.
Because by hypothesis f is not a constant in 𝔻R, it follows ∥Lq(f;z)∥r>0. Indeed, assuming the contrary, it follows that Lq(f;z)=0 for all z∈𝔻¯R that is
(3.20)∑m=1∞‍am(12-z)mzm-1+∑m=1∞‍am∑j=1m-1‍[j]qzm-1(1-z)=0,12a1+a1+∑m=1∞‍(12(m+1)am+1-am+am+1∑j=1m‍[j]q-am∑j=1m-1‍[j]q)zm=0
for all z∈𝔻¯R∖{0}. Thus am=0,  m=1,2,3,…. Thus, f is constant, which is contradiction with the hypothesis.
Now, by Theorem1.3, we have
(3.21)[n+1]q|Kn,q(f;z)-f(z)-1[n+1]qLq(f;z)|≤[n+1]q[n]q14[n]q∑m=2∞‍|am|m2(m-1)2q2mrm→0asn→∞.
Consequently, there exists n1 (depending only on f and r) such that for all n≥n1 we have
(3.22)∥Lq(f;z)∥r-[n+1]q∥Kn,q(f;z)-f(z)-1[n+1]qLq(f;z)∥r≥12∥Lq(f;z)∥r,
which implies
(3.23)∥Kn,q(f)-f∥r≥1[n+1]q12∥Lq(f;z)∥r,∀n≥n1.
For 1≤n≤n1-1, we have
(3.24)∥Kn,q(f)-f∥r≥1[n+1]q([n+1]q∥Kn,q(f)-f∥r)=1[n+1]qMr,n(f)>0,
which finally implies that
(3.25)∥Kn,q(f)-f∥r≥1[n+1]qCr,q(f),
for all n, with Cr,q(f)=min{Mr,1(f),…,Mr,n1-1(f),(1/2)∥Lq(f;z)∥r}.Proof of Theorem1.5.
Proof is similar to that of Theorem1.3 [5].



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*Source: 454579-2012-10-03.xml*