Approximation Theorems for Generalized Complex Kantorovich-Type Operators

Abstract

The order of simultaneous approximation and Voronovskaja-type results with quantitative estimate for complex q-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.

1. Introduction

For each integer 𝑘≥0, the 𝑞-integer [𝑘]_𝑞 and the 𝑞-factorial [𝑘]_𝑞! are defined by [𝑘]_𝑞∶=1−𝑞^𝑘1−𝑞,if𝑞∈𝑅⁺⧵{1},[0]𝑘,if𝑞=1,for𝑘∈𝑁,_𝑞[𝑘]=0,_𝑞!∶=[1]_𝑞[2]_𝑞⋯[𝑘]_𝑞for𝑘∈ℕ,[0]!=1. (1.1) For integers 0≤𝑘≤𝑛, the 𝑞-binomial coefficient is defined by 𝑛𝑘_𝑞[𝑛]∶=^𝑞![𝑘]_𝑞![𝑛−𝑘]_𝑞!. (1.2) For fixed 1≠𝑞>0, we denote the 𝑞-derivative 𝐷_𝑞𝑓(𝑧) of 𝑓 by 𝐷_𝑞𝑓(𝑧)=𝑓𝑞𝑧−𝑓(𝑧)𝑧𝑓𝑞−1,𝑧≠0,_′(0),𝑧=0. (1.3)

Let 𝔻_𝑅 be a disc 𝔻_𝑅∶={𝑧∈ℂ∶|𝑧|<𝑅} in the complex plane ℂ. Denote by 𝐻(𝔻_𝑅) the space of all analytic functions on 𝔻_𝑅. For 𝑓∈𝐻(𝔻_𝑅) we assume that 𝑓(𝑧)=∑^∞_𝑚=0𝑎_𝑚𝑧^𝑚.

In several recent papers, convergence properties of complex 𝑞-Bernstein polynomials, proposed by Phillips [1], defined by 𝐵_𝑛,𝑞𝑓;𝑧=^𝑛_𝑘=0𝑓[𝑘]_𝑞[𝑛]_𝑞𝑛𝑘_𝑞𝑥^{𝑘𝑛−𝑘−1}_𝑗=01−𝑞^𝑗𝑥=^𝑛_𝑘=0𝑓[𝑘]_𝑞[𝑛]_𝑞𝑝_𝑛,𝑘𝑞;𝑧 (1.4) 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, 𝐵_𝑛,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 the 𝑞-integers, in the case 𝑞>0, defined as follows: 𝐾_𝑛,𝑞𝑓;𝑧=^𝑛_𝑘=0𝑝_𝑛,𝑘𝑞;𝑧¹₀𝑓𝑞[𝑘]_𝑞+𝑡[𝑛+1]_𝑞𝑑𝑡. (1.5) Notice that in the case 𝑞=1, these operators coincide with the classical Kantorovich operators. For 0<𝑞≤1 the operator 𝐾_𝑛,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 complex 𝑞-Kantorovich-type operators attached to an analytic function in a disk of radius 𝑅>1 and center 0.

Theorem 1.1. Let 𝑓∈𝐻(𝔻_𝑅).(i)Let 0<𝑞≤1 and 1≤𝑟<𝑅. For all 𝑧∈𝔻_𝑟 and 𝑛∈ℕ, one has 𝐾_𝑛,𝑞𝑓;𝑧−𝑓(𝑧)≤3+𝑞⁻¹2[𝑛]_𝑞^∞_𝑚=1𝑎_𝑚𝑚(𝑚+1)𝑟^𝑚. (1.6)(ii)Let 1<𝑞<𝑅<∞ and 1≤𝑟<𝑅/𝑞. For all 𝑧∈𝔻_𝑟 and 𝑛∈ℕ, one has 𝐾_𝑛,𝑞(𝑧≤2𝑓;𝑧−𝑓)[𝑛]_𝑞^∞_𝑚=1𝑎_𝑚𝑚(𝑚+1)𝑞^𝑚𝑟^𝑚. (1.7)

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 𝐾_𝑛,𝑞(𝑓;𝑧) to 𝑓(𝑧). But this situation can be improved by choosing 0<𝑞=𝑞_𝑛<1 with 𝑞_𝑛↗1 as 𝑛→∞. Since in this case [𝑛]_𝑞𝑛→∞ as 𝑛→∞, from Theorem 1.1(i) we get uniform convergence in 𝔻_𝑟.
(ii) Theorem 1.1(ii) says that for functions analytic in 𝔻_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.

Let 𝑓∈𝐻(𝔻_𝑅). Let us define 𝐿_𝑞𝑓;𝑧∶=1−2𝑧2𝑓^′(𝑧)+(1−𝑧)𝐷_𝑞𝑓(𝑧)−𝑓^′(𝑧)1−𝑞⁻¹|𝑧|<𝑅,if𝑞,𝑅>𝑞>1,1−2𝑧2𝑓^′(𝑧)+𝑧(1−𝑧)2𝑓^′′(𝑧),if|𝑧|<𝑅,0<𝑞≤1. (1.8) It is not difficult to show that 𝐿_𝑞𝑓;𝑧=𝑞(1−𝑧)^∞_𝑚=1𝑎_𝑚[𝑚]_𝑞−𝑚𝑧𝑞−1^𝑚−1+1−2𝑧2^∞_𝑚=1𝑎_𝑚𝑚𝑧^𝑚−1=𝑞^∞_𝑚=1𝑎_𝑚[1]_𝑞+⋯+[𝑚−1]_𝑞𝑧^𝑚−1(1−𝑧)+1−2𝑧2^∞_𝑚=1𝑎_𝑚𝑚𝑧^𝑚−1,𝑞>1. (1.9) Here we used the identity [𝑚]_𝑞−𝑚=[1]𝑞−1^𝑞+⋯+[𝑚−1]_𝑞. (1.10) The next theorem gives Voronovskaja-type result in compact disks, for complex 𝑞-Kantorovich operators attached to an analytic function in 𝔻_𝑅, 𝑅>1 and center 0.

Theorem 1.3. Let 𝑓∈𝐻(𝔻_𝑅).(i)Let 0<𝑞≤1 and 1≤𝑟<𝑅. For all 𝑧∈𝔻_𝑟 and 𝑛∈ℕ one has 𝐾_𝑛,𝑞𝑓;𝑧−𝑓(𝑧)−1−2𝑧2[𝑛+1]_𝑞𝑓^′(𝑧)−𝑧(1−𝑧)2[𝑛+1]_𝑞𝑓^′′(𝑧)≤28+𝑞⁻¹2[𝑛]²_𝑞^∞_𝑚=2𝑎_𝑚𝑚²(𝑚−1)²𝑟^𝑚. (1.11)(ii)Let 1<𝑞<𝑅<∞ and 1≤𝑟<𝑅/𝑞². For all 𝑧∈𝔻_𝑟 and 𝑛∈ℕ, one has 𝐾_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧≤14[𝑛]²_𝑞^∞_𝑚=2𝑎_𝑚𝑚²(𝑚−1)²𝑞^2𝑚𝑟^𝑚. (1.12)

Remark 1.4. (i) In the hypothesis on 𝑓 in Theorem 1.3(i) choosing 0<𝑞_𝑛<1 with 𝑞_𝑛↗1 as 𝑛→∞, it follows that lim_𝑛→∞[𝑛+1]_𝑞𝑛𝐾_{𝑛,𝑞𝑛}𝑓;𝑧−𝑓(𝑧)=1−2𝑧2𝑓^′(𝑧)+𝑧(1−𝑧)2𝑓^′′(𝑧) (1.13) uniformly in any compact disk included in the open disk 𝔻_𝑅.
(ii) Theorem 1.3(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 Theorem 1.3(i) is 𝑂(1/[𝑛]²_𝑞𝑛) and 𝑂(1/𝑛²) for 𝑞=1, while the order given by Theorem 1.3(ii) is 𝑂(1/𝑞^2𝑛), 𝑞>1, which is essentially better.

Next theorem shows that 𝐿_𝑞(𝑓;𝑧),𝑞≥1, is continuous about the parameter 𝑞 for 𝑓∈𝐻(𝔻_𝑅), 𝑅>1.

Theorem 1.5. Let 𝑅>1 and 𝑓∈𝐻(𝔻_𝑅). Then for any 𝑟,  0<𝑟<𝑅, lim_𝑞→1+𝐿_𝑞𝑓;𝑧=𝐿₁𝑓;𝑧 (1.14) uniformly on 𝔻_𝑟.

As an application of Theorem 1.3, we present the order of approximation for complex 𝑞-Kantorovich operators.

Theorem 1.6. Let 1<𝑞<𝑅,1≤𝑟<𝑅/𝑞² (or 0<𝑞≤1,1≤𝑟<𝑅) and 𝑓∈𝐻(𝔻_𝑅). If 𝑓 is not a constant function then the estimate 𝐾_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞𝐶_𝑟,𝑞𝑓,𝑛∈ℕ (1.15) holds, where the constant 𝐶_𝑟,𝑞(𝑓) depends on 𝑓,𝑞 and 𝑟 but is independent of 𝑛.

2. Auxiliary Results

Lemma 2.1. Let 𝑞>0. For all 𝑛∈ℕ,𝑚∈ℕ∪{0}, 𝑧∈ℂ one has 𝐾_𝑛,𝑞𝑒_𝑚;𝑧=^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛+1]^𝑚_𝑞𝐵𝑚−𝑗+1^𝑛,𝑞𝑒_𝑗;𝑧, (2.1) where 𝑒_𝑚(𝑧)=𝑧^𝑚.

Proof. The recurrence formula can be derived by direct computation. 𝐾_𝑛,𝑞𝑒_𝑚;𝑧=^𝑛_𝑘=0𝑝_𝑛,𝑘(𝑧)^𝑚_𝑗=0¹₀𝑚𝑗𝑞^𝑗[𝑘]^𝑗_𝑞𝑡^𝑚−𝑗[𝑛+1]^𝑚_𝑞𝑑𝑡=^𝑛_𝑘=0𝑝_𝑛,𝑘(𝑧)^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑘]^𝑗_𝑞[𝑛+1]^𝑚_𝑞=𝑚−𝑗+1_𝑚𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛+1]^𝑚_𝑞𝑚−𝑗+1^𝑛_𝑘=0[𝑘]^𝑗_𝑞[𝑛]^𝑗_𝑞𝑝_𝑛,𝑘(𝑧)=^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛+1]^𝑚_𝑞𝐵𝑚−𝑗+1^𝑛,𝑞𝑒_𝑗;𝑧. (2.2)

Lemma 2.2. For all 𝑧∈𝔻_𝑟,𝑟≥1 one has 𝐾_𝑛,𝑞𝑒_𝑚;𝑧≤𝑟^𝑚,𝑛,𝑚∈ℕ. (2.3)

Proof. Indeed, using the inequality |𝐵_𝑛,𝑞(𝑒_𝑗;𝑧)|≤𝑟^𝑗 (see [3]), we get 𝐾_𝑛,𝑞𝑒_𝑚;𝑧≤^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛+1]^𝑚_𝑞𝐵𝑚−𝑗+1^𝑛,𝑞𝑒_𝑗≤1;𝑧[𝑛+1]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞𝑟^𝑚=1+𝑞[𝑛]_𝑞[𝑛+1]_𝑞^𝑚𝑟^𝑚=𝑟^𝑚. (2.4)

Lemma 2.3. For all 𝑛, 𝑚∈ℕ, 𝑧∈ℂ, 1≠𝑞>0 one has 𝐾_𝑛,𝑞𝑒_𝑚+1=𝑧();𝑧1−𝑧[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚;𝑧+𝑧𝐾_𝑛,𝑞𝑒_𝑚+1;𝑧[𝑛+1]_𝑞^𝑚+1𝑚+1_𝑗=0𝑗𝑞𝑚+1_𝑗[𝑛]^𝑗_𝑞1𝑗[]𝑚−𝑗+21−𝑛+1_𝑞𝑞(𝑚+1)[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧. (2.5)

Proof. We know that (see [2]) 𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧=𝐵_𝑛,𝑞𝑒_𝑗+1;𝑧−𝑧𝐵_𝑛,𝑞𝑒_𝑗;𝑧. (2.6) Taking the derivative of the formula (2.1) and using the above formula we have 𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚;𝑧=^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛+1]^𝑚_𝑞𝑧()𝑚−𝑗+11−𝑧[𝑛]_𝑞𝐷_𝑞𝐵_𝑛,𝑞𝑒_𝑗=;𝑧_𝑚𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛+1]^𝑚_𝑞𝐵𝑚−𝑗+1^𝑛,𝑞𝑒_𝑗+1;𝑧−𝑧𝐵_𝑛,𝑞𝑒_𝑗=;𝑧_𝑚+1𝑗=1𝑚𝑞𝑗−1^𝑗−1[𝑛]_𝑞^𝑗−1[𝑛+1]^𝑚_𝑞𝐵𝑚−𝑗+2^𝑛,𝑞𝑒_𝑗;𝑧−𝑧𝐾_𝑛,𝑞𝑒_𝑚;𝑧. (2.7) It follows that 𝐾_𝑛,𝑞𝑒_𝑚+1=𝑧();𝑧1−𝑧[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚;𝑧+𝑧𝐾_𝑛,𝑞𝑒_𝑚+;𝑧_𝑚+1𝑗=0𝑗𝑚+1𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛+1]_𝑞^𝑚+1𝐵𝑚−𝑗+2^𝑛,𝑞𝑒_𝑗−;𝑧_𝑚+1𝑗=1𝑚𝑞𝑗−1^𝑗−1[𝑛]_𝑞^𝑗−1[𝑛+1]^𝑚_𝑞𝐵𝑚−𝑗+2^𝑛,𝑞𝑒_𝑗=𝑧();𝑧1−𝑧[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚;𝑧+𝑧𝐾_𝑛,𝑞𝑒_𝑚+1;𝑧[𝑛+1]_𝑞^𝑚+1()+𝑚+2_𝑚+1𝑗=1𝑗𝑚+1𝑞^𝑗−1[𝑛]_𝑞^𝑗−1[𝑛+1]^𝑚_𝑞()𝑞[𝑛]𝑚−𝑗+2𝑚+1_𝑞−𝑗[𝑛+1]_𝑞(𝑚+1)[𝑛+1]_𝑞𝐵_𝑛,𝑞𝑒_𝑗=𝑧();𝑧1−𝑧[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚;𝑧+𝑧𝐾_𝑛,𝑞𝑒_𝑚+;𝑧_𝑚+1𝑗=0𝑗𝑚+1𝑞^𝑗−1[𝑛]_𝑞^𝑗−1[𝑛+1]^𝑚_𝑞()𝑞[𝑛]𝑚−𝑗+2𝑚+1_𝑞−𝑗[𝑛+1]_𝑞(𝑚+1)[𝑛+1]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧. (2.8) Here we used the identity 𝑚=𝑗𝑗𝑗−1𝑚+1(𝑚+1). (2.9)

For 𝑚∈ℕ∪{0} define 𝐸_𝑛,𝑚(𝑧)𝐾∶=^𝑛,𝑞𝑒_𝑚;𝑧−𝑒_𝑚(𝑧)−1−2𝑧2[𝑛+1]_𝑞𝑚𝑧^𝑚−1−𝑧(1−𝑧)2[𝑛+1]_𝑞𝑚(𝑚−1)𝑧^𝑚−2𝐾,if0<𝑞≤1,_𝑛,𝑞𝑒_𝑚;𝑧−𝑒_𝑚(𝑧)−1−2𝑧2[𝑛+1]_𝑞𝑚𝑧^𝑚−1−∑^𝑚−1_𝑗=1𝑗_𝑞𝑞𝑧^𝑚−1(1−𝑧)[𝑛+1]_𝑞,if𝑞>1. (2.10) Here it is assumed that ∑⁰_𝑗=1[𝑗]_𝑞=0.

Lemma 2.4. Let 𝑛, 𝑚∈ℕ.(a)If 0<𝑞<1, one has the following recurrence formula: 𝐸_𝑛,𝑚(𝑧)=𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+𝑧𝐸_{𝑛,𝑚−1}(𝑧)+𝑚−1[𝑛+1]_𝑞−[𝑚−1]_𝑞[𝑛]_𝑞𝑧^𝑚−1(1−𝑧)−1−2𝑧2[𝑛+1]_𝑞𝑧^𝑚−1+1[𝑛+1]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛]𝑚−𝑗+1𝑚𝑞_𝑞−𝑗[𝑛+1]_𝑞𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧. (2.11)(b)If 𝑞>1, one has 𝐸_𝑛,𝑚(𝑧)=𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+𝑧𝐸_{𝑛,𝑚−1}(𝑧)+[𝑚−1]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞𝑧^𝑚−1()−1−𝑧1−2𝑧2[𝑛+1]_𝑞𝑧^𝑚−1+1[𝑛+1]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛]𝑚−𝑗+1𝑚𝑞_𝑞−𝑗[𝑛+1]_𝑞𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧. (2.12)

Proof. We give the proof for the case 𝑞>1. The case 0<𝑞<1 is similar to that of 𝑞>1.
(b) It is immediate that 𝐸_𝑛,𝑚(𝑧) is a polynomial of degree less than or equal to 𝑚 and that 𝐸_𝑛,0(𝑧)=𝐸_𝑛,1(𝑧)=0.
Using the formula (2.5), we get 𝐸_𝑛,𝑚(𝑧)=𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+[𝑚−1]_𝑞[𝑛]_𝑞𝑧^𝑚−1()𝐸1−𝑧+𝑧_{𝑛,𝑚−1}(𝑧)+1−2𝑧2[𝑛+1]_𝑞(𝑚−1)𝑧^𝑚−2+^𝑚−2_𝑗=1𝑗_𝑞𝑞𝑧^𝑚−2(1−𝑧)[𝑛+1]_𝑞−1−2𝑧2[𝑛+1]_𝑞𝑚𝑧^𝑚−1−^𝑚−1_𝑗=1𝑗_𝑞𝑞𝑧^𝑚−1(1−𝑧)[𝑛+1]_𝑞+1[𝑛+1]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛]𝑚−𝑗+1𝑚𝑞_𝑞−𝑗[𝑛+1]_𝑞𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧. (2.13) A simple calculation leads us to the following relationship: 𝐸_𝑛,𝑚(𝑧)=𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+𝑧𝐸_{𝑛,𝑚−1}(𝑧)+[𝑚−1]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞𝑧^𝑚−1()−1−𝑧1−2𝑧2[𝑛+1]_𝑞𝑧^𝑚−1+1[𝑛+1]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛]𝑚−𝑗+1𝑚𝑞_𝑞−𝑗[𝑛+1]_𝑞𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧, (2.14) which is the desired recurrence formula.

Remark 2.5. Lemmas 2.3 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 case 𝑞>1. The case 0<𝑞<1 and 𝑞=1 are similar to that of 𝑞>1.

Proof of Theorem 1.1. The use of the above recurrence we obtain the following relationship: 𝐾_𝑛,𝑞𝑒_𝑚;𝑧−𝑒_𝑚(𝑧)=𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧+𝑧𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+1[𝑛+1]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛]𝑚−𝑗+1𝑚𝑞_𝑞−𝑗[𝑛+1]_𝑞𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧. (3.1) We can easily estimate the sum in the above formula as follows: 1[𝑛+1]^𝑚_𝑞^𝑚_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞1𝑗𝑚−𝑗+11−𝑚−𝑗𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗≤1;𝑧[𝑛+1]^𝑚_𝑞^𝑚−1_𝑗=0𝑗𝑚−1𝑚𝑞𝑚−𝑗^𝑗[𝑛]^𝑗_𝑞𝑗𝑚−𝑗+11−𝑚−𝑗𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗+𝑞;𝑧^𝑚−1[𝑛]_𝑞^𝑚−1[𝑛+1]^𝑚_𝑞𝑟^𝑚≤2𝑚𝑞[𝑛]_𝑞+1^𝑚−1+𝑞^𝑚−1[𝑛]_𝑞^𝑚−1[𝑛+1]^𝑚_𝑞𝑟^𝑚≤2𝑚+1[𝑛+1]_𝑞𝑟^𝑚. (3.2) It is known that by a linear transformation, the Bernstein inequality in the closed unit disk becomes 𝑃^′_𝑚(𝑧≤𝑚)𝑃𝑞𝑟^𝑚𝑞𝑟,∀|𝑧|≤𝑞𝑟,𝑟≥1, (3.3) (where ‖𝑃_𝑚‖_𝑞𝑟=max{|𝑃_𝑚(𝑧)|∶|𝑧|≤𝑞𝑟}) which combined with the mean value theorem in complex analysis implies 𝐷_𝑞𝑃_𝑚=𝑃;𝑧^𝑚𝑞𝑧−𝑃_𝑚(𝑧)≤𝑃𝑞𝑧−𝑧^{′𝑚𝑞𝑟}≤𝑚𝑃𝑞𝑟^𝑚𝑞𝑟, (3.4) for all |𝑧|≤𝑟, where 𝑃_𝑚(𝑧) is a complex polynomial of degree ≤𝑚. From the above recurrence formula (3.1), we get 𝐾_𝑛,𝑞𝑒_𝑚;𝑧−𝑒_𝑚(𝑧≤|𝑧|)||1−𝑧[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧+|𝑧|𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+2𝑚+1[𝑛+1]_𝑞𝑟^𝑚≤𝑟(1+𝑟)[𝑛]_𝑞𝑚−1𝐾𝑞𝑟^𝑛,𝑞𝑒_{𝑚−1𝑞𝑟}+𝑟𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+2𝑚+1[𝑛+1]_𝑞𝑟^𝑚≤𝑟𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧+2())𝑚−1[𝑛]_𝑞𝑞^𝑚−1𝑟^𝑚+2𝑚+1[𝑛+1]_𝑞𝑟^𝑚≤𝑟𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+4𝑚[𝑛]_𝑞𝑞^𝑚𝑟^𝑚. (3.5) By writing the last inequality for 𝑚=1,2,…, we easily obtain, step by step, the following: 𝐾_𝑛𝑒_𝑚;𝑧−𝑒_𝑚(𝑧)≤4𝑚[𝑛]_𝑞𝑞^𝑚𝑟^𝑚4()+𝑟𝑚−1[𝑛]_𝑞𝑞^𝑚−1𝑟^𝑚−1+𝑟²4(𝑚−2)[𝑛]_𝑞𝑞^𝑚−2𝑟^𝑚−2+⋯+𝑟^𝑚−14[𝑛]_𝑞=4𝑞𝑟[𝑛]_𝑞𝑞^𝑚𝑟^𝑚()≤()𝑚+𝑚−1+⋯+12𝑚𝑚+1[𝑛]_𝑞𝑞^𝑚𝑟^𝑚. (3.6) Since 𝐾_𝑛,𝑞(𝑓;𝑧) is analytic in 𝔻_𝑅, we can write 𝐾_𝑛,𝑞𝑓;𝑧=^∞_𝑚=0𝑎_𝑚𝐾_𝑛,𝑞𝑒_𝑚;𝑧,𝑧∈𝔻_𝑅, (3.7) which together with (3.6) immediately implies for all |𝑧|≤𝑟𝐾_𝑛,𝑞𝑓;𝑧−𝑓(𝑧)≤^∞_𝑚=0𝑎_𝑚𝐾_𝑛,𝑞𝑒_𝑚;𝑧−𝑒_𝑚(𝑧≤2)[𝑛]_𝑞^∞_𝑚=1𝑐_𝑚𝑚(𝑚+1)𝑞𝑟^𝑚. (3.8)

Proof of Theorem 1.3. A simple calculation and the use of the recurrence formula (2.5) lead us to the following relationship: 𝐸_𝑛,𝑚(𝑧)=𝑧(1−𝑧)[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+𝑧𝐸_{𝑛,𝑚−1}(𝑧)+[𝑚−1]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞𝑧^𝑚−1()+11−𝑧[𝑛+1]_𝑞𝑧^𝑚−𝐵_𝑛,𝑞𝑒_𝑚+1;𝑧[𝑛+1]_𝑞𝑞1−^𝑚−1[𝑛]_𝑞^𝑚−1[𝑛+1]_𝑞^𝑚−1𝐵_𝑛,𝑞𝑒_𝑚+1;𝑧2[𝑛+1]_𝑞𝑞^𝑚−1[𝑛]_𝑞^𝑚−1[𝑛+1]_𝑞^𝑚−1−1𝐵_𝑛,𝑞𝑒_𝑚−1+1;𝑧2[𝑛+1]_𝑞𝐵_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1−(𝑚−1)𝑞^𝑚−2[𝑛]_𝑞^𝑚−22[𝑛+1]^𝑚_𝑞𝐵_𝑛,𝑞𝑒_𝑚−1+1;𝑧[𝑛+1]^𝑚_𝑞^𝑚−2_𝑗=0𝑚𝑗𝑞^𝑗[𝑛]^𝑗_𝑞[𝑛]𝑚−𝑗+1𝑚𝑞_𝑞−𝑗[𝑛+1]_𝑞𝑚𝑞[𝑛]_𝑞𝐵_𝑛,𝑞𝑒_𝑗;𝑧∶=⁹_𝑘=1𝐼_𝑘. (3.9) Firstly, we estimate 𝐼₃, 𝐼₈. It is clear that 𝐼₃≤[𝑚−1]_𝑞[𝑛]_𝑞[𝑛+1]_𝑞𝑟^𝑚−1(),𝐼1+𝑟₈≤(𝑚−1)2[𝑛+1]²_𝑞𝐵_𝑛,𝑞𝑒_𝑚−1≤();𝑧𝑚−12[𝑛+1]²_𝑞𝑟^𝑚−1. (3.10) Secondly, using the known inequality 1−^𝑚_𝑘=1𝑥_𝑘≤^𝑚_𝑘=11−𝑥_𝑘,0≤𝑥_𝑘≤1, (3.11) to estimate 𝐼₅, 𝐼₆, 𝐼₉. 𝐼₅≤1[𝑛+1]_𝑞𝑞1−^𝑚−1[𝑛]_𝑞^𝑚−1[𝑛+1]_𝑞^𝑚−1𝐵_𝑛,𝑞𝑒_𝑚;𝑧≤𝑚−1[𝑛+1]²_𝑞𝑟^𝑚,𝐼₆≤12[𝑛+1]_𝑞𝑞1−^𝑚−1[𝑛]_𝑞^𝑚−1[𝑛+1]_𝑞^𝑚−1𝐵_𝑛,𝑞𝑒_𝑚−1;𝑧≤𝑚−12[𝑛+1]²_𝑞𝑟^𝑚−1,𝐼₉≤1[𝑛+1]^𝑚_𝑞^𝑚−2_𝑗=0𝑗𝑚−2𝑚(𝑚−1)𝑞𝑚−𝑗𝑚−𝑗−1^𝑗[𝑛]^𝑗_𝑞𝑗𝑚−𝑗+11−𝑚−𝑗𝑚𝑞[𝑛]_𝑞𝑟^𝑗≤2𝑚(𝑚−1)[𝑛+1]_𝑞^𝑚−2[𝑛+1]^𝑚_𝑞𝑟^𝑚=2𝑚(𝑚−1)[𝑛+1]²_𝑞𝑟^𝑚. (3.12) Finally, we estimate 𝐼₄, 𝐼₇. We use [2, Theorem 1.1.2] 𝐼₄+𝐼₇≤1[𝑛+1]_𝑞𝑧^𝑚−𝐵_𝑛,𝑞𝑒_𝑚+1;𝑧2[𝑛+1]_𝑞𝐵_𝑛𝑒_𝑚−1;𝑧−𝑧^𝑚−1≤2[𝑚−1]_𝑞(𝑚−1)[𝑛]_𝑞[𝑛+1]_𝑞𝑟^𝑚+[𝑚−2]_𝑞(𝑚−2)[𝑛]_𝑞[𝑛+1]_𝑞𝑟^𝑚−1. (3.13) Using (3.6), (3.10), (3.12), and (3.13) in (3.9) finally we have (𝑚≥3) 𝐸_𝑛,𝑚(𝑧≤𝑟())1+𝑟[𝑛]_𝑞𝐷_𝑞𝐾_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑒_𝑚−1(𝑧)+𝑟𝐸_{𝑛,𝑚−1}(𝑧+[])𝑚−1_𝑞[𝑛]_𝑞[𝑛+1]_𝑞𝑟^𝑚−1()+2[]1+𝑟𝑚−1_𝑞(𝑚−1)[𝑛]_𝑞[𝑛+1]_𝑞𝑟^𝑚+𝑚−1[𝑛+1]²_𝑞𝑟^𝑚+𝑚−12[𝑛+1]²_𝑞𝑟^𝑚−1+[𝑚−2]_𝑞(𝑚−2)[𝑛]_𝑞[𝑛+1]_𝑞𝑟^𝑚−1+(𝑚−1)2[𝑛+1]²_𝑞𝑟^𝑚−1+2𝑚(𝑚−1)[𝑛+1]²_𝑞𝑟^𝑚≤𝑟(1+𝑟)[𝑛]_𝑞𝑚−1𝐾𝑞𝑟^𝑛,𝑞𝑒_𝑚−1−𝑒_{𝑚−1𝑞𝑟}+𝑟𝐸_{𝑛,𝑚−1}(𝑧+[])10𝑚𝑚−1_𝑞[𝑛]²_𝑞𝑟^𝑚≤(𝑚−1)(1+𝑟)[𝑛]_𝑞2(𝑚−1)𝑚[𝑛]_𝑞𝑞^2(𝑚−1)𝑟^𝑚−1+𝑟𝐸_{𝑛,𝑚−1}(𝑧+[])10𝑚𝑚−1_𝑞[𝑛]²_𝑞𝑟^𝑚≤𝑟𝐸_{𝑛,𝑚−1}(𝑧+())4𝑚𝑚−1²[𝑛]²_𝑞𝑞^2𝑚𝑟^𝑚+10𝑚(𝑚−1)[𝑛]²_𝑞𝑞^𝑚𝑟^𝑚≤𝑟𝐸_{𝑛,𝑚−1}(𝑧)+14𝑚²(𝑚−1)²[𝑛]²_𝑞𝑞^2𝑚𝑟^𝑚. (3.14) As a consequence, we get 𝐸_𝑛,𝑚(𝑧)≤14𝑚²(𝑚−1)²[𝑛]²_𝑞𝑞^2𝑚𝑟^𝑚. (3.15) This inequality combined with 𝐾_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓,𝑧≤^∞_𝑚=1𝑎_𝑚𝐸_𝑛,𝑚(𝑧) (3.16) immediately implies the required estimate in statement.
Note that since 𝑓⁽⁴⁾=∑^∞_𝑚=4𝑎_𝑚𝑚(𝑚−1)(𝑚−2)(𝑚−3)𝑧^𝑚−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 Theorem 1.6. For all 𝑧∈𝔻_𝑅 and 𝑛∈ℕ, we get 𝐾_𝑛,𝑞(𝑧)=1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧+[𝑛+1]_𝑞𝐾_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧. (3.17) We apply ‖𝐹+𝐺‖_𝑟≥‖𝐹‖_𝑟−‖𝐺‖_𝑟≥‖𝐹‖_𝑟−‖𝐺‖_𝑟 (3.18) to get 𝐾_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧_𝑟−[𝑛+1]_𝑞𝐾_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧_𝑟. (3.19) Because by hypothesis 𝑓 is not a constant in 𝔻_𝑅, it follows ‖𝐿_𝑞(𝑓;𝑧)‖_𝑟>0. Indeed, assuming the contrary, it follows that 𝐿_𝑞(𝑓;𝑧)=0 for all 𝑧∈𝔻_𝑅 that is ^∞_𝑚=1𝑎_𝑚12−𝑧𝑚𝑧^𝑚−1+^∞_𝑚=1𝑎_𝑚^𝑚−1_𝑗=1𝑗_𝑞𝑧^𝑚−1()11−𝑧=0,2𝑎₁+𝑎₁+^∞_𝑚=112(𝑚+1)𝑎_𝑚+1−𝑎_𝑚+𝑎^𝑚_𝑚+1𝑗=1𝑗_𝑞−𝑎_𝑚^𝑚−1_𝑗=1𝑗_𝑞𝑧^𝑚=0 (3.20) for all 𝑧∈𝔻_𝑅⧵{0}. Thus 𝑎_𝑚=0,  𝑚=1,2,3,…. Thus, 𝑓 is constant, which is contradiction with the hypothesis.
Now, by Theorem 1.3, we have [𝑛+1]_𝑞𝐾_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞≤[]𝑓;𝑧𝑛+1_𝑞[𝑛]_𝑞14[𝑛]_𝑞^∞_𝑚=2𝑎_𝑚𝑚²(𝑚−1)²𝑞^2𝑚𝑟^𝑚→0as𝑛→∞. (3.21) Consequently, there exists 𝑛₁ (depending only on 𝑓 and 𝑟) such that for all 𝑛≥𝑛₁ we have 𝐿_𝑞𝑓;𝑧_𝑟−[𝑛+1]_𝑞𝐾_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧_𝑟≥12𝐿_𝑞𝑓;𝑧_𝑟, (3.22) which implies 𝐾_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞12𝐿_𝑞𝑓;𝑧_𝑟,∀𝑛≥𝑛₁. (3.23) For 1≤𝑛≤𝑛₁−1, we have 𝐾_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞[𝑛+1]_𝑞𝐾_𝑛,𝑞𝑓−𝑓_𝑟=1[𝑛+1]_𝑞𝑀_𝑟,𝑛𝑓>0, (3.24) which finally implies that 𝐾_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞𝐶_𝑟,𝑞𝑓, (3.25) for all 𝑛, with 𝐶_𝑟,𝑞(𝑓)=min{𝑀_𝑟,1(𝑓),…,𝑀_{𝑟,𝑛1−1}(𝑓),(1/2)‖𝐿_𝑞(𝑓;𝑧)‖_𝑟}.

Proof of Theorem 1.5. Proof is similar to that of Theorem 1.3 [5].