Approximation by Genuine 𝑞-Bernstein-Durrmeyer Polynomials in Compact Disks in the Case 𝑞>1

Abstract

This paper deals with approximating properties of the newly defined 𝑞-generalization of the genuine Bernstein-Durrmeyer polynomials in the case 𝑞>1, which are no longer positive linear operators on 𝐶[0,1]. Quantitative estimates of the convergence, the Voronovskaja-type theorem, and saturation of convergence for complex genuine 𝑞-Bernstein-Durrmeyer polynomials attached to analytic functions in compact disks are given. In particular, it is proved that, for functions analytic in {𝑧∈ℂ∶|𝑧|<𝑅}, 𝑅>𝑞, the rate of approximation by the genuine 𝑞-Bernstein-Durrmeyer polynomials (𝑞>1) is of order 𝑞^−𝑛 versus 1/𝑛 for the classical genuine Bernstein-Durrmeyer polynomials. We give explicit formulas of Voronovskaja type for the genuine 𝑞-Bernstein-Durrmeyer for 𝑞>1. This paper represents an answer to the open problem initiated by Gal in (2013, page 115).

1. Introduction

In several recent papers, convergence properties of complex 𝑞-Bernstein polynomials, proposed by Phillips [1], attached to an analytic function 𝑓 in closed disks, were intensively studied. Ostrovska [2, 3] and Wang and Wu [4, 5] have investigated convergence properies of 𝐵_𝑛,𝑞 in the case 𝑞>1. In the case 𝑞>1, the 𝑞-Bernstein polynomials are no longer positive operators; however, for a function analytic in a disc 𝔻_𝑅∶={𝑧∈ℂ∶|𝑧|<𝑅},𝑅>𝑞, it was proved in [2] that the rate of convergence of {𝐵_𝑛,𝑞(𝑓;𝑧)} to 𝑓(𝑧) has the order 𝑞^−𝑛 (versus 1/𝑛 for the classical Bernstein polynomials). Moreover, Ostrovska [3] obtained Voronovskaya-type theorem for monomials. If 𝑞≥1, then qualitative Voronovskaja-type theorem and saturation results for complex 𝑞-Bernstein polynomials were obtained by Wang and Wu [4]. Wu [5] studied saturation of convergence on the interval [0,1] for the 𝑞-Bernstein polynomials of a continuous function 𝑓 for arbitrary fixed 𝑞>1.

Genuine Bernstein-Durrmeyer operators were first considered by Chen [6] and Goodman and Sharma [7] around 1987. In recent years, the genuine Bernstein-Durrmeyer operators have been investigated intensively by a number of authors. Among the many papers written on the genuine Bernstein-Durrmeyer operators, we mention here only the ones by Gonska et al. [8], Parvanov and Popov [9], Sauer [10], Waldron [11], and the book of Păltănea [12].

On the other hand, Gal [13] obtained quantitative estimates of the convergence and of the Voronovskaja-type theorem in compact disks, for the complex genuine Bernstein-Durrmeyer polynomials attached to analytic functions. Besides, in other very recent papers, similar studies were done for complex Bernstein-Durrmeyer operators in Anastassiou and Gal [14], for complex Bernstein-Durrmeyer operators based on Jacobi weights in Gal [15], for complex genuine 𝑞-Bernstein-Durrmeyer operators (0<𝑞<1) by Mahmudov [16], and for other kinds of complex Durrmeyer operators in Mahmudov [17] and Gal et al. [18]. It should be stressed out that study of 𝑞-Durrmeyer-type operators (0<𝑞<1) in the real case was first initiated by Derriennic [19].

Also, for the case 𝑞>1, exact quantitative estimates and quantitative Voronovskaja-type results for complex 𝑞-Lorentz polynomials, 𝑞-Stancu polynomials [20], 𝑞-Stancu-Faber polynomials, 𝑞-Bernstein-Faber polynomials, 𝑞-Kantorovich polynomials [21], 𝑞-Szász-Mirakjan operators [22] obtained by different researchers are collected in the recent book of Gal [23]. In this book the definition and study of complex 𝑞-Durrmeyer-kind operators for 𝑞>1 presented an open problem. This paper presents a positive solution to this problem.

In this paper we define the genuine 𝑞-Bernstein-Durrmeyer polynomials for 𝑞>1. Note that similar to the 𝑞-Bernstein operators the genuine 𝑞-Bernstein-Durrmeyer operators in the case 𝑞>1 are not positive operators on 𝐶[0,1]. The lack of positivity makes the investigation of convergence in the case 𝑞>1 essentially more difficult than that for 0<𝑞<1. We present upper estimates in approximation and we prove the Voronovskaja-type convergence theorem in compact disks in ℂ, centered at origin, with quantitative estimate of this convergence. These results allow us to obtain the exact degrees of approximation by complex genuine 𝑞-Bernstein-Durrmeyer polynomials. Our results show that approximation properties of the complex genuine 𝑞-Bernstein-Durrmeyer polynomials are better than approximation properties of the complex Bernstein-Durrmeyer polynomials considered in [13].

2. Main Results

We begin with some notations and definitions of 𝑞-calculus; see, for example, [24, 25]. Let 𝑞>0. For any 𝑛∈ℕ∪{0}, the 𝑞-integer [𝑛]_𝑞 is defined by [𝑛]_𝑞∶=1+𝑞+⋯+𝑞^𝑛−1,[0]_𝑞∶=0; (1) and the 𝑞-factorial [𝑛]_𝑞! is defined by [𝑛]_𝑞!∶=[1]_𝑞[2]_𝑞⋯[𝑛]_𝑞,[0]_𝑞!∶=1. (2) For integers 0≤𝑘≤𝑛, the 𝑞-binomial is defined by 𝑛𝑘_𝑞[𝑛]∶=^𝑞![𝑘]_𝑞![𝑛−𝑘]_𝑞!. (3) For 𝑞=1 we obviously get [𝑛]_𝑞=𝑛, [𝑛]_𝑞!=𝑛!, and [^𝑛_𝑘]_𝑞=(^𝑛_𝑘). Moreover (1−𝑧)^𝑛_𝑞∶=^𝑛−1_𝑠=01−𝑞^𝑠𝑧,𝑝_𝑛,𝑘𝑛𝑘𝑞;𝑧∶=_𝑞𝑧^𝑘(1−𝑧)_𝑞^𝑛−𝑘,𝑧∈ℂ. (4)

For fixed 𝑞>0, 𝑞≠1, we denote the 𝑞-derivative 𝐷_𝑞𝑓(𝑧) of 𝑓 by 𝐷_𝑞𝑓(𝑧)=𝑓𝑞𝑧−𝑓(𝑧)𝑧𝑓𝑞−1,𝑧≠0,_′(0),𝑧=0. (5)

The 𝑞-analogue of integration in the interval [0,𝐴] (see [24]) is defined by ^𝐴₀𝑓(𝑡)𝑑_𝑞𝑡∶=𝐴1−𝑞^∞_𝑛=0𝑓𝐴𝑞^𝑛𝑞^𝑛,0<𝑞<1. (6) 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 for all 𝑧∈𝔻_𝑅. The norm ‖𝑓‖_𝑟∶=max{|𝑓(𝑧)|∶|𝑧|≤𝑟}. We denote 𝑒_𝑚(𝑧)=𝑧^𝑚 for all 𝑚∈ℕ∪{0}.

Definition 1. For 𝑓∶[0,1]→ℂ, the genuine 𝑞-Bernstein-Durrmeyer operator is defined as follows: 𝑈_𝑛,𝑞𝑓(0)𝑝𝑓;𝑧∶=^𝑛,0𝑞;𝑧+𝑓(1)𝑝_𝑛,𝑛+[]𝑞;𝑧𝑛−1∑_𝑞^𝑛−1_𝑘=1𝑞^1−𝑘𝑝_𝑛,𝑘𝑞;𝑧×∫¹₀𝑝_{𝑛−2,𝑘−1}𝑞;𝑞𝑡𝑓(𝑡)𝑑_𝑞𝑓(0)𝑝𝑡,0<𝑞<1,_𝑛,0(𝑧)+𝑓(1)𝑝_𝑛,𝑛(𝑧)+(𝑛−1)∑^𝑛−1_𝑘=1𝑝_𝑛,𝑘(𝑧)×∫¹₀𝑝_{𝑛−2,𝑘−1}(𝑡)𝑓(𝑡)𝑓(0)𝑝𝑑𝑡,𝑞=1,_𝑛,0𝑞;𝑧+𝑓(1)𝑝_𝑛,𝑛+[]𝑞;𝑧𝑛−1∑_𝑞−1^𝑛−1_𝑘=1𝑞^𝑘−1𝑝_𝑛,𝑘𝑞;𝑧×∫¹₀𝑝_{𝑛−2,𝑘−1}𝑞⁻¹;𝑞⁻¹𝑡𝑓𝑞^𝑘−𝑛𝑡𝑑_𝑞−1𝑡,𝑞>1, (7) where for 𝑛=1 the sum is empty; that is, it is equal to 0.

𝑈_𝑛,𝑞(𝑓;𝑧) are linear operators reproducing linear functions and interpolating every function 𝑓∈𝐶[0,1] at 0 and 1. The genuine 𝑞-Bernstein-Durrmeyer operators are positive operators on 𝐶[0,1] for 0<𝑞≤1, and they are not positive for 𝑞>1. As a consequence, the cases 0<𝑞≤1 and 𝑞>1 are not similar to each other regarding the convergence. For 𝑞→1⁻ and 𝑞→1⁺ we recapture the classical (𝑞=1) genuine Bernstein-Durrmeyer polynomials.

We start with the following quantitative estimates of the convergence for complex 𝑞-Bernstein-Durrmeyer polynomials attached to an analytic function in a disk of radius 𝑅>1 and center 0.

Theorem 2. Let 𝑓∈𝐻(𝔻_𝑅), 1≤𝑟<𝑅/𝑞, and 𝑞>1. Then for all |𝑧|≤𝑟 one has 𝑈_𝑛,𝑞(𝑧≤𝑟()𝑓;𝑧−𝑓)1+𝑟[𝑛+1]_𝑞^∞_𝑚=2𝑎_𝑚𝑚(𝑚−1)𝑞^𝑚−2𝑟^𝑚−2. (8)

Theorem 2 says that, for functions analytic in 𝔻_𝑅, 𝑅>𝑞, the rate of approximation by the genuine 𝑞-Bernstein-Durrmeyer polynomials (𝑞>1) is of order 𝑞^−𝑛 versus 1/𝑛 for the classical genuine Bernstein-Durrmeyer polynomials; see [13].

The Voronovskaja theorem for the real case with a quantitative estimate is obtained by Gonska et al. [26] in the following form: 𝑈_𝑛(𝑥)−𝑥()𝑓;𝑥−𝑓1−𝑥𝑓𝑛+1^′′(𝑧≤𝑥())1−𝑥𝜔𝑓𝑛+1^′′23√,𝑛+3 (9) and, for all 𝑛∈ℕ, 0≤𝑥≤1. For the complex genuine 𝑞-Bernstein-Durrmeyer (0<𝑞≤1) a quantitative estimate is obtained by Gal [13] (𝑞=1) and Mahmudov [16] (0<𝑞<1) in the following form: 𝑈_𝑛,𝑞(𝑧)−𝑧()𝑓;𝑧−𝑓1−𝑧[𝑛+1]_𝑞𝑓^′′(𝑧≤𝑀)^𝑟,𝑓[𝑛]²_𝑞,0<𝑞≤1, (10) and, for all 𝑛∈ℕ, |𝑧|≤𝑟.

To formulate and prove the Voronovskaja-type theorem with a quantitative estimate in the case 𝑞>1 we introduce a function 𝐿_𝑞(𝑓;𝑧).

Let 𝑅>𝑞≥1 and let 𝑓∈𝐻(𝔻_𝑅). For |𝑧|<𝑅/𝑞², we define 𝐿_𝑞()𝑞𝐷𝑓;𝑧∶=1−𝑧_𝑞𝑓(𝑧)−𝐷_𝑞−1𝑓(𝑧)𝑞−1for𝑞>1. (11) And, for 0<𝑞≤1, 𝐿_𝑞𝑓;𝑧=𝐿₁𝑓;𝑧∶=𝑓^′′(𝑧)𝑧(1−𝑧). (12)

The next theorem gives Voronovskaja-type result in compact disks; for complex 𝑞-Bernstein-Durrmeyer polynomials attached to an analytic function in 𝔻_𝑅, 𝑅>𝑞²>1 and center 0 in terms of the function 𝐿_𝑞(𝑓;𝑧).

Theorem 3. Let 𝑓∈𝐻(𝔻_𝑅), 1≤𝑟<𝑅/𝑞², and 𝑞>1. The following Voronovskaja-type result holds: 𝑈_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞≤𝑓;𝑧4𝑟²(1+𝑟)²[𝑛+1]²_𝑞^∞_𝑚=3𝑎_𝑚(𝑚−1)²(𝑚−2)²𝑞²𝑟^𝑚−2. (13) For all 𝑛∈ℕ, |𝑧|≤𝑟.

Now we are in position to prove that the order of approximation in Theorem 2 is exactly 𝑞^−𝑛 versus 1/𝑛 for the classical genuine Bernstein-Durrmeyer polynomials; see [13].

Theorem 4. Let 1<𝑞<𝑅, 1≤𝑟<𝑅/𝑞², and 𝑓∈𝐻(𝔻_𝑅). If 𝑓 is not a polynomial of degree ≤1, the estimate, 𝑈_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞𝐶_𝑟,𝑞𝑓,𝑛∈ℕ, (14) holds, where the constant 𝐶_𝑟,𝑞(𝑓) depends on 𝑓, 𝑞, and 𝑟 but is independent of 𝑛.

From Theorem 3 we conclude that, for 𝑞>1, [𝑛+1]_𝑞(𝑈_𝑛,𝑞(𝑓;𝑧)−𝑓(𝑧))→𝐿_𝑞(𝑓;𝑧) in 𝐻(𝔻_𝑅/𝑞2) and therefore 𝐿_𝑞(𝑓;𝑧)∈𝐻(𝔻_𝑅/𝑞2). Furthermore, we have the following saturation of convergence for the genuine 𝑞-Bernstein-Durrmeyer polynomials for fixed 𝑞>1.

Theorem 5. Let 1<𝑞<𝑅, 1≤𝑟<𝑅/𝑞². If a function 𝑓 is analytic in the disc 𝔻_𝑅/𝑞2, then |𝑈_𝑛,𝑞(𝑓;𝑧)−𝑓(𝑧)|=𝑜(𝑞^−𝑛) for infinite number of points having an accumulation point on 𝔻_𝑅/𝑞2 if and only if 𝑓 is linear.

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

Theorem 6. Let 𝑅>1 and 𝑓∈𝐻(𝔻_𝑅). Then, for any 𝑟,0<𝑟<𝑅, lim_𝑞→1+𝐿_𝑞𝑓;𝑧=𝐿₁𝑓;𝑧 (15) uniformly on 𝔻_𝑅.

3. Auxiliary Results

The 𝑞-analogue of beta function for 0<𝑞<1 (see [24]) is defined as 𝐵_𝑞(𝑚,𝑛)=¹₀𝑡^𝑚−11−𝑞𝑡_𝑞^𝑛−1𝑑_𝑞𝑡,𝑚,𝑛>0,0<𝑞<1. (16) Since we consider the case 𝑞>1, we need to use 𝐵_𝑞−1(𝑚,𝑛) as follows: 𝐵_𝑞−1(𝑚,𝑛)=¹₀𝑡^𝑚−11−𝑞⁻¹𝑡_𝑞^𝑛−1₋₁𝑑_𝑞−1𝑡,𝑚,𝑛>0,0<𝑞⁻¹<1. (17) Also, it is known that 𝐵_𝑞−1()=[]𝑚,𝑛𝑚−1_𝑞−1![𝑛−1]_𝑞−1![𝑚+𝑛−1]_𝑞−1!,0<𝑞⁻¹<1. (18) For 𝑚=0,1,…, we have [𝑛−1]_𝑞−1𝑞^𝑘−11₀𝑡^𝑚𝑝_{𝑛−2,𝑘−1}𝑞⁻¹;𝑞⁻¹𝑡𝑑_𝑞−1𝑡=[𝑛−1]_𝑞−1𝑛−2𝑘−1_𝑞−1𝑞^{𝑚(𝑘−𝑛)}×¹₀𝑡^{𝑘+𝑚−1}1−𝑞⁻¹𝑡_𝑞^{𝑛−𝑘−1}₋₁𝑑_𝑞−1𝑡=𝑞^{𝑚(𝑘−𝑛)}[𝑛−1]_𝑞−1![𝑘−1]_𝑞−1![𝑛−𝑘−1]_𝑞−1!𝐵_𝑞−1(𝑘+𝑚,𝑛−𝑘)=𝑞^{𝑚(𝑘−𝑛)}[𝑛−1]_𝑞−1![𝑘−1]_𝑞−1![𝑛−𝑘−1]_𝑞−1!×[𝑘+𝑚−1]_𝑞−1![𝑛−𝑘−1]_𝑞−1![𝑘+𝑚+𝑛−𝑘−1]_𝑞−1!=[𝑛−1]_𝑞![𝑘+𝑚−1]_𝑞![𝑘−1]_𝑞![𝑛+𝑚−1]_𝑞!=[𝑘+𝑚−1]_𝑞⋯[𝑘]_𝑞[𝑛+𝑚−1]_𝑞⋯[𝑛]_𝑞. (19) Thus, we get the following formula for 𝑈_𝑛,𝑞(𝑒_𝑚;𝑧): 𝑈_𝑛,𝑞𝑒_𝑚;𝑧=𝑓(0)𝑝_𝑛,0𝑞;𝑧+𝑓(1)𝑝_𝑛,𝑛+[]𝑞;𝑧𝑛−1_𝑞−1^𝑛−1_𝑘=1𝑝_𝑛,𝑘×𝑞;𝑧₁₀𝑝_{𝑛−2,𝑘−1}𝑞⁻¹;𝑞⁻¹𝑡𝑓𝑞^𝑘−𝑛𝑡𝑑_𝑞−1𝑡=𝑧^𝑛+^𝑛−1_𝑘=1𝑝_𝑛,𝑘[]𝑞;𝑧𝑘+𝑚−1_𝑞⋯[𝑘]_𝑞[𝑛+𝑚−1]_𝑞⋯[𝑛]_𝑞. (20) Note that, for 𝑚=0,1,2, we have 𝑈_𝑛,𝑞𝑒₀;𝑧=1,𝑈_𝑛,𝑞𝑒₁𝑈;𝑧=𝑧,_𝑛,𝑞𝑒₂;𝑧=𝑧²+1+𝑞𝑧(1−𝑧)[].𝑛+1 (21)

Lemma 7. 𝑈_𝑛,𝑞(𝑒_𝑚;𝑧) is a polynomial of degree less than or equal to min(𝑚,𝑛) and 𝑈_𝑛,𝑞𝑒_𝑚=[];𝑧𝑛−1_𝑞![𝑛+𝑚−1]_𝑞!^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑛]^𝑠_𝑞𝐵_𝑛,𝑞𝑒_𝑠;𝑧. (22)

Proof. From (20) it follows that 𝑈_𝑛,𝑞𝑒_𝑚=;𝑧_𝑛𝑘=1𝑝_𝑛,𝑘[]𝑞;𝑧𝑘+𝑚−1_𝑞⋯[𝑘]_𝑞[𝑛+𝑚−1]_𝑞⋯[𝑛]_𝑞=[𝑛−1]_𝑞![𝑛+𝑚−1]_𝑞!^𝑛_𝑘=1[𝑘]_𝑞[𝑘+1]_𝑞⋯[𝑘+𝑚−1]_𝑞𝑝_𝑛,𝑘𝑞;𝑧. (23) Now using [𝑘]_𝑞[𝑘+1]_𝑞⋯[𝑘+𝑚−1]_𝑞=^𝑚−1_𝑠=0𝑞^𝑠[𝑘]_𝑞+[𝑠]_𝑞=^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑘]^𝑠_𝑞, (24) where 𝑆_𝑞(𝑚,𝑠)>0, 𝑠=1,2,…,𝑚, are the constants independent of 𝑘, we get 𝑈_𝑛,𝑞𝑒_𝑚=[];𝑧𝑛−1_𝑞![𝑛+𝑚−1]_𝑞!^𝑛𝑚_𝑘=0𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑘]^𝑠_𝑞𝑝_𝑛,𝑘=[]𝑞;𝑧𝑛−1_𝑞![𝑛+𝑚−1]_𝑞!^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑛]^𝑠_𝑞𝐵_𝑛,𝑞𝑒_𝑠;𝑧. (25) Since 𝐵_𝑛,𝑞(𝑒_𝑠;𝑧) is a polynomial of degree less than or equal to min(𝑠,𝑛) and 𝑆_𝑞(𝑚,𝑠)>0, 𝑠=1,2,…,𝑚, it follows that 𝑈_𝑛,𝑞(𝑒_𝑚;𝑧) is a polynomial of degree less than or equal to min(𝑚,𝑛).

Lemma 8. The numbers 𝑆_𝑞(𝑚,𝑠),(𝑚,𝑠)∈(ℕ∪{0})×(ℕ∪{0}), given by (24), enjoy the following properties: 𝑆_𝑞(0,0)=1,𝑆_𝑞()𝑆𝑚,0=0,𝑚∈𝑁,_𝑞(𝑚+1,𝑠)=[𝑚]_𝑞𝑆_𝑞(𝑚,𝑠)+𝑞^𝑚𝑆_𝑞(𝑚,𝑠−1),𝑚∈𝑁₀𝑆,𝑠∈𝑁,_𝑞(𝑚+1,𝑚+1)=𝑞^𝑚𝑆_𝑞(),𝑆𝑚,𝑚_𝑞(𝑚,𝑠)=0𝑓𝑜𝑟𝑠>𝑚. (26)

Also, the following lemma holds.

Lemma 9. For all 𝑚,𝑛∈ℕ the identity, [𝑛−1]_𝑞![𝑛+𝑚−1]_𝑞!^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑛]^𝑠_𝑞=1, (27) holds.

Proof. It follows from end points interpolation property of 𝑈_𝑛,𝑞(𝑒_𝑚;𝑧) and 𝐵_𝑛,𝑞(𝑒_𝑠;𝑧). Indeed 1=𝑈_𝑛,𝑞𝑒_𝑚=[];1𝑛−1_𝑞![𝑛+𝑚−1]_𝑞!^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑛]^𝑠_𝑞𝐵_𝑛,𝑞𝑒_𝑠=[];1𝑛−1_𝑞![𝑛+𝑚−1]_𝑞!^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑛]^𝑠_𝑞. (28)

Lemma 9 implies that for all 𝑚,𝑛∈ℕ and |𝑧|≤𝑟 we have 𝑈_𝑛,𝑞𝑒_𝑚≤[];𝑧𝑛−1_𝑞![𝑛+𝑚−1]_𝑞!^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑛]^𝑠_𝑞𝐵_𝑛,𝑞𝑒_𝑠≤[];𝑧𝑛−1_𝑞![𝑛+𝑚−1]_𝑞!^𝑚_𝑠=1𝑆_𝑞(𝑚,𝑠)[𝑛]^𝑠_𝑞𝑟^𝑠≤𝑟^𝑚. (29)

For our purpose first we need a recurrence formula for 𝑈_𝑛,𝑞(𝑒_𝑚;𝑧).

Lemma 10. For all 𝑚,𝑛∈ℕ∪{0} and 𝑧∈ℂ one has 𝑈_𝑛,𝑞𝑒_𝑚+1=𝑞;𝑧^𝑚𝑧(1−𝑧)[𝑛+𝑚]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚+𝑞;𝑧_𝑚[𝑛]𝑧+[𝑚]_𝑞[𝑛+𝑚]_𝑞𝑈_𝑛,𝑞𝑒_𝑚;𝑧. (30)

Proof. By simple calculation we obtain (see [27]) 𝑧(1−𝑧)𝐷_𝑞𝑝_𝑛,𝑘𝑞;𝑧=[𝑘]_𝑞−[𝑛]_𝑞𝑧𝑝_𝑛,𝑘,𝑈𝑞;𝑧_𝑛,𝑞𝑒_𝑚;𝑧=𝑧^𝑛+^𝑛−1_𝑘=1𝑝_𝑛,𝑘[]𝑞;𝑧𝑘+𝑚−1_𝑞⋯[𝑘]_𝑞[𝑛+𝑚−1]_𝑞⋯[𝑛]_𝑞=𝑧^𝑛+^𝑛−1_𝑘=1𝑝_𝑛,𝑘𝑞;𝑧𝐼_𝑘,𝑚,𝐼_𝑘,𝑚[]∶=𝑘+𝑚−1_𝑞⋯[𝑘]_𝑞[𝑛+𝑚−1]_𝑞⋯[𝑛]_𝑞. (31) It follows that 𝑧(1−𝑧)𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚=[𝑛];𝑧_𝑞𝑧(1−𝑧)𝑧^𝑛−1+^𝑛−1_𝑘=1[𝑘]_𝑞−[𝑛]_𝑞𝑧𝑝_𝑛,𝑘𝑞;𝑧𝐼_𝑘,𝑚=[𝑛]_𝑞𝑧^𝑛+^𝑛−1_𝑘=1[𝑘]_𝑞𝑝_𝑛,𝑘𝑞;𝑧𝐼_𝑘,𝑚−[𝑛]_𝑞𝑧^𝑛−1_𝑘=1𝑝_𝑛,𝑘𝑞;𝑧𝐼_𝑘,𝑚−[𝑛]_𝑞𝑧^𝑛+1=[𝑛]_𝑞𝑧^𝑛+^𝑛−1_𝑘=1[𝑘]_𝑞𝑝_𝑛,𝑘𝑞;𝑧𝐼_𝑘,𝑚−𝑧[𝑛]_𝑞𝑈_𝑛,𝑞𝑒_𝑚=[𝑛];𝑧_𝑞𝑧^𝑛+𝑞^{−𝑚𝑛−1}_𝑘=1𝑝_𝑛,𝑘𝑞;𝑧𝑞^𝑚[𝑘]_𝑞+[𝑚]_𝑞−[𝑚]_𝑞𝐼_𝑘,𝑚−𝑧[𝑛]_𝑞𝑈_𝑛,𝑞𝑒_𝑚=[𝑛];𝑧_𝑞𝑧^𝑛+𝑞^{−𝑚𝑛−1}_𝑘=1𝑝_𝑛,𝑘𝑞;𝑧𝑞^𝑚[𝑘]_𝑞+[𝑚]_𝑞−[𝑚]_𝑞𝐼_𝑘,𝑚−𝑧[𝑛]_𝑞𝑈_𝑛,𝑞𝑒_𝑚;𝑧=𝑞^−𝑚𝑞^𝑚[𝑛]_𝑞+[𝑚]_𝑞−[𝑚]_𝑞𝑧^𝑛+𝑞^−𝑚[𝑛+𝑚]_𝑞^𝑛−1_𝑘=1𝑝_𝑛,𝑘𝑞;𝑧𝐼_𝑘,𝑚+1−𝑞^−𝑚[𝑚]_𝑞^𝑛−1_𝑘=1𝑝_𝑛,𝑘𝑞;𝑧𝐼_𝑘,𝑚−𝑧[𝑛]_𝑞𝑈_𝑛,𝑞𝑒_𝑚;𝑧=𝑞^−𝑚[𝑛+𝑚]_𝑞𝑈_𝑛,𝑞𝑒_𝑚+1;𝑧−𝑞^−𝑚[𝑚]_𝑞𝑈_𝑛,𝑞𝑒_𝑚[𝑛];𝑧−𝑧_𝑞𝑈_𝑛,𝑞𝑒_𝑚;𝑧, (32) which implies the recurrence in the statement.

Let Θ_𝑛,𝑚𝑞;𝑧∶=𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚−1[𝑛+1]_𝑞×𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−1(1−𝑧). (33) Using the recurrence formula (30) we prove two more recurrence formulas.

Lemma 11. For all 𝑚,𝑛∈ℕ and 𝑧∈ℂ one has 𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚=𝑞^𝑚−1𝑧(1−𝑧)[𝑛+𝑚−1]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1+𝑞;𝑧_𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞(1−𝑧)𝑧^𝑚−1,Θ_𝑛,𝑚=𝑞𝑞;𝑧_𝑚−1𝑧(1−𝑧)[𝑛+𝑚−1]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+𝑞^𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞Θ_{𝑛,𝑚−1}𝑞;𝑧+𝑅_𝑛,𝑚𝑞;𝑧, (34)(35) where 𝑅_𝑛,𝑚=[]𝑞;𝑧𝑚−1_𝑞[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞×1+𝑞^𝑚−1+𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1(𝑧+1)×𝑧^𝑚−2(1−𝑧). (36)

Proof. From the recurrence formula in Lemma 10, for all 𝑚≥2, we get 𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚=𝑞^𝑚−1𝑧(1−𝑧)[𝑛+𝑚−1]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1+𝑞;𝑧_𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+𝑞^𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞𝑧^𝑚−1−𝑧^𝑚=𝑞^𝑚−1𝑧(1−𝑧)[𝑛+𝑚−1]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1+𝑞;𝑧_𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞(1−𝑧)𝑧^𝑚−1,𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚−1[𝑛+1]_𝑞𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−1()=𝑞1−𝑧_𝑚−1𝑧(1−𝑧)[𝑛+𝑚−1]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+𝑞^𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞×𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚−1−1[𝑛+1]_𝑞×𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−2(1−𝑧)+𝑅_𝑛,𝑚𝑞;𝑧, (37) where 𝑅_𝑛,𝑚=[]𝑞;𝑧𝑚−1_𝑞[𝑛+𝑚−1]_𝑞(1−𝑧)𝑧^𝑚−1−1[𝑛+1]_𝑞𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−1()+𝑞1−𝑧_𝑚−1[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞(1−𝑧)𝑧^𝑚−1+𝑞^𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞1[𝑛+1]_𝑞×𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−2(1−𝑧)∶=𝑇_𝑛′𝑚𝑞𝑧^𝑚−1()+[]1−𝑧𝑚−1_𝑞[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞×𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−2(1−𝑧). (38) Again by simple calculation we obtain 𝑇_𝑛,𝑚𝑞=[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞−1[𝑛+1]_𝑞𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1+𝑞^𝑚−1[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞+𝑞^𝑚−1[𝑛]_𝑞[𝑛+𝑚−1]_𝑞1[𝑛+1]_𝑞×𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1−𝑞^𝑚−1[𝑛]_𝑞[𝑛+𝑚−1]_𝑞1[𝑛+1]_𝑞𝑞[𝑚−1]_𝑞+[𝑚−1]_𝑞−1=[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞+𝑞^𝑚−1[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞−𝑞^𝑚−1[𝑛]_𝑞[𝑛+𝑚−1]_𝑞×1[𝑛+1]_𝑞𝑞[𝑚−1]_𝑞+[𝑚−1]_𝑞−1+𝑞^𝑚−1[𝑛]_𝑞𝑞^𝑚−1[𝑛]_𝑞+[𝑚−1]_𝑞1−1[𝑛+1]_𝑞×𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1∶=𝑇¹_𝑛,𝑚𝑞+𝑇²_𝑛,𝑚𝑞, (39) where 𝑇¹_𝑛,𝑚(𝑞) and 𝑇²_𝑛,𝑚(𝑞) can be simplified as follows: 𝑇²_𝑛,𝑚𝑞=𝑞1−^𝑚−1[𝑛]_𝑞[𝑛+𝑚−1]_𝑞1[𝑛+1]_𝑞×𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1=[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1,𝑇¹_𝑛,𝑚𝑞=[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞+𝑞^𝑚−1[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞−𝑞^𝑚−1[𝑛]_𝑞[𝑛+𝑚−1]_𝑞1[𝑛+1]_𝑞×𝑞[𝑚−1]_𝑞+[𝑚−1]_𝑞−1=[𝑚−1]_𝑞1[𝑛+𝑚−1]_𝑞−𝑞[𝑛+1]_𝑞𝑞^𝑚−1[𝑛]_𝑞[𝑛+𝑚−1]_𝑞+[𝑚−1]_𝑞𝑞^𝑚−1[𝑛+𝑚−1]_𝑞−1[𝑛+1]_𝑞𝑞[𝑛]_𝑞[𝑛+𝑚−1]_𝑞=[𝑚−1]_𝑞[𝑛+1]_𝑞−𝑞^𝑚[𝑛]_𝑞[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞+[𝑚−1]_𝑞𝑞^𝑚−1[𝑛+1]_𝑞−𝑞[𝑛]_𝑞[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞=[𝑚−1]_𝑞1+𝑞^𝑚−1[𝑛+1]_𝑞−1+𝑞^𝑚−1𝑞[𝑛]_𝑞[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞=[𝑚−1]_𝑞1+𝑞^𝑚−1[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞. (40)

Lemma 12. Let 𝑞>1 and 𝑓∈𝐻(𝔻_𝑅). The function 𝐿_𝑞(𝑓;𝑧) has the following representation: 𝐿_𝑞𝑓;𝑧=^∞_𝑚=2𝑎_𝑚𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−1(1−𝑧),𝑧∈𝔻_𝑅. (41)

Proof. Using the following identity: [𝑚]_𝑞−𝑚=1+𝑞+𝑞²+⋯+𝑞^𝑚−1=()++𝑞−𝑚1−1𝑞−1²−1+⋯+𝑞^𝑚−1=[1]−1𝑞−1_𝑞+𝑞−1[2]_𝑞+⋯+𝑞−1[𝑚−1]_𝑞=𝑞−1[1]_𝑞+⋯+[𝑚−1]_𝑞=𝑞−1^𝑚−1_𝑖=1[𝑖]_𝑞, (42) we get 𝐿_𝑞=𝑓;𝑧_∞𝑚=2𝑎_𝑚𝑞[𝑚]_𝑞−[𝑚]_𝑞−1𝑧𝑞−1^𝑚−1()=1−𝑧_∞𝑚=2𝑎_𝑚𝑞[𝑚]_𝑞−𝑚+[𝑚]𝑞−1^𝑞−1−𝑚𝑞⁻¹𝑧−1^𝑚−1()=1−𝑧_∞𝑚=2𝑎_𝑚𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−1(1−𝑧), (43) where 𝑓(𝑧)=∑^∞_𝑚=0𝑎_𝑚𝑧^𝑚.

4. Proofs of the Main Results

Firstly we prove that 𝑈_𝑛,𝑞(𝑓;𝑧)=∑^∞_𝑚=0𝑎_𝑚𝑈_𝑛,𝑞(𝑒_𝑚,𝑧). Indeed denoting 𝑓_𝑘(𝑧)=∑^𝑘_𝑗=0𝑎_𝑗𝑧^𝑗,|𝑧|≤𝑟 with 𝑚∈ℕ, by the linearity of 𝑈_𝑛,𝑞, we have 𝑈_𝑛,𝑞𝑓_𝑘,𝑧=^𝑘_𝑚=0𝑎_𝑚𝑈_𝑛,𝑞𝑒_𝑚,𝑧, (44) and it is sufficient to show that, for any fixed 𝑛∈ℕ and |𝑧|≤𝑟 with 𝑟≥1, we have lim_𝑘→∞𝑈_𝑛,𝑞(𝑓_𝑘,𝑧)=𝑈_𝑛,𝑞(𝑓;𝑧). But this is immediate from lim_𝑘→∞‖𝑓_𝑘−𝑓‖_𝑟=0, the norm being defined as ‖𝑓‖_𝑟=max{|𝑓(𝑧)|∶|𝑧|≤𝑟}, and from the inequality 𝑈_𝑛,𝑞𝑓_𝑘,𝑧−𝑈_𝑛,𝑞≤𝑓𝑓,𝑧_𝑘(0)−𝑓(0)⋅(1−𝑧)^𝑛+𝑓_𝑘(1)−𝑓(1)⋅𝑧^𝑛+[𝑛+1]_𝑞−1^𝑛−1_𝑗=1𝑝_𝑛,𝑗𝑞;𝑧𝑞^𝑗−1×¹₀𝑝_{𝑛−2,𝑗−1}𝑞⁻¹,𝑞⁻¹𝑡𝑓_𝑘(𝑡)−𝑓(𝑡)𝑑_𝑞−1𝑡≤𝐶_𝑟,𝑛𝑓_𝑘−𝑓_𝑟, (45) valid for all |𝑧|≤𝑟, where 𝐶_𝑟,𝑛=(1+𝑟)^𝑛+𝑟^𝑛+[𝑛+1]_𝑞−1×^𝑛−1_𝑗=1𝑛𝑗_𝑞(1+𝑟)^𝑛−𝑗𝑟^𝑗𝑞^𝑗−1×¹₀𝑝_{𝑛−2,𝑗−1}𝑞⁻¹;𝑞⁻¹𝑡𝑑_𝑞−1𝑡=(1+𝑟)^𝑛+𝑟^𝑛+^𝑛−1_𝑗=1𝑛𝑗_𝑞1+𝑞^𝑛−𝑗𝑟^𝑛−𝑗𝑟^𝑗𝑞^𝑗−1. (46) Therefore we get 𝑈_𝑛,𝑞(𝑧≤𝑓;𝑧−𝑓)_∞𝑚=0𝑎_𝑚𝑈_𝑛,𝑞𝑒_𝑚,𝑧−𝑒_𝑚(𝑧)=^∞_𝑚=2𝑎_𝑚×𝑈_𝑛,𝑞𝑒_𝑚,𝑧−𝑒_𝑚(𝑧), (47) as 𝑈_𝑛,𝑞(𝑒₀,𝑧)=𝑒₀(𝑧) and 𝑈_𝑛,𝑞(𝑒₁,𝑧)=𝑒₁(𝑧).

Proof of Theorem 2. From the recurrence formula (34) and the inequality (29) for 𝑚≥2 we get 𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚≤𝑞^𝑚−1𝑧(1−𝑧)𝑞^𝑚−2[𝑛+1]_𝑞+[𝑚−2]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1+𝑞;𝑧_𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞𝑞^𝑚−1[𝑛]_𝑞+[𝑚−1]_𝑞×𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+[𝑚−1]_𝑞𝑞^𝑚−2[𝑛+1]_𝑞+[𝑚−2]_𝑞|1−𝑧||𝑧|^𝑚−1. (48) It is known that, by a linear transformation, the Bernstein inequality in the closed unit disk becomes 𝑃^′_𝑘(𝑧≤𝑘)𝑞𝑟₁𝑃_𝑘𝑞𝑟,∀|𝑧|≤𝑞𝑟,𝑟≥1, (49) which, combined with the mean value theorem in complex analysis, implies 𝐷_𝑞𝑃_𝑘;𝑧≤𝑃^′_𝑘𝑞𝑟≤𝑘𝑃𝑞𝑟^𝑘𝑞𝑟, (50) for all |𝑧|≤𝑞𝑟, where 𝑃_𝑘(𝑧) is a complex polynomial of degree ≤𝑘. It follows that 𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚≤𝑞^𝑚−1𝑟(1+𝑟)𝑞^𝑚−2[𝑛+1]_𝑞+[𝑚−2]_𝑞𝑚−1𝑈𝑞𝑟^𝑛,𝑞𝑒_{𝑚−1𝑞𝑟}+𝑟𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+[𝑚−1]_1/𝑞[𝑛+1]_𝑞(1+𝑟)𝑟^𝑚−1≤(𝑚−1)[𝑛+1]_𝑞(1+𝑟)𝑞^𝑚−1𝑟^𝑚−1+𝑟𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+[𝑚−1]_1/𝑞[𝑛+1]_𝑞(1+𝑟)𝑟^𝑚−1()𝑟()≤2𝑞𝑚−11+𝑟[𝑛+1]_𝑞𝑞𝑟^𝑚−2+𝑟𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1. (51) By writing the last inequality for 𝑚=2,3,…, we easily obtain, step by step, the following: 𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚≤𝑟𝑟𝑈_𝑛,𝑞𝑒_𝑚−2;𝑧−𝑧^𝑚−2()+2𝑚−2[𝑛+1]_𝑞𝑟(1+𝑟)𝑞𝑟^𝑚−3()+2𝑚−1[𝑛+1]_𝑞𝑟(1+𝑟)𝑞𝑟^𝑚−2=𝑟²𝑈_𝑛,𝑞𝑒_𝑚−2;𝑧−𝑧^𝑚−2𝑟()+21+𝑟[𝑛+1]_𝑞𝑟^𝑚−2()𝑟()𝑚−1+𝑚−2≤⋯≤1+𝑟[𝑛+1]_𝑞𝑚(𝑚−1)𝑞^𝑚−2𝑟^𝑚−2. (52) It follows that 𝑈_𝑛,𝑞𝑓;𝑧−𝑓(𝑧)≤^∞_𝑚=2𝑎_𝑚𝑈_𝑛,𝑞𝑒_𝑚;𝑧−𝑧^𝑚≤𝑟(1+𝑟)[𝑛+1]_𝑞^∞_𝑚=2𝑎_𝑚𝑚(𝑚−1)𝑞^𝑚−2𝑟^𝑚−2. (53)

The second main result of the paper is the Voronovskaja-type theorem with a quantitative estimate for the complex version of genuine 𝑞-Bernstein-Durrmeyer polynomials.

Proof of Theorem 3. By Lemma 11 we have Θ_𝑛,𝑚=𝑞𝑞;𝑧_𝑚−1𝑧(1−𝑧)[𝑛+𝑚−1]_𝑞𝐷_𝑞𝑈_𝑛,𝑞𝑒_𝑚−1;𝑧−𝑧^𝑚−1+𝑞^𝑚−1[𝑛]𝑧+[𝑚−1]_𝑞[𝑛+𝑚−1]_𝑞Θ_{𝑛,𝑚−1}𝑞;𝑧+𝑅_𝑛,𝑚𝑞;𝑧, (54) where 𝑅_𝑛,𝑚=[]𝑞;𝑧𝑚−1_𝑞[𝑛+𝑚−1]_𝑞[𝑛+1]_𝑞×1+𝑞^𝑚−1+𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1(𝑧+1)×𝑧^𝑚−2(1−𝑧). (55) It follows that 𝑅_𝑛,𝑚≤[]𝑞;𝑧𝑚−1_𝑞[𝑛+1]²_𝑞×1+𝑞^𝑚−1𝑟+𝑞^𝑚−2_𝑖=1[𝑖]_𝑞+^𝑚−2_𝑖=1[𝑖]_𝑞−1()×()𝑟1+𝑟1+𝑟^𝑚−2≤[𝑚−1]_𝑞[𝑛+1]²_𝑞×1+𝑞^𝑚−1+𝑞(𝑚−2)[𝑚−2]_𝑞+(𝑚−2)²×(1+𝑟)²𝑟^𝑚−2=𝑞^𝑚−2[𝑚−1]_𝑞−1[𝑛+1]²_𝑞𝑞^𝑚−2×1𝑞^𝑚−2+𝑞+(𝑚−2)[𝑚−2]_𝑞−1+1𝑞^𝑚−2(𝑚−2)²×(1+𝑟)²𝑟^𝑚−2≤3[𝑛+1]²_𝑞(𝑚−1)(𝑚−2)²(1+𝑟)²𝑞²𝑟^𝑚−2 (56) for all 𝑚≥2, 𝑛∈ℕ, and 𝑧∈ℂ. Equation (54) implies that for |𝑧|≤𝑟Θ_𝑛,𝑚Θ𝑞;𝑧≤𝑟_{𝑛,𝑚−1}+𝑞𝑞;𝑧^𝑚−1𝑟(1+𝑟)𝑞^𝑚−2[𝑛+1]_𝑞𝑚−1×𝑈𝑞𝑟_𝑛,𝑞𝑒_𝑚−1−𝑒_{𝑚−1𝑞𝑟}+3[𝑛+1]²_𝑞(𝑚−1)(𝑚−2)²(1+𝑟)²𝑞²𝑟^𝑚−2≤𝑟Θ_{𝑛,𝑚−1}+𝑟𝑞;𝑧²(1+𝑟)²[𝑛+1]²_𝑞×(𝑚−1)²(𝑚−2)𝑞²𝑟^𝑚−3+3[𝑛+1]²_𝑞(𝑚−1)(𝑚−2)²(1+𝑟)²𝑞²𝑟^𝑚−2≤𝑟Θ_{𝑛,𝑚−1}𝑞;𝑧+4𝑟²(1+𝑟)²[𝑛+1]²_𝑞×(𝑚−1)²(𝑚−2)𝑞²𝑟^𝑚−2. (57) By writing the last inequality for 𝑚=3,4,…, we easily obtain, step by step, the following: 𝑈_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞≤𝑓;𝑧4𝑟²(1+𝑟)²[𝑛+1]²_𝑞^∞_𝑚=2𝑎_𝑚𝑞²𝑟^𝑚−2×^𝑚_𝑗=2𝑗−1²𝑗−2≤4𝑟²(1+𝑟)²[𝑛+1]²_𝑞×^∞_𝑚=2𝑎_𝑚(𝑚−1)²(𝑚−2)²𝑞²𝑟^𝑚−2. (58)

Proof of Theorem 4. For all 𝑧∈𝔻_𝑅 and 𝑛∈ℕ we get 𝑈_𝑛,𝑞(𝑧)=1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧+[𝑛+1]_𝑞×𝑈_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧. (59) It follows that 𝑈_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧_𝑟−[𝑛+1]_𝑞×𝑈_𝑛,𝑞𝑓1−𝑓−[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧_𝑟. (60) Because by hypothesis 𝑓 is not a polynomial of degree ≤1 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, 𝐷_𝑞𝑓(𝑧)=𝐷_𝑞−1𝑓(𝑧) for all 𝑧∈𝔻_𝑟. Thus 𝑎_𝑚=0,𝑚=2,3,… and 𝑓 is linear, which is a contradiction with the hypothesis.
Now, by Theorem 3, we have [𝑛+1]_𝑞𝑈_𝑛,𝑞(𝑧)−1𝑓;𝑧−𝑓[𝑛+1]_𝑞𝐿_𝑞≤𝑓;𝑧4𝑟²(1+𝑟)²[𝑛+1]_𝑞^∞_𝑚=3𝑎_𝑚(𝑚−1)²(𝑚−2)²𝑞²𝑟^𝑚−2⟶0as𝑛⟶∞. (61) Consequently, there exists 𝑛₁ (depending only on 𝑓 and 𝑟) such that for all 𝑛≥𝑛₁ we have 𝐿_𝑞𝑓;𝑧_𝑟−[𝑛+1]_𝑞𝑈_𝑛,𝑞𝑓1−𝑓−[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧_𝑟≥12𝐿_𝑞𝑓;𝑧_𝑟, (62) which implies that 𝑈_𝑛,𝑞𝑓−𝑓_𝑟≥12[𝑛+1]_𝑞𝐿_𝑞𝑓;𝑧_𝑟,∀𝑛≥𝑛₁. (63) For 1≤𝑛≤𝑛₁−1 we have 𝑈_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞[𝑛+1]_𝑞𝑈_𝑛,𝑞𝑓−𝑓_𝑟=1[𝑛+1]_𝑞𝑀_{𝑟,𝑛,𝑞}𝑓>0, (64) which finally implies that 𝑈_𝑛,𝑞𝑓−𝑓_𝑟≥1[𝑛+1]_𝑞𝐶_𝑟,𝑞𝑓, (65) for all 𝑛, with 𝐶_𝑟,𝑞(𝑓)=min{𝑀_𝑟,1,𝑞(𝑓),…,𝑀_{𝑟,𝑛1−1,𝑞}(𝑓),(1/2)‖𝐿_𝑞(𝑓;𝑧)‖_𝑟}, which ends the proof.

Proof of Theorem 6. Let 1≤𝑟<𝑅, 1<𝑞₀<𝑅/𝑟 be fixed. Then, by Lemma 12 for any 1≤𝑞≤𝑞₀ and |𝑧|≤𝑟, we have 𝐿_𝑞𝑓;𝑧=^∞_𝑚=2𝑎_𝑚𝑞^𝑚−1_𝑖=1[𝑖]_𝑞+^𝑚−1_𝑖=1[𝑖]_𝑞−1𝑧^𝑚−1(),𝐿1−𝑧₁𝑓;𝑧=^∞_𝑚=2𝑎_𝑚𝑚(𝑚−1)𝑧^𝑚−1(1−𝑧). (66) Using the inequality 𝑞^𝑚−1_𝑖=1[𝑖]_𝑞−𝑚(𝑚−1)2=𝑞^𝑚−1_𝑖=2[𝑖]_𝑞+𝑚()−𝑖𝑞−1𝑚−12=𝑞𝑞−1^𝑚−1𝑖_𝑖=2𝑗=1𝑗_𝑞+𝑚()𝑞−1𝑚−12≤𝑞𝑞−1[𝑚−1]_𝑞𝑚(𝑚−1)2+𝑚()𝑞−1𝑚−12=𝑚()𝑞−1𝑚−12𝑞[𝑚−1]_𝑞≤𝑞+1𝑞−1^𝑚−1𝑚²(𝑚−1)2_{𝑚−1𝑖=1},[𝑖]_𝑞−1−𝑚(𝑚−1)2=^𝑚−1_𝑖=2𝑖−[𝑖]_𝑞−1=1−𝑞^{−1𝑚−1𝑖}_𝑖=2𝑗=1𝑗_𝑞−1≤1−𝑞⁻¹𝑚(𝑚−1)²2, (67) we get, for 1≤𝑞≤𝑞₀ and |𝑧|≤𝑟, 𝐿_𝑞𝑓;𝑧−𝐿₁≤𝑓;𝑧_{𝑁−1𝑚=2}𝑎_𝑚𝑞^𝑚−1_𝑖=1[𝑖]_𝑞−𝑚(𝑚−1)2𝑧^𝑚−1−𝑧^𝑚+^∞_𝑚=𝑁𝑎_𝑚𝑞^𝑚−1_𝑖=1[𝑖]_𝑞−𝑚(𝑚−1)2𝑧^𝑚−1−𝑧^𝑚+^𝑁−1_𝑚=2𝑎_𝑚^𝑚−1_𝑖=1[𝑖]_𝑞−1−𝑚(𝑚−1)2𝑧^𝑚−1−𝑧^𝑚+^∞_𝑚=𝑁𝑎_𝑚^𝑚−1_𝑖=1[𝑖]_𝑞−1−𝑚(𝑚−1)2𝑧^𝑚−1−𝑧^𝑚≤𝑞−1^𝑁−1_𝑚=2𝑎_𝑚𝑚²(𝑚−1)𝑞₀^𝑚−1𝑟^𝑚+4^∞_𝑚=𝑁𝑎_𝑚(𝑚−1)²𝑞^𝑚₀𝑟^𝑚+1−𝑞^−1𝑁−1_𝑚=2𝑎_𝑚𝑚(𝑚−1)²𝑟^𝑚+2^∞_𝑚=𝑁𝑎_𝑚𝑚(𝑚−1)𝑟^𝑚. (68) Since 𝑓∈𝐻(𝔻_𝑅), we can find that 𝑁=𝑁_𝜀∈ℕ such that 4^∞_𝑚=𝑁𝑎_𝑚(𝑚−1)²𝑞^𝑚₀𝑟^𝑚+2^∞_𝑚=𝑁𝑎_𝑚𝑚(𝑚−1)𝑟^𝑚<𝜀2. (69) Thus, for 𝑞 sufficiently close to 1 from the right, we conclude that lim_𝑞→1+1𝐿_𝑞𝑓;𝑧=𝐿₁𝑓;𝑧 (70) uniformly on 𝔻_𝑟. The proof is finished.

Proof of Theorem 5. Then, by Theorem 3, we get 𝐿_𝑞(𝑓;𝑧)=lim_𝑛→∞[𝑛+1]_𝑞(𝑈_𝑛,𝑞(𝑓;𝑧)−𝑓(𝑧))=0 for infinite number of points having an accumulation point on 𝔻_𝑅/𝑞2. Since 𝐿_𝑞(𝑓;𝑧)∈𝐻(𝔻_𝑅/𝑞2), by the unicity Theorem for analytic functions, we get 𝐿_𝑞(𝑓;𝑧)=0 in 𝔻_𝑅/𝑞2, and, therefore, by (11), 𝑎_𝑚=0, 𝑚=2,3,…. Thus, 𝑓 is linear. Theorem 5 is proved.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author dedicates this paper to Professor Agamirza E. Bashirov at his 60th anniversary.