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Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk

  • Larisa Cheregi and Mohammad Mursaleen EMAIL logo
Published/Copyright: May 30, 2024
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

The purpose this study is to present and investigate the q -Beta-Baskakov-Szasz-Stancu operator. The operators are accompanied by Voronovskaja-type consequences, which include both an exact approximation order and a quantitative assessment, specifically within compact disks.

Keywords: q-Beta-Baskakov-Schurer-Szasz-Stancu operator; q-integers; Voronovskaja estimate; approximation in compact disks; simultaneous approximation; overconvergence
MSC 2010: 30E10; 41A25; 41A28; 41A35

1 Introduction

Over the years, there have been numerous discussions and proposals surrounding the use of complex approximations. The field of approximation theory has seen significant advancements, with noteworthy contributions to its knowledge base. One particular area of research that has garnered attention is the work of Aslan, which centers around their exploration of [1]. Aslan’s research delves into the intricate nature of approximation methods that involve operators. Their article provides valuable insights and significant findings regarding the behavior of these operators, ultimately contributing to our overall understanding of their ability to approximate. On the other hand, Agrawal et al.’s research [2] takes a different approach by studying the interrelationships among various types of operators. This enriches the scope of approximation theory and provides valuable resources for effectively approximating functions.

Cicek and İzgi’s study [3] is centered on bivariate operators and their ability to estimate functions within triangular regions. The findings of this research carry significant weight for numerous fields and practical uses.

The past few years have seen a proliferation of talks and propositions regarding the effective employment of intricate approximation operators. Q -calculus holds a crucial importance in the realm of mathematical research and has been emphasized in the seminal book by Aral et al. [4]. This work serves as a fundamental reference in the realm of q -calculus, revealing the profound utility of q -calculus in operator theory. Various researchers have introduced and examined several q -operators, focusing on their properties in approximation. One notable contribution to this field came in 1987 from Lupaş [5], who presented the initial q -analogue of the traditional Bernstein polynomials. Additionally, Phillips [6] made a significant contribution by introducing another important innovation. The q -analogue of Bernstein polynomials is a mathematical concept that seeks to extend the traditional Bernstein polynomials to a q -parameterized version. This extension introduces a new level of complexity and versatility to the polynomials, allowing for a broader range of applications and calculations within the realm of mathematical analysis. In 2012, there were two notable studies conducted by Maheshwari and Sharma [7] and Govil and Gupta [8]. These studies focused on examining the approximation properties of q -Baskakov-Beta-Stancu operators and q -Beta-Szász-Stancu operators. The researchers also provided estimations of the moments and successfully established a direct correlationbetween the outcomes in relation to the modulus of continuity and a formula that approaches infinity for the q -operators.

Yüksel and Dinlemez provided a Voronovskaja theorem for the q -analogue of a specific family called Szász-Beta in their work published in [9]. The use of type operators is discussed in the literature. Gupta and Agrawal [10] provide information on the rate at which the Szász-Beta operators converge, using q -integers. Dinlemez [11] explores the approximation properties of q -Szász-Beta-Stancu-type operators and presents a weighted approximation theorem for these operators. Following these studies, further research has been conducted to expand upon these findings. Numerous captivating generalizations concerning the q -calculus of the Beta operator have been presented in various sources, including [12–14].

The main focus of [15] revolves around the q analogue of the widely recognized Stancu-Beta operators. This includes an examination of their moments, recurrence relation, and the presentation of various direct outcomes related to the modulus of continuity of the q -Stancu-Beta operators. In a separate study, Yüksel and Dinlemez introduced a positive linear operator of the q -Baskakov-Schurer-Szasz variety, along with its Stancu generalization [16]. Furthermore, Cheregi explored the complex q -Baskakov-Schurer-Szasz-Stancu operators on compact disks [17]. The current research expands upon prior studies and introduces a modified version of the Beta-type generalization of complex q -Baskakov-Szasz-Stancu operators on compact disks. The focus of this study is specifically on the modified Beta operators, as mentioned in [18]. For recent related work, we refere to [19–23].

In order to address the restricted abilities of q -Beta operators to solely replicate constant functions, we present a resolution to improve the convergence of approximating functions on compact disks. Notwithstanding this constraint, the operators retain their versatility as a result of their capacity to handle intricate variables and adapt to diverse weight functions. This adaptability renders them a valuable instrument for a range of applications.

In programming languages, the aforementioned operators play a vital role in data manipulation and transformation. These operators are specifically designed to carry out a range of operations on different data types, including arithmetic calculations, logical comparisons, and assignment of values. Programmers rely on these operators to develop intricate algorithms and incorporate specific functionalities into their programs. It is imperative to utilize these operators correctly and efficiently to write code that is both functional and efficient. Additionally, type operators are indispensable for ensuring the proper handling and manipulation of data in programming.

The terminologies that we introduce in our work are specifically linked to the analysis of q .

Each integer l ≥ 0 , q > 0 , [ l ] q is accompanied by a set of corresponding definitions [17]:

(1) [ l ] q ≔ 1 − q k 1 − q , q ∈ R * \ { 1 } k , q = 1 .

For l ∈ N and [ 0 ] q = 0 , the definition of the q -factorial is as follows:

(2) [ l ] q ! ≔ [ 1 ] q [ 2 ] q ⋯ [ l ] q , ∀ l ∈ N * , 1 , l = 0 ,

and in their respective order

(3) ( 1 + z ) q l ≔ ∏ i = 0 n − 1 ( 1 + q i z ) , ∀ l ∈ N * , 1 , l = 0 .

The q -binomial coefficient is defined for integers 0 ≤ l ≤ m as follows:

(4) n l q = [ m ] q ! [ l ] q ! [ m − l ] q ! .

Let us denote the q -derivative of D q f q ( z ) of f q for a fixed value of q > 1 as follows:

(5) D q f q ( z ) ≔ f q ( q z ) − f q ( z ) ( q − 1 ) z , z ≠ 0 , f q ′ ( 0 ) , z = 0 .

Also, D q 0 f q ≔ f and D q m f q ≔ D q ( D q m − 1 f q ) , ∀ m ∈ N .

We have presented two different forms for the q -exponential function. Jackson defined the q -exponential function e q ( z ) using ∣ q ∣ < 1 and ∣ z ∣ < 1 1 − q :

(6) e q ( z ) ≔ ( 1 − z + q z ) q − ∞ .

Assuming e q ( z ) is an entire function for ∣ q ∣ > 1 ,

(7) e q ( z ) = ∏ m = 0 ∞ 1 + 1 q m − z q m + 1 .

To acquire an alternative exponential function, we need to invert the base in equation (6), for values of q such that ∣ q ∣ is less than 1:

(8) E q ( z ) = ( 1 − z + q z ) q ∞ .

Upon requesting it, we are promptly provided with

(9) E q ( z ) = ∏ m = 0 ∞ ( 1 + z q m − z q m + 1 ) , ∣ q ∣ ∈ ( 0 , 1 ) ,

by (7).

Our study focuses on the scrutiny of an analytical function that lies within a disk centered at the origin, denoted as O , and has a specific radius of R .

Consider a disk D R = { z ∈ C ∣ ∣ z ∣ < R } in the complex plane C .

Let H ( D R ) represent the set of analytic functions on D R , where f q maps to f : [ R , ∞ ) ∪ D R ¯ → C continuously in ( R , ∞ ) ∪ D R ¯ .

In our writing regarding f q ∈ H ( D R ) , it is permissible to utilize the expression f q ( z ) = ∑ m = 0 ∞ c m z m for any value of z within the domain D R .

Assume that there exists real-valued continuous functions f q and f q defined on the interval [ 0 , ∞ ) , q ∈ ( 0 , 1 ) , p , l ∈ N , m ∈ N *

(10) B m , p , q ( α q , β q ) ( f q ) ( x ) = [ m − 1 ] q [ m ] q ∑ l = 0 ∞ b m , p l ( x ) ∫ 0 ∞ s m , p l ( t ) f q [ m ] q t + α q [ m ] q + β q d q t ,

where

(11) b m , p l ( x ) = m + l l q q l 2 x l ( 1 + x ) m + j + 1

and

(12) s m , p l ( x ) = ( [ m + 1 ] q t ) l [ l ] q ! e q − [ m + 1 ] q t .

2 Additional outcomes

Lemma 1

The recurrence formula is defined as

(13) T q , m , p , l + 1 ( α q , β q ) ( z ) = z ( 1 + z ) [ m + 1 ] q D q T q , m , p , l ( α q , β q ) ( z ) + [ m + 1 ] q z + l + 1 [ m + 1 ] q T q , m , p , l ( α q , β q ) ( z ) ,

where T q , m , p , l ( α q , β q ) ( z ) ≔ B m , p , q ( α q , β q ) ( e l ) ( z ) and N 0 represents the collection of all non-negative integers. It can be observed that the said recurrence formula applies to all integers within m , p , l ∈ N 0 , 0 ≤ α q ≤ β q , and z ∈ C .

Proof

By denoting e l ( z ) = z l and writing,

z ( 1 + z ) D q T q , m , p , l ( α q , β q ) ( z ) = [ m − 1 ] q [ m ] q ∑ l = 1 ∞ z ( z + 1 ) D q b m , p l ( z ) ∫ 0 ∞ s m , p l ( t ) t l d t , = [ m − 1 ] q [ m ] q ∑ l = 1 ∞ ( l − [ m + 1 ] q z ) b m , p l ( z ) ∫ 0 ∞ s m , p l ( t ) t l d t , = [ m − 1 ] q [ m ] q ∑ l = 1 ∞ b m , p l ( z ) ∫ 0 ∞ [ ( l − [ m + 1 ] q t ) + ( [ m + 1 ] q t − [ m + 1 ] q z ) ] s m , p l ( t ) t l d t , = [ m − 1 ] q [ m ] q ∑ l = 1 ∞ b m , p l ( z ) ∫ 0 ∞ D q s m , p l ( t ) t l + 1 d t + [ m + 1 ] q T q , m , p , l + 1 ( α q , β q ) ( z ) − [ m + 1 ] q z T q , m , p , l ( α q , β q ) ( z ) ,

□ z ( 1 + z ) D q T q , m , p , l ( α q , β q ) ( z ) = − ( l + 1 ) T q , m , p , l ( α q , β q ) ( z ) − [ m + 1 ] q z T q , m , p , l ( α q , β q ) ( z ) + [ m + 1 ] q T q , m , p , l + 1 ( α q , β q ) ( z ) , [ m + 1 ] q T q , m , p , l + 1 ( α q , β q ) ( z ) = z ( 1 + z ) D q T q , m , p , l ( α q , β q ) ( z ) + ( l + 1 + [ m + 1 ] q z ) T q , m , p , l ( α q , β q ) ( z ) , T q , m , p , l + 1 ( α q , β q ) ( z ) = z ( 1 + z ) [ m + 1 ] q D q T q , m , p , l ( α q , β q ) ( z ) + [ m + 1 ] q z + l + 1 [ m + 1 ] q T q , m , p , l ( α q , β q ) ( z ) .

We were able to obtain the desired result for z ∈ C .

Lemma 2

Denoting B m , p , q ( 0 , 0 ) ( e j ) ( z ) as B m , p , q ( e j ) ( z ) , for all m , p , l ∈ N 0 , we have established a recursive relation for the images of monomials e l under B m , p , q ( α q , β q ) in terms of B m , p , q ( e j ) , j = 0 , 1 , ⋯ , l :

(14) T q , m , p , l ( α q , β q ) ( z ) = ∑ j = 0 l l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ B m , p , q ( e j ) ( z ) .

Proof

When l equals zero, equation (15) remains true. For the specific value

(15) T q , m , p , t ( α q , β q ) ( z ) = ∑ j = 0 t t j q [ m + 1 ] q j α q t − j ( [ m + 1 ] q + β q ) t ⋅ B m , p , q ( e j ) ( z ) ,

let it be true that l equals t .

Using equation (14),

(16) T m , p , t + 1 ( α q , β q ) ( z ) = z ( 1 + z ) [ m + 1 ] q ∑ j = 0 t t j q [ m + 1 ] q j α q t − j ( [ m + 1 ] q + β q ) t D q B m , p , q ( e j ) ( z )

(17) + [ m + 1 ] q z + l + 1 [ m + 1 ] q ∑ j = 0 t t j q [ m + 1 ] q j α q t − j ( [ m + 1 ] q + β q ) t B m , p , q ( e j ) ( z ) ,

(18) = ∑ j = 0 t t j q [ m + 1 ] q j + 1 α q t − j ( [ m + 1 ] q + β q ) t + 1

(19) × z ( 1 + z ) [ m + 1 ] q D q B m , p , q ( e j ) ( z ) + [ m + 1 ] q z + l + 1 [ m + 1 ] q B m , p , q ( e j ) ( z ) ,

which has been obtained.

By deriving a recurrence relation for the complex q -Beta-Baskakov-Szasz-Stancu operator, it becomes evident that

(20) T m , p , t + 1 ( α q , β q ) ( z ) = ∑ j = 1 t t j − 1 q [ m + 1 ] q j α q t − j + 1 ( [ m + 1 ] q + β q ) t + 1 B m , p , q ( e j ) ( z ) + ∑ j = 0 t t j q [ m + 1 ] q j α q t − j + 1 ( [ m + 1 ] q + β q ) t + 1 ⋅ B m , p , q ( e j ) ( z ) , = ∑ j = 0 t + 1 t + 1 j q [ m + 1 ] q j α q t − j + 1 ( [ m + 1 ] q + β q ) t + 1 B m , p , q ( e j ) ( z ) ,

providing proof for the lemma.□

3 Approximation of complex q-Beta-Baskakov-Szasz-Stancu operators

We proceed to provide numerical approximations in the subsequent section, accompanied by the following theorem.

Theorem 1

Let R be a real number greater than 1, and let f q be a function defined on [ R , ∞ ) ∪ D R ¯ that is continuous and bounded in [ 0 , ∞ ) and analytic in the disk D R . In particular, we can write f q as a power series ∑ l = 0 ∞ c l z l that converges for all z in D R and for all non-negative integers l . For any 0 ≤ α q ≤ β q and 1 < r < R assuming arbitrary but constant values, we present an estimate that holds true for all ∣ z ∣ ≤ r and q > 1 , m ∈ N ,

(21) ∣ B m , p , q ( α q , β q ) ( f q ) ( z ) − f q ( z ) ∣ ≤ [ m + 1 ] q ( β q + 1 ) + β q [ m + 1 ] q ( [ m + 1 ] q + β q ) ,

where

C r = 2 ( r + 2 ) ∑ l = 1 ∞ ∣ c l ∣ ( l + 2 ) ! r l − 1 < ∞ .

Proof

Through the use of Lemma 2, we are able to obtain

T q , m , p , l ( α q , β q ) ( e l ) ( z ) − e l ( z ) = ∑ j = 0 l − 1 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ ( B m , p , q ( e j ) ( z ) − e j ( z ) ) + ∑ j = 0 l − 1 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ ( e j ) ( z ) + [ m + 1 ] q l ( [ m + 1 ] q + β q ) l B m , p , q ( e l ) ( z ) − e l ( z ) .

The utilization of Lemma 1 presents a potential opportunity for the acquisition of

(22) T q , m , p , l ( α q , β q ) ( z ) − z l = z ( 1 + z ) [ m + 1 ] q D q ( T q , m , p , l − 1 ( α q , β q ) ( z ) − z l − 1 ) + [ m + 1 ] q z + l [ m + 1 ] q ( T q , m , p , l − 1 ( α q , β q ) ( z ) − z l − 1 ) + ( 2 l − 1 ) + ( l − 1 ) r [ m + 1 ] q z l − 1 , ∀ z ∈ C , l , m , p ∈ N .

We denote the norm ‖ ⋅ ‖ in C ( D R ¯ ) for 1 ≤ r ≤ R . Here, D R ¯ is defined as the set of complex numbers z where ∣ z ∣ ≤ r . Within the closed unit disk, we establish an inequality that states ∣ D q P l ( z ) ∣ ≤ l r ‖ P l ‖ r for all ∣ z ∣ ≤ r . This is applicable to P l ( z ) , which is a polynomial of a degree less than or equal to l . Using the recurrence relation mentioned earlier, we are able to derive the following result:

∣ T q , m , p , l ( α q , β q ) ( z ) − z l ∣ ≤ r ( 1 + r ) [ m + 1 ] q l − 1 r ‖ T m , p , l − 1 ( α q , β q ) ‖ r + [ m + 1 ] q r + l [ m + 1 ] q ∣ T m , p , l − 1 ( α q , β q ) ( z ) − z l − 1 ∣ + l ( r + 2 ) [ m + 1 ] q r l − 1 .

The suggestion made using the notation ι = r + 2 is that

∣ T q , m , p , l ( α q , β q ) ( z ) − z k ∣ ≤ r + ( 2 + r ) k [ m + 1 ] q ‖ T q , m , p , k − 1 ( α q , β q ) ‖ r + k [ m + 1 ] q ( 2 + r ) r k − 1 = r + ι k [ m + 1 ] q ‖ T q , m , p , k − 1 ( α q , β q ) ‖ r + k [ m + 1 ] q ι r k − 1 .

The process of deriving outcomes involves utilizing induction in regards to k, and taking into account the condition of m + 1 ≥ ι :

(23) ∣ T q , m , p , l ( α q , β q ) ( z ) − z l ∣ ≤ ι ( l + 2 ) ! [ m + 1 ] q r l − 1 , ∀ l ≥ 1 .

The aforementioned recurrence transforms into

∣ T q , m , p , l + 1 ( α q , β q ) ( z ) − z l + 1 ∣ ≤ r + ι ( l + 1 ) [ m + 1 ] q ( l + 2 ) ! [ m + 1 ] q ι r l − 1 + ι ( l + 1 ) [ m + 1 ] q r l ≤ r + ι ( l + 1 ) [ m + 1 ] q ( l + 2 ) ! + r ( l + 1 ) .

Given that the value of m added to 1 is greater than or equal to ι , we can conclude that:

∣ T q , m , p , l + 1 ( α q , β q ) ( z ) − z l + 1 ∣ ≤ ( r + l + 1 ) ( l + 2 ) ! + r ( l + 1 ) ≤ ( l + 3 ) ! r .

For any value of l greater than or equal to one and a fixed value of r greater than or equal to one, the validity of this is evident.

In the course of our work, we employed the equation denoted as (23):

(24) ‖ T q , m , p , l ( α q , β q ) ( e l ) − e l ‖ r ≤ ∑ j = 0 l − 1 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ ‖ B m , p , q ( e j ) − e j ‖ r + ∑ j = 0 l − 1 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ ( r j ) ( z ) + [ m + 1 ] q l ( [ m + 1 ] q + β q ) l ‖ B m , p , q ( e l ) − e l ‖ r + 1 − [ m + 1 ] q l ( [ m + 1 ] q + β q ) l ⋅ r l ≤ ( [ m + 1 ] q + α q ) l ( [ m + 1 ] q + β q ) l ⋅ ι ( l + 2 ) ! [ m + 1 ] q ⋅ r l − 1 + r l ⋅ ( [ m + 1 ] q + α q ) l ( [ m + 1 ] q + β q ) l − [ m + 1 ] q l ( [ m + 1 ] q + β q ) l + [ m + 1 ] q l ( [ m + 1 ] q + β q ) l ⋅ ι ( l + 2 ) ! [ m + 1 ] q ⋅ r l − 1 + 1 − [ m + 1 ] q l ( [ m + 1 ] q + β q ) l ⋅ r l ≤ 2 ι ( l + 2 ) ! [ m + 1 ] q ⋅ r l − 1 + 2 r l ⋅ l β q [ m + 1 ] q + β q ≤ ( [ m + 1 ] q ) ( β q + 1 ) + β q ( [ m + 1 ] q ) ( [ m + 1 ] q + β q ) ⋅ 2 ( l + 2 ) ! ⋅ ι ⋅ r l − 1 .

Incorporating the inequality 1 − ∏ j = 1 l x j ≤ ∑ j = 1 l ( 1 − x j ) is a common practice in mathematical discourse. This equation is applicable when considering variables x j within the range of 0 to 1, where j is any integer between 1 and l .

Now, we write B m , p , q ( α q , β q ) ( f q ) ( z ) = ∑ l = 1 ∞ c k T q , m , p , l ( α q , β q ) ( z ) , which implies

(25) ∣ B m , p , q ( α q , β q ) ( f q ) ( z ) − f q ( z ) ∣ ≤ ∑ l = 1 ∞ ∣ c l ∣ ⋅ ∣ T q , m , p , l ( α q , β q ) ( z ) − z l ∣ ≤ ∑ l = 1 ∞ ∣ c l ∣ ⋅ 2 ι ( [ m + 1 ] q ) ( β q + 1 ) + β q ( [ m + 1 ] q ) ( [ m + 1 ] q + β q ) ⋅ ( k + 2 ) ! ⋅ r l − 1 = 2 ( r + 2 ) ⋅ ( [ m + 1 ] q ) ( β q + 1 ) + β q ( [ m + 1 ] q ) ( [ m + 1 ] q + β q ) ∑ l = 1 ∞ ∣ c l ∣ ⋅ ( l + 2 ) ! ⋅ r l − 1 .

We denote that, for every value of r from one to R , and for every value of z smaller than or equal to r , the following notation holds

C r = 2 ( r + 2 ) ∑ l = 1 ∞ ∣ c l ∣ ( l + 2 ) ! r l − 1 < ∞ .

As a result, the demonstration has reached its conclusion.□

Lemma 3

We can establish that the given inequality holds true for all values of  ∣ z ∣ less than or equal to r, where r is greater than or equal to 1, q is greater than 1, and m , p , and l are all natural numbers, provided that m is greater than or equal to r + 1 . The absolute value ∣ T q , m , p , l ( α q , β q ) ( e l ) ( z ) ∣ ≤ 4 r l ( l + 2 ) ! .

Proof

We start with the following inequality for the norm in the disk of radius r :

‖ T q , m , p , l ( α q , β q ) ( e l ) ‖ r ≤ ‖ T q , m , p , l ( α q , β q ) ( e l ) − e l ‖ r + ‖ e l ‖ r .

This is just the triangle inequality for the norm.

Next, we employ the relationship indicated in equation (23)

‖ T q , m , p , l ( α q , β q ) ( e l ) ‖ r ≤ ( r + 2 ) ( l + 2 ) ! [ m + 1 ] q ⋅ r l − 1 + r l = ( r + 2 ) ( l + 2 ) ! r ( [ m + 1 ] q ) r l + r .

Now, we can simplify the aforementioned expression:

‖ T q , m , p , l ( α q , β q ) ( e l ) ‖ r ≤ r l 1 + 2 r ( l + 2 ) ! + r l ≤ 3 r l ( l + 2 ) ! + r l ≤ 4 r l ( l + 2 ) ! ,

for all r ≥ 1 , m , p , l ∈ N and m ≥ r + 1 .□

Theorem 2

Assuming that the Theorem 1  conditions are met, where 0 ≤ α q ≤ β q , 1 < r < R , and q > 1 , the Voronovskaja-type finding applies to all ∣ z ∣ ≤ r and m , p ∈ N . The inequality expression

(26) B m , p , q ( α q , β q ) ( f q ) ( z ) − f q ( z ) − α q + 1 − β q ( z − 1 ) [ m + 1 ] q f q ′ ( z ) − z ( z + 2 ) 2 [ m + 1 ] q f q ″ ( z ) ≤ M r ( f q ) ( [ m + 1 ] q ) 2 + N r ( f q ) [ m + 1 ] q + P r ( f q ) ( [ m + 1 ] q + β q ) 2

holds true. The next formulas illustrate a complex mathematical equation with various contributing factors

M r ( f q ) = 2 ( r + 1 ) l + 1 ∑ l = 2 ∞ ∣ c l ∣ r ( l − 1 ) 2 − l 4 ( l − 1 ) + ( r + 2 ) 2 ( l + 3 ) ! < ∞

and

N r ( f q ) = ∑ l = m + 2 ∞ ∣ c l ∣ r 2 + 1 ( l + 2 ) ! r l − 1 < ∞ ,

P r ( f q ) = ∑ l = 0 ∞ ∣ c l ∣ [ ( l − 1 ) ( 2 α q 2 + β q 2 − α q β q + 3 β q ) + ( β q + 1 ) ( α q + 1 ) + ( r + 1 ) ( α q + β q ) ] ( l + 2 ) ! l r l < ∞ .

Proof

Considering all z ∈ D R , we examine

B m , p , q ( α q , β q ) ( f q ) ( z ) − f q ( z ) − α q + 1 − β q ( z − 1 ) [ m + 1 ] q f q ′ ( z ) − z ( z + 2 ) 2 [ m + 1 ] q f q ″ ( z ) = B m , p , q ( f q ) ( z ) − f q ( z ) − 1 [ m + 1 ] q f q ′ ( z ) − z ( z + 2 ) 2 [ m + 1 ] q f q ″ ( z ) + B m , p , q ( α q , β q ) ( f q ) ( z ) − B m , p , q ( f q ) ( z ) − α q − β q ( z − 1 ) [ m + 1 ] q f q ′ ( z ) .

Using the expression f q ( z ) = ∑ l = 0 ∞ c l z l in the second term, we are able to obtain our desired result:

B m , p , q ( α q , β q ) ( f q ) ( z ) − f q ( z ) − α q + 1 − β q ( z − 1 ) [ m + 1 ] q f q ′ ( z ) − ( z + 2 ) z 2 [ m + 1 ] q f q ″ ( z ) ≤ B m , p , q ( f q ) ( z ) − f q ( z ) − 1 [ m + 1 ] q f q ′ ( z ) − ( z + 2 ) z 2 [ m + 1 ] q f q ″ ( z ) + ∑ l = 0 ∞ ∣ c l ∣ B m , p , q ( α q , β q ) ( e l ) ( z ) − B m , p , q ( e l ) ( z ) − α q − β q ( z − 1 ) [ m + 1 ] q l z l − 1 .

The equation involves several absolute values of the difference between certain functions and their approximations, as well as a sum of absolute values of coefficients and differences between certain modified and unmodified functions, along with a polynomial expression with coefficients that depend on the function being approximated.

We have developed a method to estimate the first sum by expressing B m , p , q ( f q ) ( z ) as a summation of ∣ c l ∣ multiplied by B m , p , q ( e l ) ( z ) , with l ranging from 0 to infinity.

Thus,

B m , p , q ( f q ) ( z ) − f q ( z ) − 1 [ m + 1 ] q f q ′ ( z ) − z ( z + 2 ) 2 [ m + 1 ] q f q ″ ( z ) ≤ ∑ l = 0 ∞ ∣ c l ∣ B m , p , q ( e l ) ( z ) − e l ( z ) − 1 [ m + 1 ] q l z l − 1 − ( z + 2 ) z 2 [ m + 1 ] q l ( l − 1 ) z l − 2 , B m , p , q ( f q ) ( z ) − f q ( z ) − 1 [ m + 1 ] q f q ′ ( z ) − ( z + 2 ) z 2 [ m + 1 ] q f q ″ ( z ) ≤ ∑ l = 0 ∞ ∣ c l ∣ B m , p , q ( e l ) ( z ) − e l ( z ) − [ l ( l − 1 ) z + 2 l ( 2 − l ) ] z l − 1 2 [ m + 1 ] q .

For any natural numbers m , p , and l , along with any complex number z , we possess Lemma 1.

B m , p , q ( e l + 1 ) ( z ) = ( 1 + z ) z [ m + 1 ] q D q B m , p , q ( e l ) ( z ) + [ m + 1 ] q z + l + 1 [ m + 1 ] q B m , p , q ( e l ) ( z ) .

We currently denote

(27) O m , l ( z ) = B m , p , q ( e l ) ( z ) − e l ( z ) − [ l ( l − 1 ) z + 2 l ( 2 − l ) ] z l − 1 2 [ m + 1 ] q .

The polynomial O m , l ( z ) has a degree that does not exceed l; after some basic computation and utilizing the aforementioned recurrence relation, we arrive at their conclusion:

O m , l ( z ) = z ( 1 + z ) [ m + 1 ] q D q O m , l − 1 ( z ) + [ m + 1 ] q z + l [ m + 1 ] q O m , l − 1 ( z ) + E m , l ( z ) ,

where

E m , l ( z ) = 2 z l − 2 [ m + 1 ] q 2 ( l − 1 ) 2 ( l − 2 ) ( l 2 − 2 l + 2 ) + z l − 1 ( l − 1 ) 2 ( l − 2 ) 2 ( z + 1 ) 2 [ m + 1 ] q 2 .

Given l ≥ 2 , m , p ∈ N and ∣ z ∣ ≤ r , the statement holds true. Using theorem 1, we were able to derive an estimate. The following inequality is true for the polynomial function

∣ B m , p , q ( e l ) ( z ) − e l ( z ) ∣ ≤ ( r + 2 ) ( l + 2 ) ! [ m + 1 ] q r l − 1 .

The inequality expression states that

∣ O m , l ( z ) ∣ ≤ r ( r + 1 ) [ m + 1 ] q ∣ D q O m , l − 1 ( z ) ∣ + r + l [ m + 1 ] q ∣ O m , l − 1 ( z ) ∣ + ∣ E m , l ( z ) ∣ .

Due to the fact that O m , l − 1 ( z ) is a polynomial with a degree of less than or equal to l − 1 , we are able to obtain

∣ D q O m , l − 1 ( z ) ∣ ≤ l − 1 r ‖ O m , l − 1 ( z ) ‖ r .

From (27),

∣ D q O m , l − 1 ( z ) ∣ ≤ l − 1 r ‖ B m , p , q ( e l + 1 ) − e l + 1 ‖ r + [ ( l − 1 ) ( l − 2 ) e 1 + 2 ( l − 1 ) ( 3 − l ) ] e l − 2 2 [ m + 1 ] q ≤ l − 1 r ( r + 2 ) ( l + 2 ) ! [ m + 1 ] q r l − 2 + [ ( l − 1 ) ( l − 2 ) r + 2 ( l − 1 ) ( 3 − l ) ] r l − 2 2 [ m + 1 ] q ≤ l − 1 r ⋅ r l − 2 [ m + 1 ] q ⋅ [ ( r + 2 ) ( l + 2 ) ! + ( l − 1 ) ( l − 2 ) ( r + 2 ) ] = ( l − 1 ) r l − 2 ( r + 2 ) [ ( l + 2 ) ! + ( l − 1 ) ( l − 2 ) ] r [ m + 1 ] q ≤ r l − 3 ( r + 2 ) ( l + 3 ) ! [ m + 1 ] q .

The equation can be simplified to

r ( 1 + r ) [ m + 1 ] q ∣ D q O m , l − 1 ( z ) ∣ ≤ r l − 2 ( r + 1 ) ( r + 2 ) ( l + 3 ) ! [ m + 1 ] q 2

and

∣ O m , l ( z ) ∣ ≤ r l − 2 ( r + 1 ) ( r + 2 ) ( l + 3 ) ! [ m + 1 ] q 2 + r + l [ m + 1 ] q ∣ O m , l − 1 ( z ) ∣ + ∣ E m , l ( z ) ∣ .

Assuming l is greater than or equal to two, and with m , p , and r being natural numbers, it is required that m is greater than r + 1 . We were able to obtain the following result for l ≤ m , m > r + 1 , and ∣ z ∣ ≤ r , by using the inequality r + l [ m + 1 ] q ≤ r + 1 .

∣ O m , l ( z ) ∣ ≤ r l − 2 ( r + 1 ) ( r + 2 ) ( l + 3 ) ! [ m + 1 ] q 2 + ( r + 1 ) ∣ O m , l − 1 ( z ) ∣ + ∣ E m , l ( z ) ∣ ,

with

∣ E m , l ( z ) ∣ ≤ 2 r l − 2 [ m + 1 ] q 2 ( l − 1 ) 2 ( l − 2 ) ( l 2 − 2 l + 2 ) + r l − 1 ( l − 1 ) 2 ( l − 2 ) 2 ( r + 1 ) 2 [ m + 1 ] q 2 .

We were able to derive

∣ O m , l ( z ) ∣ ≤ 2 ( r + 1 ) l [ m + 1 ] q 2 ∑ j = 2 l ( j − 1 ) 2 ( j − 2 ) ( j 2 − 2 j + 2 ) + r ( r + 1 ) l + 1 2 [ m + 1 ] q 2 ∑ j = 2 l ( j − 1 ) 2 ( j − 2 ) 2 + ( r + 1 ) l + 1 ( r + 2 ) [ m + 1 ] q 2 ∑ j = 2 l ( j + 3 ) ! ≤ 2 ( l − 1 ) 2 r ( r + 1 ) l + 1 [ m + 1 ] q ( l − 2 ) ( l 2 − 2 l + 2 ) + ( l − 2 ) 2 4 r + ( l − 1 ) ( r + 2 ) ( r + 1 ) l + 1 [ m + 1 ] q ( l + 3 ) ! ,

for any z ∈ C and 2 ≤ l ≤ m + 1 , where ∣ O m , 0 ( z ) ∣ = ∣ O m , 1 ( z ) ∣ = 0 .

To deduce or conclude something, one needs to have a premise or condition, which is typically represented by

(28) B m , p , q ( f q ) ( z ) − f q ( z ) − 1 [ m + 1 ] q f q ′ ( z ) − z ( z + 2 ) 2 [ m + 1 ] q f q ″ ( z ) ≤ ∑ l = 2 n + 2 ∣ c l ∣ ∣ O m , l ( z ) ∣ + ∑ l = m + 3 ∞ ∣ c l ∣ ∣ O m , l ( z ) ∣ ≤ 1 [ m + 1 ] q 2 ∑ l = 2 ∞ 2 ∣ c l ∣ ( l − 1 ) 2 r ( r + 1 ) l + 1 ( l − 2 ) ( l 2 − 2 l + 2 ) + ( l − 2 ) 2 4 r + 1 [ m + 1 ] q 2 ∑ l = 2 ∞ ( l − 1 ) ( r + 2 ) ( r + 1 ) l + 1 ( l + 3 ) ! + ∑ l = m + 3 ∞ ∣ c l ∣ ∣ B m , p , q ( e l ) ( z ) − e l ( z ) ∣ − [ l ( l − 1 ) z − 2 l ( l − 2 ) ] z l − 1 2 [ m + 1 ] q ≤ 1 [ m + 1 ] q 2 ∑ l = 2 ∞ ∣ c l ∣ ( l − 1 ) 2 r ( r + 1 ) l + 1 ( l − 2 ) ( l 2 − 2 l + 2 ) + ( l − 2 ) 2 4 r + 1 [ m + 1 ] q 2 ∑ l = 2 ∞ ( l − 1 ) ( r + 2 ) ( r + 1 ) l + 1 ( l + 3 ) ! + ∑ l = m + 3 ∞ ∣ c l ∣ ( r + 2 ) ( l + 2 ) ! [ m + 1 ] q r l − 1 ≤ T r ( f q ) ( [ m + 1 ] q ) 2 + N r ( f q ) [ m + 1 ] q ,

where

M r ( f q ) = 2 ( r + 1 ) l + 1 ∑ l = 2 ∞ ∣ c l ∣ r ( l − 1 ) 2 − l 4 ( l − 1 ) + ( r + 2 ) 2 ( l + 3 ) ! < ∞

and

N r ( f q ) = ∑ l = m + 2 ∞ ∣ c l ∣ r 2 + 1 ( l + 2 ) ! r l − 1 < ∞ .

We proceeded to estimate the second sum by using Lemma 2, which involved rewriting the sum in the following manner:

B m , p , q ( α q , β q ) ( e l ) ( z ) − B m , p , q ( e l ) ( z ) + β q ( z − 1 ) − α q [ m + 1 ] q l z l − 1 = ∑ j = 0 l − 1 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ B m , p , q ( e j ) ( z ) − 1 − [ m + 1 ] q [ m + 1 ] q + β q l ⋅ B m , p , q ( e l ) ( z ) + β q ( z − 1 ) − α q [ m + 1 ] q + β q l z l − 1 = ∑ j = 0 l − 2 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ B m , p , q ( e j ) ( z ) + l α q [ m + 1 ] q l − 1 ( [ m + 1 ] q + β q ) l B m , p , q ( e l − 1 ) ( z ) − ∑ j = 0 l − 1 l j q [ m + 1 ] q j β q l − j ( [ m + 1 ] q + β q ) l ⋅ B m , p , q ( e l ) ( z ) + β q ( z − 1 ) − α q [ m + 1 ] q + β q l z l − 1 = ∑ j = 0 l − 2 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l B m , p , q ( e j ) ( z ) + l α q [ m + 1 ] q l − 1 ( [ m + 1 ] q + β q ) l ( B m , p , q ( e l − 1 ) ( z ) − z l − 1 ) − ∑ j = 0 l − 2 l j q [ m + 1 ] q j β q l − j ( [ m + 1 ] q + β q ) l B m , p , q ( e l ) ( z ) − l β q [ m + 1 ] q l − 1 ( [ m + 1 ] q β q ) l ( B m , p , q ( e l ) ( z ) − z l ) + l α q [ m + 1 ] q + β q z l − 1 ⋅ [ m + 1 ] q l − 1 ( [ m + 1 ] q + β q ) l − 1 − 1 + l β q [ m + 1 ] q + β q z l 1 − [ m + 1 ] q l − 1 ( [ m + 1 ] q + β q ) l − 1 − β q ( α q + β q + β q z + [ m + 1 ] q ) [ m + 1 ] q ( [ m + 1 ] q + β q ) .

Using Lemma 3 in conjunction with the subsequent inequality, we can proceed as follows:

(29) 1 − [ m + 1 ] q [ m + 1 ] q + β q l ≤ ∑ j = 1 l 1 − [ m + 1 ] q [ m + 1 ] q + β q = l β q [ m + 1 ] q + β q .

We obtain

∑ j = 0 l − 2 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l B m , p , q ( e j ) ( z ) ≤ ∑ j = 0 l − 2 l j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ ∣ B m , p , q ( e j ) ( z ) ∣ = ∑ j = 0 l − 2 l ( l − 1 ) ( l − j ) ( l − 1 − j ) l − 2 j q [ m + 1 ] q j α q l − j ( [ m + 1 ] q + β q ) l ⋅ ∣ B m , p , q ( e j ) ( z ) ∣ ≤ l ( l − 1 ) 2 ⋅ α q 2 ( [ m + 1 ] q + β q ) 2 ⋅ 4 l ! r l − 2 ∑ j = 0 l − 2 l − 2 j q [ m + 1 ] q j α q l − j − 2 ( [ m + 1 ] q + β q ) l − 2 ≤ 2 l ( l − 1 ) l ! r l − 2 α q 2 ( [ m + 1 ] q + β q ) 2 .

Now, we have obtained

B m , p , q ( α q , β q ) ( e l ) ( z ) − B m , p , q ( e l ) ( z ) + β q ( z − 1 ) − α q [ m + 1 ] q + β q l z l − 1 ≤ 2 l ( l − 1 ) l ! r l − 2 α q 2 ( [ m + 1 ] q + β q ) 2 + l ( r + 2 ) α q ( [ m + 1 ] q + β q ) 2 ( l + 1 ) ! r l − 2 + 2 l ( l − 1 ) β q 2 ( [ m + 1 ] q + β q ) 2 l ! r l − 2 + l ( l − 1 ) α q β q ( [ m + 1 ] q + β q ) 2 r l − 1 + l β q ( r + 2 ) ( l + 2 ) ! ( [ m + 1 ] q + β q ) 2 r l − 1 + l ( l − 1 ) β q 2 ( [ m + 1 ] q + β q ) 2 r l + β q ( α q + β q + β q r + [ m + 1 ] q ) l r l − 1 ( [ m + 1 ] q + β q ) 2 ,

which led to

B m , p , q ( α q , β q ) ( e l ) ( z ) − B m , p , q ( e l ) ( z ) + β q ( z − 1 ) − α q [ m + 1 ] q + β q l z l − 1 ≤ H r ( f q ) ( [ m + 1 ] q + β q ) 2 ,

P r ( f q ) = ∑ l = 0 ∞ ∣ c l ∣ [ ( l − 1 ) ( 2 α q 2 + β q 2 − α q β q + 3 β q ) + ( β q + 1 ) ( α q + 1 ) + ( r + 1 ) ( α q + β q ) ] ( l + 2 ) ! l r l < ∞ .

Using (28) in conjunction with this, we are able to establish the intended outcome.□

Theorem 3

Assuming that q is greater than 1 and 0 ≤ α q ≤ β q , and the conditions of the function f are met as described in Theorem 1, suppose that 1 < r < R is a constant. Then, for any m and p that belong to the set of natural numbers, where m is greater than r + 1 and ∣ z ∣ is less than or equal to r, we have demonstrated that

(30) ‖ B m , p , q ( α q , β q ) ( f q ) − ( f q ) ‖ r ∼ 1 [ m + 1 ] q .

The equivalence’s constants are solely dependent on the values of f q , α q , β q , and r , provided that f q is not a polynomial of degree less than or equal to zero.

Proof

We state that, for any natural numbers n and p , as well as for any complex number z with an absolute value less than or equal to r , the following holds:

B m , p , q ( α q , β q ) ( f q ) ( z ) − f q ( z ) = α q − β q + 1 − β q z [ m + 1 ] q f q ′ ( z ) + z ( z + 2 ) 2 [ m + 1 ] q f q ″ ( z ) + 1 [ m + 1 ] q 2 [ m + 1 ] q 2 × B m , p , q ( α q , β q ) ( f q ) ( z ) − f q ( z ) + β q z − β q − α q − 1 [ m + 1 ] q f q ′ ( z ) − z ( z + 2 ) 2 [ m + 1 ] q f q ″ ( z ) .

By making use of the inequality that follows

‖ F q + G q ‖ r ≥ ∣ ‖ F q ‖ r − ‖ G q ‖ r ∣ ≥ ‖ F q ‖ r − ‖ G q ‖ r ,

we were able to obtain the following result:

‖ B m , p , q ( α q , β q ) ( f q ) − f q ‖ r ≥ 1 [ m + 1 ] q ⋅ ( α q + 1 − β q e 1 + β q ) f q ′ + e 1 ( e 1 + 2 ) 2 f q ″ ( z ) r − 1 [ m + 1 ] q 2 [ m + 1 ] q 2 ⋅ ⋅ B m , p , q ( α q , β q ) ( f q ) − f + β q e 1 − β q − α q − 1 [ m + 1 ] q f q ′ − ( e 1 + 2 ) e 1 2 f q ″ ( z ) r .

We are able to write, based on our hypotheses on f , that f cannot be considered a polynomial with a degree of ≤ 1 in D R :

( α q + 1 − β q e 1 + β q ) f ′ + e 1 ( e 1 + 2 ) 2 f q ″ ( z ) r > 0 .

The application of Theorem 2 leads to the conclusion that:

[ m + 1 ] q 2 B m , p , q ( α q , β q ) ( f q ) − f q + β q e 1 − ( α q + β q + 1 ) [ m + 1 ] q f q ′ − e 1 ( e 1 + 2 ) 2 f q ″ ( z ) r ≤ C .

An independent constant C with a positive value is present.

We have established that, given f q , α q , β q , q , and r , there exists an index m 0 such that for all n greater than or equal to m 0 , the condition 1 [ m + 1 ] q → 0 as m → ∞ is satisfied.

( α q + 1 − β q ( e 1 − 1 ) ) f ′ + ( e 1 + 2 ) e 1 2 f q ″ ( z ) r − [ m + 1 ] q ⋅ B m , p , q ( α q , β q ) ( f q ) − f + β q e 1 − ( α q + β q + 1 ) [ m + 1 ] q + β q f ′ − ( e 1 + 2 ) e 1 2 [ m + 1 ] q f q ″ ( z ) r ≥ 1 2 ( α q + 1 − β q ( e 1 − 1 ) ) f ′ + ( e 1 + 2 ) e 1 2 f q ″ ( z ) r .

This suggests

‖ B m , p , q ( α q , β q ) ( f q ) − f q ‖ r ≥ 1 2 [ m + 1 ] q ( α q + 1 − β q ( e 1 − 1 ) ) f q ′ + ( e 1 + 2 ) e 1 2 f q ″ ( z ) r .

Whenever the value of the parameter m is greater than or equal to the baseline value of m 0 , the following statement applies for all m . On the other hand, if the value of m is between 1 and m 0 − 1 , inclusive, we obtain the following outcome:

(31) ‖ B m , p , q ( α q , β q ) ( f q ) − f q ‖ r ≥ M r , n ( f q ) [ m + 1 ] q .

If M r , m ( f q ) = [ m + 1 ] q ‖ B m , p , q ( α q , β q ) ( f q ) − f ‖ r is taken into consideration, it is evident that the value obtained will always be greater than zero. Ultimately, we are able to acquire.

(32) ‖ B m , p , q ( α q , β q ) ( f q ) − f q ‖ r ≥ M r α q , β q ( f q ) [ m + 1 ] q .

Using the function M r α q , β q ( f q ) , defined as the minimum of M r , 1 ( f q ) through M r , m 0 − 1 ( f q ) , and also incorporating 1 2 ‖ ( α q − β q e 1 + β q ) ⋅ f ′ + ( e 1 + 2 ) e 1 2 ⋅ f ″ ‖ r , we, as stated in Theorem 1, are able to attain the desired conclusion.

Theorem 4

If q > 1 , 0 ≤ α q ≤ β q , and the conditions of f in Theorem 1  are met, then suppose 1 < r < r 1 < R are the constants. In this scenario, and for all m , p , u ∈ N , where m > r + 2 and ∣ z ∣ ≤ r , we have established the following:

(33) ‖ ( B m , p , q ( α q , β q ) ( f q ) ) ( u q ) − f q ( u q ) ‖ r ∼ 1 [ m + 1 ] q .

The constants present in the equivalence are dependent solely on f q , q , u q , r 1 , α q , β q , and r , provided that f q is not a polynomial of degree less than or equal to u q − 1 .

Proof

Let ι be the circle with center O and a radius of r 1 , where 1 ≤ r < r 1 < R . Because ∣ z ∣ ≤ r and ι is a part of Γ , it follows that ∣ ι − z ∣ ≥ r 1 − r . Applying Cauchy’s formulas, we can deduce that for any ∣ z ∣ ≤ r as well as n m , p ∈ N where m > r + 2 , the following is true:

(34) ( B m , p , q ( α q , β q ) ( f q ) ( z ) ) ( u q ) − f q ( u q ) ( z ) ≤ u q ! 2 π ∫ ι B m , p , q ( α q , β q ) ( f q ) ( ι ) − f q ( ι ) ( ι − z ) u q + 1 d ι ≤ M r 1 ( α q , β q ) ( f q ) [ m + 1 ] q ⋅ u q ! 2 π ⋅ 2 π r 1 ( r 1 − r ) u q + 1 = M r 1 ( α q , β q ) ( f q ) [ m + 1 ] q ⋅ u q ! r 1 ( r 1 − r ) u q + 1 .

This serves as confirmation for one of the inequalities present in the equivalence.

We obtain a result from Cauchy’s formula

(35) ( B m , p , q ( α q , β q ) ( f q ) ( z ) ) ( u q ) − f q ( u q ) ( z ) = u q ! 2 π i ∫ ι B m , p , q ( α q , β q ) ( f q ) ( ι ) − f q ( ι ) ( ι − z ) u q + 1 d ι .

We have considered all values of ι in the set Γ , along with natural numbers p and m :

(36) B m , p , q ( α q , β q ) ( f q ) ( ι ) − f q ( ι ) = 1 [ m + 1 ] q ( α q + 1 − β q ι ) f q ′ ( ι ) + ι ( ι + 2 ) 2 f q ″ ( ι ) + 1 [ m + 1 ] q [ m + 1 ] q 2 B m , p , q ( α q , β q ) ( f q ) ( ι ) − f q ( ι ) − ( α q + 1 − β q ι ) [ m + 1 ] q + β q f q ′ ( ι ) − ι ( ι + 2 ) 2 [ m + 1 ] q f q ″ ( ι ) .

Using Cauchy’s formula, we were able to derive our results:

(37) B m , p , q ( α q , β q ) ( f q ) ( z ) ( u q ) − f q ( u q ) ( z ) = 1 [ m + 1 ] q ( α q + 1 − β q ι ) f q ′ ( z ) + z ( z + 2 ) 2 f q ″ ( z ) ( u q ) + 1 [ m + 1 ] q u q ! 2 π i ∫ ι [ m + 1 ] q 2 ( ι − z ) u q + 1 B m , p , q ( α q , β q ) ( f q ) ( ι ) − f ( ι ) + β q ι − α q − 1 [ m + 1 ] q f q ′ ( ι ) − ι ( ι + 2 ) 2 [ m + 1 ] q f q ″ ( ι ) d ι .

By transitioning to the norm ‖ ⋅ ‖ r , we were able to obtain the following result:

(38) ‖ B m , p , q ( α q , β q ) ( f q ) ( u q ) − f q ( u q ) ‖ r ≥ 1 [ m + 1 ] q ( α q + 1 − β q e 1 ) f q ′ + e 1 ( e 1 + 2 ) 2 f q ″ ( z ) ( u q ) r − 1 [ m + 1 ] q u q ! 2 π i ∫ ι [ m + 1 ] q 2 ( ι − z ) u q + 1 ( B m , p , q ( α q , β q ) ( f q ) ( ι ) − f ( ι ) + β q ι − α q − 1 [ m + 1 ] q + β q f q ′ ( ι ) − ι ( ι + 2 ) 2 [ m + 1 ] q f q ″ ( ι ) ) d ι r .

We have, according to Theorem 2, established that for any natural numbers p and m , the statement holds true:

(39) u q ! 2 π i ∫ ι [ m + 1 ] q 2 ( ι − z ) u q + 1 ( B m , p , q ( α q , β q ) ( f q ) ( ι ) − f ( ι ) + β q ι − α q − 1 [ m + 1 ] q + β q f q ′ ( ι ) − ι ( ι + 2 ) 2 [ m + 1 ] q f q ″ ( ι ) ) d ι r ≤ u q ! 2 π ⋅ 2 π r 1 ( r 1 − r ) u q + 1 ∑ j = 1 5 M r 1 , j ( α q , β q ) ( f q ) .

It is possible to express f in D R by means other than a polynomial of degree ≤ 0 , given that it does not belong to that category:

(40) ( α q + 1 − β q ( e 1 − 1 ) ) f q ′ + ( e 1 + 2 ) e 1 2 [ m + 1 ] q f q ″ ( u ) r > 0 .

One can obtain the remaining portion of the proof in a similar manner as demonstrated in the proof for Theorem 3 (see [12]).□

4 Conclusions

The utilization of q -calculus has enabled the creation of q -analogues of numerous traditional operators and has been explored in terms of their approximation properties. In the subsequent article, we introduced a Beta-type generalization that had been modified for complex Baskakov-Szasz-Stancu-type operators and applied it to their q -analogues on compact disks. We also confirmed the convergence of these operators and provided estimates of the errors in the modified Beta-type generalization of complex Baskakov-Szasz-Stancu-type operators and their corresponding q -analogues. The properties of our approximations have been a subject of interest. In response to the limited capabilities of q -Beta operators to only replicate constant functions, we propose a solution to enhance the convergence of approximating functions on compact disks. Despite this limitation, the operators remain versatile due to their ability to handle complex variables and conform to various weight functions. This flexibility makes them a useful tool for various applications.

Acknowledgement

We are extremely thankful to the referees for a very careful reading of the manuscript and making valuable suggestions and comments leading to a better presentation of this article.

  1. Funding information: Not applicable.

  2. Author contributions: All the authors contributed equally and significantly in writing this article.

  3. Conflict of interest: The authors declare that they have no competing interests.

  4. Data availability statement: Not applicable.

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Received: 2023-06-13
Revised: 2023-11-08
Accepted: 2024-02-20
Published Online: 2024-05-30

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

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  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
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  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
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  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
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  21. Results on solutions of several systems of the product type complex partial differential difference equations
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  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
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  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
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https://doi.org/10.1515/dema-2023-0158
Keywords for this article
q-Beta-Baskakov-Schurer-Szasz-Stancu operator; q-integers; Voronovskaja estimate; approximation in compact disks; simultaneous approximation; overconvergence
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Articles in the same Issue

  1. Regular Articles
  2. On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity
  3. L-Fuzzy fixed point results in ℱ -metric spaces with applications
  4. Solutions of a coupled system of hybrid boundary value problems with Riesz-Caputo derivative
  5. Nonparametric methods of statistical inference for double-censored data with applications
  6. LADM procedure to find the analytical solutions of the nonlinear fractional dynamics of partial integro-differential equations
  7. Existence of projected solutions for quasi-variational hemivariational inequality
  8. Spectral collocation method for convection-diffusion equation
  9. New local fractional Hermite-Hadamard-type and Ostrowski-type inequalities with generalized Mittag-Leffler kernel for generalized h-preinvex functions
  10. On the asymptotics of eigenvalues for a Sturm-Liouville problem with symmetric single-well potential
  11. On exact rate of convergence of row sequences of multipoint Hermite-Padé approximants
  12. The essential norm of bounded diagonal infinite matrices acting on Banach sequence spaces
  13. Decay rate of the solutions to the Cauchy problem of the Lord Shulman thermoelastic Timoshenko model with distributed delay
  14. Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters
  15. An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces
  16. An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution
  17. On the existence and Ulam-Hyers stability for implicit fractional differential equation via fractional integral-type boundary conditions
  18. Some properties of a class of holomorphic functions associated with tangent function
  19. The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain
  20. On the continuity in q of the family of the limit q-Durrmeyer operators
  21. Results on solutions of several systems of the product type complex partial differential difference equations
  22. On Berezin norm and Berezin number inequalities for sum of operators
  23. Geometric invariants properties of osculating curves under conformal transformation in Euclidean space ℝ3
  24. On a generalization of the Opial inequality
  25. A novel numerical approach to solutions of fractional Bagley-Torvik equation fitted with a fractional integral boundary condition
  26. Holomorphic curves into projective spaces with some special hypersurfaces
  27. On Periodic solutions for implicit nonlinear Caputo tempered fractional differential problems
  28. Approximation of complex q-Beta-Baskakov-Szász-Stancu operators in compact disk
  29. Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions
  30. Jordan left derivations in infinite matrix rings
  31. Nonlinear nonlocal elliptic problems in ℝ3: existence results and qualitative properties
  32. Invariant means and lacunary sequence spaces of order (α, β)
  33. Novel results for two families of multivalued dominated mappings satisfying generalized nonlinear contractive inequalities and applications
  34. Global in time well-posedness of a three-dimensional periodic regularized Boussinesq system
  35. Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator
  36. Some applications and maximum principles for multi-term time-space fractional parabolic Monge-Ampère equation
  37. On three-dimensional q-Riordan arrays
  38. Some aspects of normal curve on smooth surface under isometry
  39. Mittag-Leffler-Hyers-Ulam stability for a first- and second-order nonlinear differential equations using Fourier transform
  40. Topological structure of the solution sets to non-autonomous evolution inclusions driven by measures on the half-line
  41. Remark on the Daugavet property for complex Banach spaces
  42. Decreasing and complete monotonicity of functions defined by derivatives of completely monotonic function involving trigamma function
  43. Uniqueness of meromorphic functions concerning small functions and derivatives-differences
  44. Asymptotic approximations of Apostol-Frobenius-Euler polynomials of order α in terms of hyperbolic functions
  45. Hyers-Ulam stability of Davison functional equation on restricted domains
  46. Involvement of three successive fractional derivatives in a system of pantograph equations and studying the existence solution and MLU stability
  47. Composition of some positive linear integral operators
  48. On bivariate fractal interpolation for countable data and associated nonlinear fractal operator
  49. Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix
  50. Online makespan minimization for MapReduce scheduling on multiple parallel machines
  51. The sequential Henstock-Kurzweil delta integral on time scales
  52. On a discrete version of Fejér inequality for α-convex sequences without symmetry condition
  53. Existence of three solutions for two quasilinear Laplacian systems on graphs
  54. Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
  55. Nilpotent perturbations of m-isometric and m-symmetric tensor products of commuting d-tuples of operators
  56. Characterizations of transcendental entire solutions of trinomial partial differential-difference equations in ℂ2#
  57. Fractional Sturm-Liouville operators on compact star graphs
  58. Exact controllability for nonlinear thermoviscoelastic plate problem
  59. Improved modified gradient-based iterative algorithm and its relaxed version for the complex conjugate and transpose Sylvester matrix equations
  60. Superposition operator problems of Hölder-Lipschitz spaces
  61. A note on λ-analogue of Lah numbers and λ-analogue of r-Lah numbers
  62. Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions
  63. A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1
  64. Fixed point results for generalized convex orbital Lipschitz operators
  65. Asymptotic model for the propagation of surface waves on a rotating magnetoelastic half-space
  66. Multiplicity of k-convex solutions for a singular k-Hessian system
  67. Poisson C*-algebra derivations in Poisson C*-algebras
  68. Signal recovery and polynomiographic visualization of modified Noor iteration of operators with property (E)
  69. Approximations to precisely localized supports of solutions for non-linear parabolic p-Laplacian problems
  70. Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces
  71. Pseudo compact almost automorphic solutions to a family of delay differential equations
  72. Periodic measures of fractional stochastic discrete wave equations with nonlinear noise
  73. Asymptotic study of a nonlinear elliptic boundary Steklov problem on a nanostructure
  74. Cramer's rule for a class of coupled Sylvester commutative quaternion matrix equations
  75. Quantitative estimates for perturbed sampling Kantorovich operators in Orlicz spaces
  76. Review Articles
  77. Penalty method for unilateral contact problem with Coulomb's friction in time-fractional derivatives
  78. Differential sandwich theorems for p-valent analytic functions associated with a generalization of the integral operator
  79. Special Issue on Development of Fuzzy Sets and Their Extensions - Part II
  80. Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index
  81. Binary relations applied to the fuzzy substructures of quantales under rough environment
  82. Algorithm selection model based on fuzzy multi-criteria decision in big data information mining
  83. A new machine learning approach based on spatial fuzzy data correlation for recognizing sports activities
  84. Benchmarking the efficiency of distribution warehouses using a four-phase integrated PCA-DEA-improved fuzzy SWARA-CoCoSo model for sustainable distribution
  85. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part II
  86. A study on a type of degenerate poly-Dedekind sums
  87. Efficient scheme for a category of variable-order optimal control problems based on the sixth-kind Chebyshev polynomials
  88. Special Issue on Mathematics for Artificial intelligence and Artificial intelligence for Mathematics
  89. Toward automated hail disaster weather recognition based on spatio-temporal sequence of radar images
  90. The shortest-path and bee colony optimization algorithms for traffic control at single intersection with NetworkX application
  91. Neural network quaternion-based controller for port-Hamiltonian system
  92. Matching ontologies with kernel principle component analysis and evolutionary algorithm
  93. Survey on machine vision-based intelligent water quality monitoring techniques in water treatment plant: Fish activity behavior recognition-based schemes and applications
  94. Artificial intelligence-driven tone recognition of Guzheng: A linear prediction approach
  95. Transformer learning-based neural network algorithms for identification and detection of electronic bullying in social media
  96. Squirrel search algorithm-support vector machine: Assessing civil engineering budgeting course using an SSA-optimized SVM model
  97. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part I
  98. Some fixed point results on ultrametric spaces endowed with a graph
  99. On the generalized Mellin integral operators
  100. On existence and multiplicity of solutions for a biharmonic problem with weights via Ricceri's theorem
  101. Approximation process of a positive linear operator of hypergeometric type
  102. On Kantorovich variant of Brass-Stancu operators
  103. A higher-dimensional categorical perspective on 2-crossed modules
  104. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part I
  105. On parameterized inequalities for fractional multiplicative integrals
  106. On inverse source term for heat equation with memory term
  107. On Fejér-type inequalities for generalized trigonometrically and hyperbolic k-convex functions
  108. New extensions related to Fejér-type inequalities for GA-convex functions
  109. Derivation of Hermite-Hadamard-Jensen-Mercer conticrete inequalities for Atangana-Baleanu fractional integrals by means of majorization
  110. Some Hardy's inequalities on conformable fractional calculus
  111. The novel quadratic phase Fourier S-transform and associated uncertainty principles in the quaternion setting
  112. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part I
  113. A novel iterative process for numerical reckoning of fixed points via generalized nonlinear mappings with qualitative study
  114. Some new fixed point theorems of α-partially nonexpansive mappings
  115. Generalized Yosida inclusion problem involving multi-valued operator with XOR operation
  116. Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph
  117. Convergence of Peaceman-Rachford splitting method with Bregman distance for three-block nonconvex nonseparable optimization
  118. Topological structure of the solution sets to neutral evolution inclusions driven by measures
  119. (α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces
  120. Solvability of infinite system of integral equations of Hammerstein type in three variables in tempering sequence spaces c 0 β and ℓ 1 β
  121. Special Issue on Nonlinear Evolution Equations and Their Applications - Part I
  122. Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative
  123. Klein-Gordon potential in characteristic coordinates
  124. Asymptotic analysis of Leray solution for the incompressible NSE with damping
  125. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part I
  126. Long time decay of incompressible convective Brinkman-Forchheimer in L2(ℝ3)
  127. Numerical solution of general order Emden-Fowler-type Pantograph delay differential equations
  128. Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation
  129. Spectral properties for a system of Dirac equations with nonlinear dependence on the spectral parameter
  130. A memory-type thermoelastic laminated beam with structural damping and microtemperature effects: Well-posedness and general decay
  131. The asymptotic behavior for the Navier-Stokes-Voigt-Brinkman-Forchheimer equations with memory and Tresca friction in a thin domain
  132. Absence of global solutions to wave equations with structural damping and nonlinear memory
  133. Special Issue on Differential Equations and Numerical Analysis - Part I
  134. Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks
  135. Limiting dynamics for stochastic complex Ginzburg-Landau systems with time-varying delays on unbounded thin domains
  136. A comparison of two nonconforming finite element methods for linear three-field poroelasticity
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