Abstract and Applied Analysis

Volume 2013, Article ID 732069, 8 pages

http://dx.doi.org/10.1155/2013/732069

## Degree of Approximation by Hybrid Operators

^{1}Department of Applied Mathematics, Delhi Technological University (Formerly Delhi College of Engineering), Bawana Road, Delhi 110042, India^{2}Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea^{3}Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 8 May 2013; Accepted 12 July 2013

Academic Editor: Irena Rachůnková

Copyright © 2013 Naokant Deo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider hybrid (Szász-beta) operators, which are a general sequence of integral type operators including beta function, and we give the degree of approximation by these Szász-beta-Durrmeyer operators.

#### 1. Introduction

The Lupaş-Durrmeyer operators were introduced by Sahai and Prasad [1] who studied the asymptotic formula for simultaneous approximation, and many mathematicians have given different results for the Durrmeyer operators (see [2–6]). Now we consider here a sequence of linear positive operators, which was introduced by Gupta et al. [7] as follows. Let and be positive integers. For satisfying , where is a positive integer, Let . For a function on , we define the norm by Recently Jung and Sakai [8] investigated the Lupaş-Durrmeyer operators and studied the circumstances of convergence. Motivated with the idea of Jung and Sakai [8], we give the degree of approximation by Szász-Beta-Durrmeyer operators in this paper.

#### 2. Basic Results

Lemma 1 (cf. [7]). *Let , , , and be integers with , , and :
**
Then one has*(i)* and ,*(ii)*for *(iii)*where is a polynomial of degree such that the coefficients of are bounded independently of .*

*Proof. *Let . Then (i)
Using
we see that
(ii) Using , we obtain
Since we know that
we have
Then substituting (12) into (10), we consider the following:
Then since we see
we have
Here the last equation follows from integration by parts. Furthermore, we easily see
Therefore, we conclude
Consequently, (ii) is proved.

(iii) For , (6) holds. Let us assume (6) for . We note
So, we have, by the assumption of induction,
Here, if is even, then
and if is odd, then
Hence we have
and here we see that is a polynomial of degree such that the coefficients of are bounded independently of .

Lemma 2 (cf. [7]). *Let , , and be integers with . Let satisfy for a positive integer **
Then one has, for ,
**
where
*

*Proof. *Using
we have

#### 3. Main Results

Theorem 3. *Let , and let and be nonnegative integers. Let and be integers with . Let satisfy
**
Then one has uniformly, for and ,
*

*Proof. *Let and . By the second inequality of (28),
Let . Then using Lemma 2 and
we have
From (30) and Lemma 1, we have
Next, we estimate . By the use of the first inequality in (28), we have
Now using and the notation
we have
Then, with ,
Here for , we get
because
Finally we get
From (32),
If we put , then we get

In the following, we let ,.

Theorem 4. *Let and be nonnegative integers. Let and be integers with . Let satisfy
**
Then one has uniformly, for and ,
*

*Proof. *For , we have
From (45), (46), and Lemma 2, we get
Using , we obtain
Therefore, we have
For , we have , , and . Hence

Let us define the weighted modulus of smoothness by where

Theorem 5. *Let and be nonnegative integers. Let and be integers with . Then one has, for ,
*

To prove Theorem 5, we need the following theorem.

Theorem 6. *Let and be nonnegative integers. Let and be integers with . Let satisfy
**
Then one has uniformly, for , , and ,
*

*Proof. *Using , we have
Therefore, by Lemma 1 (6), we have
Since is uniformly bounded on , we have with Lemma 2 and (59)
Therefore, we have the result.

The Steklov function for is defined as follows: Then for the Steklov function with respect to , we have the following properties.

Lemma 7 (see [8, Lemma 2.4]). *Let , and let be a positive and nonincreasing function on . Then*(i)*,*(ii)*(iii)**(iv)*

Now, we prove Theorem 5.

*Proof of Theorem 5. *We know that, for ,
Then first, we split it as follows:
Then for the first term, we have, using Theorem 6, (62), and (65),
Here, we suppose , and then we know that
For the second term, from Theorem 4, (65), (63), and (64) of Lemma 7,
Therefore, we have
If we let , then
because .

#### Acknowledgment

The authors thank the referees for many kind suggestions and comments.

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