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 [26]). 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.