The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 1

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Some approximation results on modified positive linear operators

Dr. R .P. Pathak and Shiv Kumar Sahoo

ABSTRACT:Recently Deo N.et.al. (Appl. Maths. Compt., 201(2008), 604 -612.) introduced a new Bernstein type special operators. Motivated by Deo N.et.al., in this paper we introduce special class of positive linear operators and shall study some approximation results on it.

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1. INTRODUCTION

Recently Deo N.et.al. [1] Introduced new Bernstein type special operators defined as,

Again Deo N.et.al. [1] gave the integral modification of the operators (1.1) which are defined as ,

and prove some approximation results on the operators
(1.2).
Singh S.P. [4] studied some approximation results on a

sequence of Szȧ sz type operators defined as,

which map the space of bounded continuous funtions into itself following [3].
Kasana H.S. et. el. [2] obtained a sequence of modified

Szậsz operators for integrable function on defined as,
Motivated by Deo N.et.al.[1] we introduce a sequence of positive linear operators which are defined as,

we shall study some approximation results on the operators
(1.5).

Again following Kasana H.S. et. el. [2] we introduce a sequence of positive linear operators which are defined as,

and shall study some approximation results on the operators (1.6).

2. BASIC RESULTS-I

In order to prove our main result, the following basic results are needed.

1.

2.
........(2.2)

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The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 2

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3. 5.

4. 6.

)3 3+7( + )2 2+( + ) ….(2.4) 7.

PROOF OF BASIC RESULTS-I

We know that

Differentiating with respect to , we get

Multiplying both sides, we get

This completes the proof of (2.1).

Again differentiating (2.6) with respect to , we get

Multiplying both sides, we get

This completes the proof of (2.2).

In the same way after differentiations and calculations, we get required result s (2.3) and (2.4).

3. BASIC RESULTS-II

1.

2.

3.
4.
8.
9.

Proof of Basic Results-II.

By putting in equation (1.5), we get

This completes the proof of (2.8).

By putting in equation (1.5), we get

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The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 3

ISSN 2229-5518

This completes the proof of (2.9).
By putting in equation (1.5), we get

This completes the proof of (2.10).

In the same way by taking respectively in (1.5) and after little calculations we get required results (2.11) to (2.16).
This completes the proof.

4. MAIN RESULTS

In this section we shall give our main result.

Reference Theorem :- Let be the integrable and

bounded in the interval and let if exists at a point in , then one gets that

where are defined in (1.5).

Theorem : Let be the integrable and bounded in the

interval and let if exists at a point in

, then one gets that

where are defined in (1.6).

Proof : Since exists at a point in , then
by using Taylor’s expansion, we write

where
Now for each , there corresponds such that

Again for then there exist a positive number such that

Thus for all and , we get

Applying on (3.6), we get

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Since is arbitrary and small, we get

Thus

Multiplying both sides, we get

17−12 2−6 +6 2 +12 +2 +3 + ,

Here we write,

Using (3.2) in equation (3.4), we get
This completes the proof.

5. ACKNOWLEDGMENT

The authors are thankful to Director of National Institute of
Technology, Raipur (C.G.) for encouragement.

6. REFERENCES

By choosing we get that
Dr. R .P. Pathak Associate Professor Department of Mathematics, National Institute of Technology, Raipur
Shiv Kumar Sahoo Department of Mathematics, National Institute of Technology, Raipur
G.E. Road, Raipur -492010

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The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 5

ISSN 2229-5518

G.E. Road, Raipur -492010 (C.G.) India.
e-mail id:
r.p.pathak8@gmail.com.
(C.G.) India. e-mail id:
maths10sks@gmail.com.

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