A note on Chebychev-Gruss type inequalities for differentiable functions *.

Abstract

In the present note new Chebychev-Gruss type integral inequalities are established by using integral representation for n-time differentiable functions.

Keywords and Phrases: Chebychev-Gruss type inequalities, Differentiable functions, Absolutely continuous, Integral representation, Properties of modulus.

1. Introduction

In [4] G.Gruss proved the following well known integral inequality (see also [5,p.296]) :

|T (f,g;a,b)| [less than or equal to] ([PHI] - [phi])([GAMMA] - [gamma]), (1.1)

where f,g : [a, b] [right arrow] R are integrable on [a, b] and satisfy the condition

[phi] [less than or equal to] f(x) [less than or equal to] [PHI], [gamma] [less than or equal to] g(x) [less than or equal to] [GAMMA]

for each x [member of] [a, b], where [phi],[PHI],[gamma], [GAMMA] are given real constants and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Another less known inequality of this type derived in 1882 is due to Chebychev [1] (see, also [10, p.207]), which states that :

|T (f,g;a,b)|[less than or equal to] [(b-a).sup.2] / 12 ||f'||[infinity]||g'||[infinity], (1,2)

provided f, g are absolutely continuous and f', g' [member of] L[infinity] [a, b].

Since the publication of [4] in 1935, a number of authors have obtained various generalizations, extensions and variants of the inequalities (1.1) and (1.2). For some other integral inequalities of the above type, see the books [5,10] and the recent papers [2, 6-9] and also the papers appeared in RGMIA Research Report Collections. The main purpose of this note is to establish two new inequalities similar to the inequalities given in (1.1) and (1.2) involving n-time differentiable functions. The analysis used in the proofs is elementary and based on the integral representation for n-time differentiable function given in [3, p.291].

2. Statement of Results

In what follows, we denote by R the set of real numbers and [a, b] [subset] R, a < b. We use the following notations to simplify the details of presentation.

For the functions f, g : [a, b] [right arrow] R such that [f.sup.(n-1)], [g.sup.(n-1)] are absolutely continuous on [a, b] we set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

A Note on Type Inequalities for Differentiable Functions 31

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and n [greater than or equal to] 1 is an integer. For n = 1 we take the sum to be zero. It is easy to observe that

A(f,g;a,b;1) = B (f,g;a,b;1) = T (f,g;a,b).

Our main results are given in the following theorems.

Theorem 1. Let the functions f, g : [a, b] [right arrow] R be such that [f.sup.(n-1)], [g.sup.(n-1)] are absolutely continuous on [a, b] and [f.sup.(n)], [g.sup.(n)] [member of] L[infinity] [a, b]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

for x [member of] [a,b], in which

r(t,x) = {t-a, if a [less than or equal to] t [less than or equal to] x [less than or equal to]b, t-a, if a [less than or equal to] x [less than or equal to] t [less than or equal to] b. (2.3)

Remark 1. In the special case, when n = 1 the inequality (2.1) in Theorem 1 reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

which is similar to the Chebychev inequality in (1.2).

Theorem 2. Let the functions f, g : [a, b] [right arrow] R be such that [f.sup.(n-1)], [g.sup.(n-1)] are absolutely continuous on [a, b] and [f.sup.(n)], [g.sup.(n)] [member of] L[infinity] [a, b]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5)

Remark 2. If we take n = 1 in Theorem 2, then we get the following Gruss type inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

For the inequalities of the type (2.6), see [8].

A Note on Chebychev-Gruss Type Inequalities for Differentiable Functions 33

3. Proofs of Theorems 1 and 2

From the hypotheses on f, g in Theorems 1 and 2 we have the following representations (see [3, p.291]) :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.2)

Multiplying the left sides and right sides of (3.1) and (3.2) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.3)

Integrating both sides of (3.3) with respect to x over [a, b] and rewriting we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.4)

From (3.4) and using the properties of modulus we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

which is the required inequality in (2.1). The proof of Theorem 1 is complete.

Multiplying both sides of (3.1) and (3.2) by [g(x) + [n-1.summation over (k=1)] [G.sub.k] (x)]

And [f(x) [n-1.summation over (k=1)] [F.sub.k] (x)] [respectively, adding the resulting identities and rewriting we have

[f (x) + [n-1.summation over (k=1)] [F.sub.k] (x)] [g (x) + [n-1.summation over (k=1)] [G.sub.k] (x)]

A Note on [Chebychev-Gruss Type Inequalities for Differentiable Functions 35

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.5)

Integrating both sides of (3.5) with respect to x over [a, b] and rewriting we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.6)

From (3.6) and using the properties of modulus and by following the proof of Theorem 1 we get the required inequality in (2.5). This completes the proof of Theorem 2.

In concluding, we note that in the hypotheses of Theorems 1 and 2, if we assume [f.sup.(k)] (a) = [f.sup.(k)] (b) = 0,[g.sup.(k)] (a) = [g.sup.(k)] (b) = 0 for k = 0, 1,..., n-1, then we get the variants of the inequalities in (2.1) and (2.5), which we believe are also of independent interest. For similar discussion concerning the varient of generalization of Ostrowsk's inequality, see [3].

Received January 13, 2005, Accepted March 7, 2005.

References

[1] P. L. Chebyshev, Sue les expressions approximatives des integrales definies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov 2 (1882), 93-98.

[2] S. S. Dragomir, Some integral inequalities of Gruss type, Indian J. Pure and Appl. Math. 31 (2000), 379-415.

[3] A. M. Fink, Bounds on the deviation of a function from its averages, Czechoslovak Math. J. 42 (1992), 289-310.

[4] G.Gruss, Uber das maximum des absoluten Betrages von

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Math.Z. 39 (1935), 215-226.

[5] D. S. Mitrinovic, J.E.Pecaric and A. M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht, 1993.

[6] B. G. Pachpatte, On Gruss type inequalities for double integrals, J. Math. Anal. Appl. 267 (2002), 454-459.

[7] B. G. Pachpatte, On Gruss type integral inequalities, J. Inequal. Pure and Appl. Math. 3(1) Art.11, 2002.

[8] B. G. Pachpatte, On Trapezoid and Gruss type integral inequalities, Tamkang J. Math. 34 (2003), 365-369.

[9] B. G. Pachpatte, New weighted multivariate Gruss type inequalities, J. Inequal. Pure and Appl. Math. 4(5) Art.108, 2003.

[10] J. Pecaric, F. Prochan and Y. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, San Diego, 1992.

* 2000 Mathematics Subject Classification. 26D15, 26D20.

B. G. Pachpatte ([dagger])

Shri Niketan Colony, Near Abhinay, Talkies

Aurangabad 431 001 (Maharashtra), India

([dagger]) E-mail: [e-mail]