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On sharp perturbed trapezoidal inequalities for the harmonic sequence of polynomials.

Abstract

The main purpose of this paper is to use a variant of Gruss inequality to obtain some perturbed trapezoid inequality with bounded derivatives of n-th order.

Keywords and Phrases: Gruss inequality, Trapezoid inequality, Perturbed trapezoid inequality, Harmonic sequence of polynomials, n-convex function.

1. Introduction

Let f(x) be a convex function on the closed interval [a, b]. The inequality

f ([a+b]/2) [less than or equal to] [1/[b - a]] [[integral].sub.a.sup.b] f(x)dx [less than or equal to] [[f(a) + f(b)]/2] (1.1)

is well known in the literature as the Hermite-Hadamard inequality [16].

A function f(x) is said to be r-convex on [a, b] with r [greater than or equal to] 2 if and only if [f.sup.(r)](x) exists and [f.sup.(r)](x) [greater than or equal to] 0.

In terms of a trapezoid formula for a real function f(x) defined and integerable on [a, b], using the first and second Euler-Maclaurin summation formulas, inequality (1.1) was generalized for (2r)-convex function functions on [a, b] with r [greater than or equal to] 1 in [2, 6].

In [5], Lj. Dedic et al. established the following trapezoidal Gruss type inequality for n-time differentiable function:

Let f : [a, b] [right arrow] R be such that [f.sup.(n)] is a continuous function for some n [greater than or equal to] 1 and

[m.sub.n] [less than or equal to] [f.sup.(n)](t) [less than or equal to] [M.sub.n], t [member of] [a, b], [m.sub.n], [M.sub.n] [member of] R.

Then, we have

|[[integral].sub.a.sup.b] f(t)dt - [[b - a]/2][f(a) + f(b)] - [S.sub.n.sup.T](a, b)| [less than or equal to] [1/2](b-a)[.sup.n+1]([M.sub.n] - [m.sub.n])[square root of (|[B.sub.2n]|/(2n)!)]. (1.2)

where [B.sub.2n] is the Bernoulli numbers, [S.sub.1.sup.T] (a, b) = 0 and

[S.sub.n.sup.T](a, b) = - [[n/2].summation over (j=1)] [[(b - a)[.sup.2j]]/(2j)!][B.sub.2j][[f.sup.(2j-1)](b) - [f.sup.(2j-1)](a)],

for n [greater than or equal to] 2. The other trapezoidal Gruss type inequality for n-time differentiable function, see [9, 11, 12, 14, 17]. In this paper, using concept of the harmonic sequence of polynomials, we shall establish some new generalizations of trapezoidal Gruss type for n-time differentiable function.

2. Definitions and Lemmas

Definition 1. A sequence of polynomials {[P.sub.i](t)}[.sub.i=0.sup.[infinity]] is called harmonic if it satisfies the following condition

[P'.sub.i](t) = [P.sub.i-1](t) (2.1)

and [P.sub.0](t) = 1 for all defined t and i [member of] N.

It is well-known that Bernoulli's polynomials [B.sub.i](t) can be defined by the following expansion

[[x[e.sup.tx]]/[[e.sup.x] - 1]] = [[infinity].summation over (i=0)] [[[B.sub.i](t)]/i!][x.sup.i], |x| < 2[pi], t [member of] R,

and are uniquely determined by the following formulae

[B'.sub.i](t) = i[B.sub.i-1](t), [B.sub.0](t) = 1; (2.2)

[B.sub.i](t + 1) - [B.sub.i](t) = [it.sup.i-1]. (2.3)

Similarly, Euler's polynomials can be defined by

[[2[e.sup.tx]]/[[e.sup.x] + 1]] = [[infinity].summation over (i=0)] [[[E.sub.i](t)]/i!][x.sup.i], |x| < [pi], t [member of] R,

and are uniquely determined by the following properties

[E'.sub.i](t) = i[E.sub.i-1](t), [E.sub.0](t) = 1; (2.4)

[E.sub.i](t + 1) + [E.sub.i](t) = 2[t.sup.i]. (2.5)

For further details about Bernoulli's polynomials and Euler's polynomials, please refer to [1, 23.1.5 and 23.1.6] or [18]. Moreover, some new generalizations of Bernoulli's numbers and polynomials can be found in [10, 13].

If i is a nonegative integer, t, s, [theta] [member of] R and s [not equal to] [theta], then

[P.sub.i,E](t) = [[(s - [theta])[.sup.i]]/i!][E.sub.i]([t - [theta]]/[s - [theta]])

is a harmonic sequences of polynomials.

As usual, let [B.sub.i] = [B.sub.i](0), i [member of] N, denote Bernoulli's numbers. From properties (2.2) and (2.3), (2.4) and (2.5) of Bernoulli's and Euler's polynomials respectively, we can obtain easily that, for i [greater than or equal to] 1,

[B.sub.i+1](0) = [B.sub.i+1](1) = [B.sub.i+1], [B.sub.1](0) = -[B.sub.1](1) = -[1/2], [B.sub.2](0) = [B.sub.2] = 1/6 (2.6)

and, for j [member of] N,

[E.sub.j](0) = -[E.sub.j](1) = -[2/[j + 1]]([2.sup.j+1] - 1)[B.sub.j+1]. (2.7)

It is also a well known fact that [B.sub.2i+1] = 0 for all i [member of] N.

In 1935, G. Gruss proved the following integral inequality which gives an approximation for the integral of the product of two functions in terms of the product of the integrals of the two functions [15, P.296].

Let f, g : [a, b] [right arrow] R be two integrable functions such that [phi] [less than or equal to] f(x) [less than or equal to] [PHI] and [gamma] [less than or equal to] g(x) [less than or equal to] [GAMMA] for all x [member of] [a, b], where [phi], [PHI], [gamma] and [GAMMA] are real numbers. Then we have

|[1/[b - a]] [[integral].sub.a.sup.b] f(x)g(x)dx - [1/[b - a]] [[integral].sub.a.sup.b] f(x)dx x [1/[b - a]] [[integral].sub.a.sup.b] g(x)dx| [less than or equal to] [1/4]([PHI] - [phi])([GAMMA] - [gamma]),

and the inequality is sharp, in the sense that the constant 1/4 can not be replaced by a smaller one.

The above inequality is well known in the literature as Gruss inequality. In [4], X. L. Cheng and J. Sun proved the following variant of the Gruss inequality.

Lemma 2. Let f, g : [a, b] [right arrow] R be two integrable functions such that [gamma] [less than or equal to] g(x) [less than or equal to] [GAMMA] for all x [member of] [a, b], where [gamma], [GAMMA] [member of] R are constants. Then

|[[integral].sub.a.sup.b] f(x)g(x)dx - [1/[b - a]] [[integral].sub.a.sup.b] f(x)dx x [[integral].sub.a.sup.b] g(x)dx| [less than or equal to] [([GAMMA] - [gamma])/2] [[integral].sub.a.sup.b] |f(x) - [1/[b - a]] [[integral].sub.a.sup.b] f(t)dt|dx. (2.8)

Further, Cerone and Dragomir [3] proved that the constant 1/2 in (2.8) is sharp.

3. Mail Results

Theorem 3. Let {[P.sub.i](t)}[.sub.i=0.sup.[infinity]] be a harmonic sequence of polynomials, let f(t) be n-time differentiable on the closed interval [a, b] such that [m.sub.n] [less than or equal to] [f.sup.(n)](t) [less than or equal to] [M.sub.n] for t [member of] [a, b], n [member of] N and [m.sub.n], [M.sub.n] [member of] R. Then

|(-1)[.sup.n] [[infinity].sub.a.sup.b] f(t)dt + [n.summation over (i=1)](-1)[.sup.n+i][[P.sub.i](b)[f.sup.(i-1)](b) - [P.sub.i](a)[f.sup.(i-1)](a)] - [1/[b - a]][[P.sub.n+1](b) - [P.sub.n+1](a)][[f.sup.(n-1)](b) - [f.sup.(n-1)](a)]| [less than or equal to] [([M.sub.n] - [m.sub.n])/2] [[integral].sub.a.sup.b] |[P.sub.n](t) - [1/[b - a]][[P.sub.n+1](b) - [P.sub.n+1](a)]|dt. (3.1)

Proof. By successive integrabtion by parts and mathematical induction, we have

(-1)[.sup.n] [[integral].sub.a.sup.b] [P.sub.n](t)[f.sup.(n)](t)dt - [[integral].sub.a.sup.b] f(t)dt = [n.summation over (i=1)](-1)[.sup.i][[P.sub.i](b)[f.sup.(i-1)](b) - [P.sub.i](a)[f.sup.(i-1)](a)]. (3.2)

Using the definition of the harmonic sequence of polynomials yields

[[integral].sub.a.sup.b] [P.sub.n](t)dt = [P.sub.n+1](b) - [P.sub.n+1](a), (3.3)

By Lemma 2, we have

|[[integral].sub.a.sup.b] [P.sub.n](t)[f.sup.(n)](t)dt - [1/[b - a]] [[integral].sub.a.sup.b] [P.sub.n](t)dt [[integral].sub.a.sup.b] [f.sup.(n)](t)dt| [less than or equal to] [([M.sub.n] - [m.sub.n])/2] [[integral].sub.a.sup.b] |[P.sub.n](t) - [1/[b - a]][[integral].sub.a.sup.b] [P.sub.n](x)dx| dt. (3.4)

From combining of (3.2), (3.3) and (3.4) we obtain (3.1). This completes the proof.

Remark 4. If taking [P.sub.1](t) = t and n = 1 in (3.1), then we obtain

|[[integral].sub.a.sup.b] f(t)dt - [[b - a]/2][f(a) + f(b)]| [less than or equal to] [([M.sub.1] - [m.sub.1])/8](b - a)[.sup.2]. (3.5)

The constant 1/8 in inequality (3.5) is better than the constant 1/[4[square root of 3]] in inequality (1.2) for n = 1. In fact, the constant 1/8 is sharp (see [7], [8]).

4. Application

Using Theorem 3, we have the following Theorem.

Theorem 5. Let {[E.sub.i](t)}[.sub.i=0.sup.[infinity]] be the Euler's polynomials and {[B.sub.i]}[.sub.i=0.sup.[infinity]] the Bernoulli's numbers. Let f(t) be n-time differentiable on the closed interval [a, b] such that [m.sub.n] [less than or equal to] [f.sup.(n)](t) [less than or equal to] [M.sub.n] for t [member of] [a, b], n [member of] N and [m.sub.n], [M.sub.n] [member of] R. Then

|(-1)[.sup.n] [[integral].sub.a.sup.b] f(t)dt + 2[[[n+1]/2].summation over (i=1)](-1)[.sup.n](1 - [4.sup.i])[[(b - a)[.sup.2(i-1)]]/(2i)!][[f.sup.2(i-1)](a) + [f.sup.2(i-1)](b)][B.sub.2i] - [[4([2.sup.n+2] - 1)(b - a)[.sup.n][B.sub.n+2]]/[(n + 2)!]][[f.sup.(n-1)](b) - [f.sup.(n-1)](a)]| [less than or equal to] [[([M.sub.n] - [m.sub.n])(b - a)[.sup.n]]/2n!] [[integral].sub.a.sup.b] |[E.sub.n]([t - a]/[b - a]) - [[4([2.sup.n+2] - 1)]/[(n + 2)(n + 1)]][B.sub.n+2]| dt (3.6)

where [x] denotes the Gauss function, whose value is the largest integer not more than x.

Proof. Let

[P.sub.i](t) = [P.sub.i,E](t; b; a) = [[(b - a)[.sup.i]]/i!][E.sub.i]([t - a]/[b - a]) (3.7)

Then, we have

[[P.sub.n+1](b) - [P.sub.n+1](a)]/[b - a] = [4([2.sup.n+2] - 1)(b - a)[.sup.n][B.sub.n+2]]/[(n + 2)!]. (3.8)

Using formula (2.7) and straightforward calculating yields

[n.summation over (i=1)](-1)[.sup.n+i][[P.sub.i](b)[f.sup.(i-1)](b) - [P.sub.i](a)[f.sup.(i-1)](a)]

= [n.summation over (i=1)](-1)[.sup.n+i][[(b - a)[.sup.i]]/i!][[E.sub.i](1)[f.sup.(i-1)](b) - [E.sub.i](0)[f.sup.(i-1)](a)]

= [n.summation over (i=1)](-1)[.sup.n+i][[(b - a)[.sup.i]]/i!][[E.sub.i](1)[f.sup.(i-1)](a) + [f.sup.(i-1)](b)]

= 2[n.summation over (i=1)](-1)[.sup.n+i][[(b - a)[.sup.i]]/[(i + 1)!]][[f.sup.(i-1)](a) + [f.sup.(i-1)](b)]([2.sup.i+1] - 1)[B.sub.i+1]

= 2[[[n+1]/2].summation over (i=1)](-1)[.sup.n](1 - [4.sup.i])[[(b - a)[.sup.2.sup.i-1]]/(2i)!][[f.sup.2(i-1)](a) + [f.sup.2(i-1)](b)][B.sub.2i]. (3.9)

Substituting (3.7), (3.8) and (3.9) into (3.1) lead to (3.6). The proof is complete.

Remark 6. If taking [E.sub.1](t) = t - [1/2] and n = 1 in (3.6), then we recapture (3.5) again.

References

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[2] G. Allasia, C. Diodano, and J. Pecaric, Hadamard-type inequalities for (2r)-convex functions with application, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 133 (1999), 187-200.

[3] P. Cerone and S. S. Dragomir, A refinement of Gruss inequality and applications, RGMIA Res. Rep. Coll. 5 (2) (2002), Article 14. Available on line at: http://rgmia.vu.edu.au/v5n2.html

[4] Xiao-liang Cheng and Jie Sun, Note on the perturbed trapezoid inequality, Journal of Inequalities in Pure and Applied Mathematics, 3 (2) (2002), Article 29. Available on line at: http://jipam.vu.edu.au

[5] Lj. Dedic, M. Matic, and J. Pecaric, Some inequalities of Euler-Gruss type, Computers Math. Applic. 41 (2001), 843-856.

[6] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Type Inequalities and Applications, RGMIA Monographs, 2000. Available on line at: http://rgmia.vu.edu.au/monographs/hermite_hadamard.html

[7] S. S. Dragomir, Improvements of Ostrowski and generalized trapezoid inequality in terms of the upper and lower bounds of the first derivative. Tamkang J. Math. 34 (2003), no. 3, 213-222.

[8] S. S. Dragomir and Th. M. Rassias (ED.), Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht, 2002.

[9] Hillel Gauchman, Some integeral inequalities involving Taylor's Remainder II, Journal of Inequalities in pure and Applied Mathematics 4 (1) (2003), Article 1. [http://jipam.vu.edu.au/]

[10] Bai-Ni Guo and Feng Qi, Generalization of Bernoulli polynomials, Internat. J. Math. Ed. Sci. Tech. 33 (2002), no. 3, 428-431.

[11] Dah-Yan Hwang, Improvements of some integral inequalities involving Taylor's remainder. J. Appl. Math. Comput. 16 (2004), no. 1-2, 151-163.

[12] Dah-Yan Hwang, Kuei-Lin Tseng, and Gou-Sheng Yang, Improvements and generalizations of some Euler Gruss type inequalities and applications, Tamsui Oxford Journal of Mathematical Sciences, submitted.

[13] Oiu-Ming Luo, Bai-Ni, Feng Qi, and Lokenath Debnath, Generalizations of Bernoulli numbers and polynomials. Int. J. Math. Math. Sci. (2003), no. 59, 3769-3776.

[14] M. Matic, J. Pecaric, and N. Ujevic, Improvement and further generalization of some inequalities of Ostrowski-Gruss type, Computers Math. Applic. 39 (3/4) (2000), 161-175.

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

[16] J. E. Pecaric, F. Proschan, and Y. L. Tong, Convex Function, Partial Ordering and Statistical Applications, Academic Press, New York, 1991.

[17] Feng Qi, Zong- Li Wei, and Qiao Yang, Generalizations and refinements of Hermite-Hadamard's inequality. Rocky Mountain J. Math. 35 (2005), no. 1, 235-251.

[18] Zhu-Xi Wang and Dun-Ren Guo, Teshu Hanshu Gailun (introduction to Sepecial Function), The Series of Advanced Physics of Peking University, Peking University Press, Beijing, China, 2000. (Chinese)

Dah-Yan Hwang ([dagger])

Center for General Education, Kuang Wu Institute of Technology, Peito, Taipei, 11271 TAIWAN.

and

Gou-Sheng Yang

Department of Mathematics, Tamkang University, Tamsui, 25137 TAIWAN.

Received Septemer 21, 2005, Accepted December 12, 2006.

* 2001 Mathematics Subject Classification. 26D15, 41A55.

([dagger]) E-mail: dyhuang@tsint.edu.tw
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Author:Yang, Gou-Sheng
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Date:Nov 1, 2007
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