A generalisation of Cerone's identity and applications.

Abstract

An identity due to P. Cerone for the Cebysev functional is extended for Stieltjes integrals. A sharp inequality and its application in approximating Stieltjes integrals are also given.

Keywords and Phrases: Integral inequalities, Stieltjes integrals.

1. Introduction

In 2001, P. Cerone [1] established the following identity for the Cebysev functional:

T (f, g; p) [colon, equals] [1/[[[integral].sub.a.sup.b] p(s) ds]] [[integral].sub.a.sup.b] p(t) f(t) g(t) dt - [1/[[[integral].sub.a.sup.b] p(s) ds]] [[integral].sub.a.sup.b] p(t) f(t) dt x [1/[[[integral].sub.a.sup.b] p(s) ds]] [[integral].sub.a.sup.b] p(t) g(t) dt = [1/[[([[integral].sub.a.sup.b] p(s) ds)[.sup.2]]] [[integral].sub.a.sup.b] [[[integral].sub.a.sup.t] p(s) ds [[integral].sub.t.sup.b] p(s) g(s) ds - [[integral].sub.t.sup.b] p(s) ds [[integral].sub.a.sup.t] p(s) g(s) ds] df (t), (1.1)

provided f is of bounded variation on [a, b] and g is continuous on [a, b]. He proved (1) on utilising the auxiliary function [PSI]: [a, b] [right arrow] R,

[PSI](t) [colon, equals] (t - a) [[integral].sub.t.sup.b] g(s) ds - (b - t) [[integral].sub.a.sup.t] g(s) ds (1.2)

and integrating by parts in the Stieltjes integral [[integral].sub.a.sup.b] [PSI] (t) df (t), which exists, since f is of bounded variation and [PSI] is differentiable on (a, b).

One may observe that the result remains valid if one assumes that g is Lebesgue integrable on [a, b] and f is of bounded variation. This follows by the fact that, in this case [PSI] becomes absolutely continuous on [a, b], the Stieltjes integral [[integral].sub.a.sup.b] [PSI] (t) df (t) still exists and the argument will follow as in [1].

The weighted version of this inequality has been obtained in the same paper [1] and can be stated as:

T (f, g; p) [colon, equals] [1/[[[integral].sub.a.sup.b] p(s) ds]] [[integral].sub.a.sup.b] p(t) f(t) g(t) dt - [1/[[[integral].sub.a.sup.b] p(s) ds]] [[integral].sub.a.sup.b] p(t) f(t) dt x [1/[[[integral].sub.a.sup.b] p(s) ds]] [[integral].sub.a.sup.b] p(t) g(t) dt = [1/[([[integral].sub.a.sup.b] p(s) ds)[.sup.2]]] [[integral].sub.a.sup.b] [[[integral].sub.a.sup.t] p(s) ds [[integral].sub.t.sup.b] p(s) g(s) ds - [[integral].sub.t.sup.b] p(s) ds [[integral].sub.a.sup.t] p(s) g(s) ds] df (t), (1.3)

provided f is of bounded variation on [a, b] and p, g are continuous on [a, b] with [[integral].sub.a.sup.b] p(s) ds > 0. The same remark for the extension of the identity in the case that p, g are Lebesgue integrable on [a, b] so that pg is also integrable, may apply.

The above two identities have been applied in [1] to obtain some interesting new bounds for the Cebysev functionals T (f, g) and T (f, g; p) from which we only mention the following:

|T (f, g)|

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.4)

where [[disjunction].sub.a.sup.b](f) is the total variation of f on [a, b], [PSI] (t) is given by (1), and

|T (f, g; p)|

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.5)

where in this case the wighted auxiliary mapping [[PSI].sub.p] is defined as [[PSI].sub.p]: [a, b] [right arrow] R,

[[PSI].sub.p](t) [colon, equals] [[integral].sub.a.sup.t] p(s) ds [[integral].sub.t.sup.b] p(s) g (s) ds - [[integral].sub.t.sup.b] p(s) ds [[integral].sub.a.sup.t] p(s) g(s) ds.

For other inequalities and applications for moments, see [1].

For further results, see the follow up paper [2] where various lower and other upper bounds were established.

2. A Related Functional

In [4], the authors have considered the following functional

D(f; u) [colon, equals] [[integral].sub.a.sup.b] f(x) du (x) - [u (b) - u (a)] x [1/[b - a]] [[integral].sub.a.sup.b] f (t) dt, (2.1)

provided that the Stieltjes integral [[integral].sub.a.sup.b] f (x) du (x) exists.

This functional palys an important role in approximating the Stieltjes integral [[integral].sub.a.sup.b] f (x) du (x) in terms of the Riemann integral [[integral].sub.a.sup.b] f (t) dt and the divided diference of the integrator u. Therefore, further bounds on D(f; u) will generate a flow of different error estimates for the approximation of the Stieltjes integral that plays an important role in various fields of Analysis, Numerical Analysis, Integral Operator Theory, Probability & Statistics and other fields of Modern Mathematics.

In [4], the following result in estimating the above functional D(f; u) has been obtained:

|D(f; u)| [less than or equal to] [1/2]L (M - m) (b - a), (2.2)

provided u is L-Lipschitzian and f is Riemann integrable and with the property that there exists the constants m, M [member of] R such that

m [less than or equal to] f (x) [less than or equal to] M for any x [member of] [a, b]. (2.3)

The constant 1/2 is best possible in (3) in the sense that it cannot be replaced by a smaller quantity.

If one assumes that u is of bounded variation and f is K-Lipschitzian, then D(f,u) satisfies the inequality [5]

|D(f;u)| [less than or equal to] [1/2]K (b - a) [b.[disjunction].a] (u). (2.4)

Here the constant 1/2 is also best possible.

The above inequalities have been used in [4] and [5] for obtaining inequalities between special means and on estimating the error in approximating the Stieltjes integral [[integral].sub.a.sup.b] f (x) du (x) in terms of the Riemann integral for the function f and the divided difference of u.

Now, for the function u : [a, b] [right arrow] R, consider the following auxiliary mappings [PHI], [GAMMA] and [DELTA] [3]:

[PHI](t) [colon, equals] [[(t - a) u (b) + (b - t) u (a)]/[b - a]] - u (t), t [member of] [a, b],

[GAMMA](t) [colon, equals] (t - a) [u (b) - u (t)] - (b - t) [u (t) - u (a)], t [member of] [a, b],

[DELTA](t) [colon, equals] [u; b, t] - [u; t, a], t [member of] (a, b),

where [u; [alpha], [beta]] is the divided difference of u in [alpha], [beta], i.e.,

[u; [alpha], [beta]] [colon, equals] [u ([alpha]) - u([beta])]/[[alpha] - [beta]].

The following representation of D(f, u) may be stated.

Theorem 1. Let f, u : [a, b] [right arrow] R be such that the Stieltjes integral [[integral].sub.a.sup.b] f (t) du (t) and the Riemann integral [[integral].sub.a.sup.b] f (t) dt exist. Then

D(f, u) = [[integral].sub.a.sup.b] [PHI](t) df (t) = [1/[b - a]] [[integral].sub.a.sup.b] [GAMMA] (t) df (t)

= [1/[b - a]] [[integral].sub.a.sup.b] (t - a) (b - t) [DELTA](t) df (t). (2.5)

Proof. Since [[integral].sub.a.sup.b] f (t) du (t) exists, hence [[integral].sub.a.sup.b] [PHI](t) df (t) also exists, and the integration by parts formula for Stieltjes integrals gives that

[[integral].sub.a.sup.b] [PHI](t) df (t) = [[integral].sub.a.sup.b] [[[(t - a) u (b) + (b - t) u (a)]/[b - a]] - u (t)] df (t)

= [[[(t - a) u (b) + (b - t) u (a)]/[b - a]] - u (t)] f (t)[|.sub.a.sup.b] - [[integral].sub.a.sup.b] f (t) d [[[(t - a) u (b) + (b - t) u (a)]/[b - a]] - u (t)]

= -[[integral].sub.a.sup.b] f (t) [[[u (b) - u (a)]/[b - a]]dt - du (t)] = D(f, u),

proving the required identity.

Remark 1. The identity (1) has been established in [3]. There were some typographical errors in [3] that have been corrected above.

Remark 2. If u is an integral, i.e., u (t) = [[integral].sub.a.sup.t] g (s) ds, t [member of] [a, b], then

[PHI](t) = [[t - a]/[b - a]] [[integral].sub.a.sup.b] g (s) ds - [[integral].sub.a.sup.t] g (s) ds,

[GAMMA](t) = (t - a) [[integral].sub.t.sup.b] g (s) ds - (b - t) [[integral].sub.a.sup.t] g (s) ds,

[DELTA](t) = [[[[integral].sub.t.sup.b] g (s) ds]/[b - t]] - [[[[integral].sub.a.sup.t] g (s) ds]/[t - a]],

and then, from (1), one may recapture Cerone's identity (1) for the Cebysev functional T (f, g).

Since it well known that u is an integral if and only if u is absolutely continuous, and in this case g (s) = u' (s) for s [member of] [a, b], hence (1) is indeed a proper generalisation of (1) holding for larger classes of functions than the absolutely continuous functions.

Remark 3. If one chooses u: [a, b] [right arrow] R,

u (t) = [[[integral].sub.a.sup.t] p (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds], t [member of] [a, b],

where p, g are Lebesgue integrable with pg is also integrable and [[integral].sub.a.sup.b] p (s) ds [not equal to] 0, then the identity (1) produces the representation:

E (f, g; p) [colon, equals] [[[[integral].sub.a.sup.b] p (s) f (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds]] - [[[[integral].sub.a.sup.b] p (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds]] x [1/[b - a]] [[integral].sub.a.sup.b] f (t) dt

= [[integral].sub.a.sup.b] [[PHI].sub.p] (t) df (t) = [1/[b - a]] [[integral].sub.a.sup.b] [[GAMMA].sub.p] (t) df (t)

= [1/[b - a]] [[integral].sub.a.sup.b] (t - a) (b - t) [[DELTA].sub.p] (t) df (t), (2.6)

where

[[PHI].sub.p] (t) [colon, equals] [[t - a]/[b - a]] x [[[[integral].sub.a.sup.b] p (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds]] - [[[[integral].sub.a.sup.t] p (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds]],

[[GAMMA].sub.p] (t) [colon, equals] (t - a) [[[[integral].sub.a.sup.b] p (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds]] - (b - t) [[[[integral].sub.a.sup.t] p (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds]]

and

[[DELTA].sub.p] (t) [colon, equals] [[[[integral].sub.t.sup.b] p (s) g (s) ds]/[(b - t) [[integral].sub.a.sup.b] p (s) ds]] - [[[[integral].sub.a.sup.t] p (s) g (s) ds]/[(t - a) [[integral].sub.a.sup.b] p (s) ds]].

One must observe that the identity (3) is not the same as Cerone's identity for weighted integrals (1).

For recent inequalities related to D(f; u) for various pairs of functions (f, u), see [3, pp. 112-118].

3. A Bound for f of Bounded Variation and u Continuous

It is known that if u is continuous on [a, b] and f is of bounded variation on [a, b], then the Stieltjes integral [[integral].sub.a.sup.b] f (t) du (t) exists. This integral may exists even for larger clases of integrators f, for instance, piecewise continuous functions for which the discontinuities of the integrand f do not overlap with those of the integrator u.

The following result may be stated:

Theorem 2. Let f: [a, b] [right arrow] R be of bounded variation on [a, b] and u: [a, b] [right arrow] R such that there exist the constants [gamma], [GAMMA] [member of] R with:

[gamma] [less than or equal to] u (t) [less than or equal to] [GAMMA] for any t [member of] [a, b] (3.1)

and the Stieltjes integral [[integral].sub.a.sup.b] f (t) du (t) exists. Then

|D(f; u)| [less than or equal to] ([GAMMA] - [gamma])[b.[disjunction].a](f). (3.2)

The multiplicative constant 1 in front of [GAMMA] - [gamma] cannot be replaced by a smaller quantity.

Proof. By (1), we obviously have:

[gamma] (b - t) [less than or equal to] (b - t) u (a) [less than or equal to] (b - t) [GAMMA],

[gamma] (t - a) [less than or equal to] (t - a) u (b) [less than or equal to] (t - a) [GAMMA],

-(b - a) [GAMMA] [less than or equal to] -(b - a) u (t) [less than or equal to] -(b - a) [gamma],

which gives by addition and division with b - a that

-([GAMMA] - [gamma]) [less than equal to] [[(b - t) u (a) + (t - a) u (b)]/[b - a]] - u (t) [less than equal to] [GAMMA] [gamma],

showing that |[PHI](t)| [less than or equal to] [GAMMA] [gamma] for any t [member of] [a, b].

Taking into account that for [phi] bounded and [psi] of bounded variation on [a, b] one has

|[[integral].sub.a.sup.b] [phi] (t) d[psi] (t)| [less than or equal to] [sup.[t[member of][a,b]]] |[psi] (t)|[b.[disjunction].a]([phi]),

provided the Stieltjes integral exists, we have by (1) that

|D(f; u)| [less than or equal to] [sup.[t[member of][a,b]]]| - [phi] (t)| [b.[disjunction].a] (f) [less than or equal to] ([GAMMA] [gamma])[b.[disjunction].a] (f),

proving the required inequality (7).

Now, for the sharpness of the inequality.

Assume that there exists a c > 0 such that

|D(f; u)| [less than or equal to] c ([GAMMA] - [gamma]) [b.[disjunction].a] (f), (3.3)

where u and f are as in the hypothesis of the theorem.

Consider u, f : [a, b] [right arrow] R with

u(t) = [1/2] (t - [[a+b]/2])[.sup.2], f (t) = sgn (t - [[a+b]/2]), t [member of] [a, b]). Then u is continuous, f is of bounded variation, the integral [[integral].sub.a.sup.b] f (t) du (t) exists and

[b.[disjunction].a](f) = 2, [b.[disjunction].a] f (t) dt = 0,

[GAMMA] = [sup.[t[member of][a,b]]] u (t) = [(b - a)[.sup.2]]/8, [gamma] = [inf.[t[member of][a,b]]] u (t) = 0,

[[integral].sub.a.sup.b] f (t) du (t) = [[integral].sub.a.sup.b] sgn (t - [[a + b]/2])(t - [[a + b]/2]) dt

= [[integral].sub.a.sup.b]|t - [[a + b]/2]| dt = [(b - a)[.sup.2]]/4.

Substituting into (8) we get [(b-a)[.sup.2]]/4 [less than or equal to] [c(b-a)[.sup.2]]/4, which implies that c [greater than or equal to] 1.

Corollary 1. Let f: [a, b] [right arrow] R be of bounded variation and u: [a, b] [right arrow] R continuous on [a, b]. Then:

|D(f; u)| [less than or equal to] [[max.[t[member of][a,b]]] u (t) - [min.[t[member of][a,b]]] u (t)] [b.[disjunction].a] (f). (3.4)

The inequality (9) is sharp.

If we consider the Cebysev functional T (f, g), then we can state the following corollary as well:

Corollary 2. Let f : [a, b] [right arrow] R be of bounded variation and g: [a, b] [right arrow] R a Lebesgue integrable function such that there exists the constants m and M with

m [less than or equal to] g (s) [less than or equal to] M for a.e. s [member of] [a, b]. (3.5)

Then

|T (f, g)| [less than or equal to] (b - a) (M - m) [b.[disjunction].a] (f). (3.6)

Proof. We choose u (t) [colon, equals] [[integral].sub.a.sup.t] g (s) ds which is continuous on [a, b] and satisfies the inequality (6) with [gamma] = (b - a) m and [GAMMA] = (b - a) M and apply Theorem 2.

Remark 4. If we assume that for the Lebesgue integrable function g, [[integral].sub.a.sup.[dot]]g (s) ds satisfies the condition

[gamma] [less than or equal to] [[integral].sub.a.sup.t] g (s) ds [less than or equal to] [GAMMA] for any t [member of] [a, b],

then

|T (f, g)| [less than or equal to] ([GAMMA] - [gamma]) [b.[disjunction].a] (f)

and the inequality is sharp. The equality case is realised for g (t) = t - [[a+b]/2] and f (t) = sgn (t - [[a+b]/2]), t [member of] [a, b].

It is an open problem wether or not the bound in (11) is sharp.

Remark 5. If p, g [member of] L [a, b] so that pg [member of] L [a, b] and [[integral].sub.a.sup.b] p (s) ds [not equal to] 0 and there exists the constants, [delta], [DELTA] so that

[delta] [less than or equal to] [[[integral].sub.a.sup.t] p (s) g (s) ds]/[[[integral].sub.a.sup.b] p (s) ds] [less than or equal to] [DELTA]

for any t [member of] [a, b], then

|E (f, g; p)| [less than or equal to] ([DELTA] - [delta]) [b.[disjunction].a] (f).

The last inequality is sharp.

4. Application for Approximating the Stieltjes Integral

Let us consider the partition of the interval [a, b] given by

[I.sub.n] : a = [t.sub.0] < [t.sub.1] < ... < [t.sub.n-1] < [t.sub.n] = b.

Denote v ([I.sub.n]) [colon, equals] max {[h.sub.i]|i = 0,..., n - 1}, where [h.sub.i] [colon, equals] [t.sub.i+1] - [t.sub.i], i = 0,..., n - 1. If u: [a, b] [right arrow] R is continuous on [a, b] and if we define

[M.sub.i] [colon, equals] [sup.[t[member of][[t.sub.i],[t.sub.i+1]]]] u (t), [m.sub.i] [colon, equals] [inf.[t[member of][[t.sub.i],[t.sub.i+1]]]] u (t)

and

v (u, [I.sub.n]) [colon, equals] [max.[0[less than or equal to]i[less than or equal or]n-1]] ([M.sub.i] - [m.sub.i]),

then, obviously, by the continuity of u on [a, b], for any [epsilon] [greater than or equal to] 0, there exists a [delta]> 0 and a division [I.sub.n] with norm v ([I.sub.n]) < [delta] such that v (u, [I.sub.n]) < [epsilon].

Consider now the quadrature rule

[S.sub.n] (f, u, [I.sub.n]) [colon, equals] [n-1.summation over (i=0)] [[u ([t.sub.i+1]) - u ([t.sub.i])]/[[t.sub.i+1] - [t.sub.i]]] x [[integral].sub.[t.sub.i].sup.[t.sub.i+1]] f (t) dt, (4.1)

provided u is continuous on [a, b] and f is of bounded variation on [a, b].

We may state the following result in approximating the Stieltjes integral:

Theorem 3. Let f, u : [a, b] [right arrow] R be such that f is of bounded variation on [a, b] and u is continuous on [a, b]. Then for any division [I.sub.n] as above,

[[integral].sub.a.sup.b] f (t) du (t) = [S.sub.n] (f, u, [I.sub.n]) + [R.sub.n] (f, u, [I.sub.n]), (4.2)

where the remainder [R.sub.n] (f, u, [I.sub.n]) satisfies the estimate:

|[R.sub.n] (f, u, [I.sub.n])| [less than or equal to] v (u, [I.sub.n]) [b.[disjunction].a] (f). (4.3)

Proof. Applying Theorem 2 on the intervals [[t.sub.i], [t.sub.i+1]], i = 0,..., n - 1, we have sucessively:

|[R.sub.n] (f, u, [I.sub.n])| = |[n-1.summation over (i=0)] [[integral].sub.[t.sub.i].sup.[t.sub.i+1]] f (t) du (t) - [[u ([t.sub.i+1]) - u ([t.sub.i])]/[[t.sub.i+1] - [t.sub.i]]] [[integral].sub.[t.sub.i].sup.[t.sub.i+1]] f (t) dt|

[less than or equal to] [n-1.summation over (i=0)] |[[integral].sub.[t.sub.i].sup.[t.sub.i+1]] f (t) du (t) - [[u ([t.sub.i+1]) - u ([t.sub.i])][[t.sub.i+1] - [t.sub.i]]] [[integral].sub.[t.sub.i].sup.[t.sub.i+1]] f (t) dt|

[less than or equal to] [n-1.summation.over (i=0)] ([M.sub.i] - [m.sub.i]) [[t.sub.i+1].[disjunction].[t.sub.i]] (f) [less than or equal to] v (u, [I.sub.n]) [b.[disjunction].a] (f)

and the estimate (14) is obtained.

References

[1] P. Cerone, On an identity for the Chebychev functional and some ramifications, J. Inequal. Pure & Appl. Math., 3(1) (2002), Article 2.

[2] P. Cerone and S. S. Dragomir, New upper and lower bounds for the Cebysev functional, J. Inequal. Pure & Appl. Math., 3(5) (2002), Article 77.

[3] S. S. Dragomir, Inequalities of Gruss type for the Stieltjes integral and applications, Kragujevac J. Math., 26 (2004), 89-112.

[4] S. S. Dragomir and I. Fedotov, An inequality of Gruss type for Riemann-Stieltjes integral and applications for special means, Tamkang J. Math., 29(4) (1998), 287-292.

[5] S. S. Dragomir and I. Fedotov, A Gruss type inequality for mappings of bounded variation and applications to numerical analysis, Non. Funct. Anal. & Appl., 6(3) (2001), 425-437.

S. S. Dragomir ([dagger])

School of Computer Science and Mathematics Victoria University of Technology PO Box 14428, Melbourne City, VIC 8001, Australia.

Received July 27, 2005, Accepted October 5, 2005.

*2000 Mathematics Subject Classification. Primary 26D15, 26D10.

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