# A companion for the generalized Ostrowski and the generalized trapezoid type inequalities.

1. Introduction

In 2000, Dragomir (2) answered to the problem of approximating the Stieltjes integral [[integral].sub.a.sup.b]f(x)du(x) by the quantity [u(b) - u(a)]f (x), which is a natural generalization of the Ostrowski problem (3) analysed in 1937. He obtained the following result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

for each x [member of] [a, b], provided f is of bounded variation on [a, b], while u: [a, b] [right arrow] R is r-H-Holder continuous, i.e., we recall that:

|u(x) - u(y)| [less than or equal to] H[|x - y|.sup.r] for each x, y [member of] [a, b].

From a different view point, the problem of approximating the Stieltjes integral [[integral].sub.a.sup.b]f(x)du(x) by the generalized trapezoid rule [(u(b) - u(x))f (b) + (u(x) - u(a))f (a)] was considered by Dragomir et al. (4). The following inequality was obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for each x [member of] [a, b], provided f is of bounded variation on [a, b] while u: [a, b] [right arrow] R is r-H-Holder continuous.

For a Riemann-Stieltjes integrable function f: [a, b] [right arrow] R and for a given x [member of] [a, b], it is natural to investigate the distances between the quantities

f (x), 1/u(b) - u(a) [[integral].sub.a.sup.b]f(x)du(x) and [(u(b) - u(x))f(b) + (u(x) - u(a))f(a)]/u(b) - u(a) (1.2)

respectively, and to seek sharp upper bounds for these distances in terms of different measure that can be associated with f, where f is restricted to particular classes of functions including functions of bounded variation, Lipschitzian, convex and absolutely continuous functions.

The authors of (2), (4) have been given sharp upper bounds for absolute value between the first quantity and the second, the second and the third in (1.2).

The main aim of this paper is to provide sharp upper bounds for absolute value of the remaining difference between the first quantity and the third in (1.2), that is,

[[PSI].sub.f](x): = f(x) - [(u(b) - u(x))f(b) + (u(x) - u(a))f(a))]/[(u(b) - u(a))], x [member of] [a, b]. (1.3)

As applications, some bounds for the absolute value of the difference

[[PHI].sub.f](x):= [N.summation over (i= 1)] [p.sub.i]f([x.sub.i]) - ((u(b) - [[SIGMA].sub.i=1.sup.n] [p.sub.i]u([x.sub.i])f(b) + ([[SIGMA].sub.i=1.sup.n] [p.sub.i]u([x.sub.i]) - u(a))f(a))/(u(b) - u(a)) (1.4)

where [x.sub.i] [member of] [a, b], pi [greater than or equal to] 0, i [member of] {1, 2, ... , n} and [[SIGMA].sub.i=1.sup.n] [p.sub.i] = 1, are also given.

Remark Using the Stieltjes integral by Dragomir (2), generalization of the Ostrowski problem (3) was considered, so our results are natural to generalize some results obtained by Barnett et al.'s some results (1).

The case when f is of bounded variation and u Holder continuous

The following representation holds.

Lemma 2.1 Let f is of bounded function on [a, b] and let T: [[a, b].sup.2] [right arrow] R be given by

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

Then we have the following representation,

[[PSI].sub.f](x): = [1/[(u(b) - u(a))]][[integral].sub.a.sup.b]T(x, s)df(s), x [member of] [a, b], (2.2)

where the integral is considered in the Riemann-Stieltjes sense.

Proof. If f is bounded on [a, b], the for any t [member of] [a, b], the Riemann-Stieltjes Integral [[integral].sub.a.sup.x]df(s) = f(x) - f(a), [[integral].sub.x.sup.b]df(s) = f(b) - f(x). It follows that

[[integral].sub.a.sup.b]T(x, s)df(s) = (u(x) - u(a))[[integral].sub.a.sup.x]df(s) + (u(x) - u(b)) [[integral].sub.x.sup.b]df(s) = (u(b) - u(a))[[PSI].sub.f](x),

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

The following provides a sharp bound for the absolute value of [[PSI].sub.f] where f is of bounded variation and u is r-H-Holder continuous.

Theorem 2.2 If f: [a, b] [right arrow] R is of bounded variation and u: [a, b] [right arrow] R is r-H-Holder continuous on the interval [a, b], i.e.,

|u(x) - u(y)| [less than or equal to] H[|x - y|.sup.r] for each x, y [member of] [a, b].

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

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

for any x [member of] [a, b]. The constant 1/2 is also the best possible in both branches of (2.5).

Proof. Utilizing the representation (2.2), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies the inequalities (2.3) and (2.4).

Combination inequality (1.1) and the above inequality, we have inequality (2.5).

Now, we prove that The constant 1/2 is also the best possible in both branches of (2.5). Consider the function [f.sub.0](t) = |t - (a + b)/2| which is of bounded variation on [a, b], with [f.sub.0](a) = [f.sub.0](b) = (b - a)/2 and [V.sub.a.sup.b]([f.sub.0]) = b - a. And [u.sub.0](x) = x which is 1-1-Holder continuous. According to the proof of the best possibility of the constant in Theorem 1 in (1), the sharpness of the constant 1/2 in the inequality (2.5) is the best possible.

As application, we give the case when f and u have some slight variations as follows.

Corollary 2.3 If f: [a, b] [right arrow] R is [L.sub.1]-Lipschitzian on [a, x] and [L.sub.2]-Lipschitzian on [x, b], [L.sub.1], [L.sub.2] > 0, x [member of] [a, b], while the function u: [a, b] [right arrow] R satisfies some local Holder continuous, namely,

|u(t) - u(a)| [less than or equal to] [H.sub.1][|t - a|.sup.r1] for any t [member of] [a, x] (2.6)

and

|u(b) - u(t)| [less than or equal to] [H.sub.2][|b - t|.sup.r2] for any t [member of] [x, b] (2.7)

where [H.sub.1], [H.sub.2] > 0, [r.sub.1], [r.sub.2] [member of] (-1, +[infinity]), then

|[[psi].sub.f](x)| [less than or equal to] 1/|u(b) - u(a)| [[L.sub.1]|u(x) - u(a)|(x - a) + [L.sub.2]|u(x) - u(b)|(b - x)] (2.8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.10)

for any x [member of] [a, b]

Proof. It is well known that if g: [[alpha], [beta]] [right arrow] R is L-Lipschitzian, then g is of bounded variation and [V.sub.[alpha].sup.[beta]](g) [less than or equal to] L([beta] - [alpha]). Therefore, by the first inequality (2.4), we get the corresponding inequality (2.8). Using the local Holder continuity of the function u, we have inequality (2.9) from (2.8). The other inequalities follow by the Holder inequality and the details are omitted.

Corollary 2.4 If f: [a, b] [right arrow] R is monotonic nondecreasing, while u: [a, b] [right arrow] R is L-Lipschitzian on [a, b], where L > 0, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any x [member of] [a, b].

Proof. It is easy to observe that we obtain Corollary 2.4 by using Theorem 2.2 and Holder inequality, so the details are omitted.

3. The case when f is absolutely continuous and u Holder continuous

When f is absolutely continuous, the following representation holds.

Lemma 3.1 If f is of bounded function on [a, b]. Then we have the following representation,

[[PSI].sub.f] (x) = 1/(u(b) - u(a))[[integral].sub.a.sup.b]T(x, s)f'(s)ds, x [member of] [a, b], (3.1)

where the integral is considered in the Lebesgue sense and where the kernel T: [[a, b].sup.2] [right arrow] R has been defined in (2.1).

We cite the following Lebesgue norms defined in Section 3 in (1) as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 3.2 If f is absolutely continuous on [a, b], u: [a, b] [right arrow] R is r-H-Holder continuous on [a, b], where H >0 and r [member of] (-1, [infinity]). Then we have the following inequalities:

|[[PSI].sub.f] (x)| [less than or equal to] 1/|u(b) - u(a)| [|u(x) - u(a)| || f' ||[.sub.[a,x],1] +|u(b) - u(x)| || f' ||[.sub.[x,b],1] (3.2)

[less than or equal to] H/|u(b) - u(a)|[[(x - a).sup.r] || f' ||[.sub.[a,x],1] + [(b - x).sup.r] || f' ||[.sub.[x,b],1] (3.3)

[less than or equal to] H/|u(b) - u(a)|W(x), x [member of] [a, b], (3.4)

where W(x) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and W(x) should be seen as all four possible combinations.

Proof. By Lemma 3.1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for x [member of] [a, b], which implies inequalities (3.2) and (3.3).

Utilizing (3.4) and (3.5) in (1) and the above inequality, we obtain the desired inequality (3.4).

Corollary 3.3 If f is absolutely continuous on [a, b], the function u: [a, b] [right arrow] R satisfies some local Holder continuous defined by (2.6) and (2.7). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where W(x) is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and W(x) should be seen as all four possible combinations.

Proof. It is similar to the proof of Theorem 3.2, so the details are omitted.

4. The case when f is convex and u monotonic nondecreasing and bi-Holder

Before giving the case when f is convex and u is monotonic nondecreasing and bi-Holder, we establish sharp lower and upper bounds for the remaining differences as follows:

[[OMEGA].sub.1](x) := [[integral].sub.a.sup.b] f(x)du(x) - (u(b) - u(a))f(x) (4.1)

and

[[OMEGA].sub.2](x) := [(u(b) - u(x))f(b) + (u(x) - u(a))f(a)] -[[integral].sub.a.sup.b] f(x)du(x). (4.2)

Theorem 4.1 If f: [a, b] [right arrow] R is a convex function on [a, b] with [f'.sub.-] (b) and [f'.sub.+] (a) finite, and u: [a, b] [right arrow] R is monotonic nondecreasing and bi-Holder function on [a, b], that is,

[L.sub.1][[y - x].sup.r] [less than or equal to] u(y) - u(x) [less than or equal to] [L.sub.2][[y - x].sup.r], for x [less than or equal to] y, x, y [member of] [a, b], (4.3)

where [L.sub.1], [L.sub.2] > 0 and r > -1. Then we have the following inequalities:

[1/[r+1]][[L.sub.1][(b - x).sup.r+1][f'.sub.+](x) - [L.sub.2][(x - a).sup.r+1][f'.sub.-](x)] [less than or equal to] [[OMEGA].sub.1](x) [less than or equal to] [1/[r+1]][[L.sub.2][(b - x).sup.r+1][f'.sub.-](b) - [L.sub.1][(x - a).sup.r+1][f'.sub.+](a)] (4.4)

and

[1/[r+1]][[L.sub.1][(b - x).sup.r+1][f'.sub.+](x) - [L.sub.2][(x - a).sup.r+1][f'.sub.-](x)] [less than or equal to] [[OMEGA].sub.2](x) [less than or equal to] [1/[r+1]][[L.sub.2][(b - x).sup.r+1][f'.sub.-](b) - [L.sub.1][(x - a).sup.r+1][f'.sub.+](a)], (4.5)

where [[OMEGA].sub.1] (x) and [[OMEGA].sub.2] (x) are defined by (4.1) and (4.2). The constant 1/(r + 1) is sharp in both inequalities.

Proof. First of all, we give the proof of inequality (4.4). It is easy to see that for any locally absolutely continuous function f: [a, b] [right arrow] R, we have the identity

[[integral].sub.a.sup.x] (u(t) - u(a))f'(t)dt + [[integral].sub.x.sup.b] (u(t) - u(b))f'(t)dt = (u(b) - u(a))f(x) - [[integral].sub.a.sup.b] f(t)du(t) (4.6)

for any x [member of] (a, b), where f' is the derivation of f which exists a.e. on (a, b).

Since f is convex, then it is locally Lipschitzian and thus (4.6) holds. Moreover, for any x [member of] (a, b), we have the inequalities

f'(t) [less than or equal to] [f'.sub.-](x) for a.e. t [member of] [a, x] (4.7)

and

f'(t) [greater than or equal to] [f'.sub.+](x) for a.e. t [member of] [x, b]. (4.8)

If we multiply (4.7) by u(t) - u(a) [greater than or equal to] 0, t [member of] [a, x] and integrate on [a, x], by (4.3), we get

[[integral].sub.a.sup.x](u(t) - u(a))f'(t)dt [less than or equal to] [f'.sub.-](x) [[integral].sub.a.sup.x] (u(t) - u(a))dt [less than or equal to] 1/[r+1][L.sub.2][(x - a).sup.r+1][f'.sub.-](x) (4.9)

and if we multiply (4.8) by u(b) - u(x) [greater than or equal to] 0, t [member of] [x, b] and integrate on [x, b], by (4.3), we get

[[integral].sub.x.sup.b] (u(b) - u(t))f'(t)dt [greater than or equal to] [f'.sub.+](x) [[integral].sub.x.sup.b] (u(b) - u(t))dt [greater than or equal to] [1/[r+1]][L.sub.1][(b - x).sup.r+1][f'.sub.+](x). (4.10)

If we subtract (4.10) from (4.9) and use the representation (4.6), we deduce the first inequality in (4.4).

Since f is convex, then we have the inequalities

f'(t) [greater than or equal to] [f'.sub.+](a) (a) for a.e. t [member of] [a, x] (4.11)

and

f'(t) [less than or equal to] [f'.sub.-](b) for a.e. t [member of] [x, b]. (4.12)

If we multiply (4.11) by u(t) - u(a) [greater than or equal to] 0, t [member of] [a, x] and integrate on [a, x], by (4.3), we get

[[integral].sub.a.sup.x] (u(b) - u(t))f'(t)dt [greater than or equal to] [f'.sub.+](a) [[integral].sub.a.sup.x] (u(b) - u(t))dt [greater than or equal to] [1/[r+1]][L.sub.1][(x - a).sup.r+1][f'.sub.+](a). (4.13)

and if we multiply (4.12) by u(b) - u(x) [greater than or equal to] 0, t [member of] [x, b], integrate on [x, b] and integrate on [a, x], by (4.3), we get

[[integral].sub.x.sup.b] (u(b) - u(t))f'(t)dt [less than or equal to] [f'.sub.-](b) [[integral].sub.x.sup.b] (u(b) - u(t))dt [less than or equal to] [1/[r+1]][L.sub.2][(b - x).sup.r+1][f'.sub.-](b). (4.14)

If we subtract (4.14) from (4.13) and use the representation (4.6), we deduce the second inequality in (4.4).

Now we prove that the constant 1/(r + 1) is also the best possible in inequalities (4.4). Consider the function [f.sub.0](t) = k |t - (a + b)/2| which is a convex function on the interval [a, b], where k > 0, t [member of] [a, b]. Then

[f'.sub.0] - ([a + b]/2) = -k, [f.sub.0+]'([a + b]/2) = k and [f.sub.0] ([a + b]/2) = 0.

And [u.sub.0](x) = x, then [L.sub.1] = [L.sub.2] = r = 1. Thus we have [[integral].sub.a.sup.b] [f.sub.0](t)dt = k[(b-a).sup.2]/2. If in (4.4) we choose f = [f.sub.0], u = [u.sub.0] and x = (a + b)/2. According to the proof of the best possibility of the constant in Lemma 2.1 in (5), the sharpness of the constant 1/(r + 1) in the inequality (4.4) is the best possible.

Secondly, we give the proof of inequality (4.5). It is easy to see that for any locally absolutely continuous function f: [a, b] [right arrow] R, we have the identity

[[integral].sub.a.sup.b] (u(t) - u(x))f '(t)dt = (u(b) - u(x))f (b) +(u(x) - u(a))f(a) -[[integral].sub.a.sup.b] f(t)du(t) (4.15)

for any x [member of] (a, b), where f' is the derivation of f which exists a.e. on (a, b).

Since f is convex, then it is locally Lipschitzian and thus (4.15) holds. The following proof is similar to the proof of inequalities (4.4) and Lemma 2.1 in (6), so the details are omitted.

In the following we give sharp lower and upper bounds for the remaining difference (1.4) when f is convex and u is monotonic nondecreasing and bi-Holder.

Theorem 4.2 If f: [a, b] [right arrow] R is a convex function on [a, b] with [f'.sub.-](b) and [f'.sub.+] (a) finite, and u: [a, b] [right arrow] R is monotonic nondecreasing and bi-Holder function on [a, b] defined by (4.3). Then we have the following inequalities:

[1/(u(b) - u(a))][[L.sub.1][(x - a).sup.r+1][f'.sub.+](a) - [[L.sub.2][(b - x).sup.r+1][f'.sub.-](b)] [less than or equal to] [[PSI].sub.f](x) [less than or equal to] 1/(u(b) - u(a))[[L.sub.2][(x - a).sup.r+1][f'.sub.-](x) - [[L.sub.1][(b - x).sup.r+1][f'.sub.+](x)] (4.16)

where [[PSI].sub.f] (x) is defined by (1.3). The constant 1 is the best possible on both sides of (4.16).

Proof. From Lemma 2.1,

u(b) - u(a)) [[PHI].sub.f] (x) = (u(x) - u(a))(f(x) - f(a)) -(u(b) - u(x))(f(b) - f(x)), x[member of] [a, b]. (4.17)

Let x [member of] (a, b), then, by the convexity of f, we have

(x - a)[f'.sub.-] (x) [greater than or equal to] f(x) - f(a) [greater than or equal to] (x - a)[f'.sub.+] (a) (4.18)

and

(b - x)[f'.sub.-] (b) [greater than or equal to] f(b) - f(x) [greater than or equal to] (b - x)[f'.sub.+] (x). (4.19)

If we multiply (4.18) by u(x) - u(a) > 0 and (4.19) by u(b) - u(x) > 0, we obtain

(u(x) - u(a))(x - a)[f'.sub.-](x) [greater than or equal to] (u(x) - u(a))(f(x) - f(a)) [greater than or equal to] (u(x) - u(a))(x - a) [f'.sub.+] (a) (4.20)

and

(u(b) - u(x))(b - x) [f'.sub.-](b) [greater than or equal to] (u(b) - u(x))(f(b) - f(x)) [greater than or equal to] (u(b) - u(x))(b - x) [f'.sub.+] (x). (4.21)

By (4.3), the above inequalities can rewrite

[[L.sub.2][(x - a).sup.r+1][f'.sub.-](x) [greater than or equal to] (u(x) - u(a))(f(x) - f(a)) [greater than or equal to] [[L.sub.1][(x-a).sup.r+1][f'.sub.+](a) (4.22)

and

-[L.sub.1][(b - x).sup.r+1][f'.sub.+](x) [greater than or equal to] -(u(b) - u(x))(f (b) - f (x)) [greater than or equal to] [-L.sub.2][(b - x).sup.r+1][f'.sub.-](b). (4.23)

Finally, on adding (4.22) to (4.23), we deduce the desired result (4.16).

Now we prove that The constant 1 is also the best possible in inequalities (4.16). Consider the function [f.sub.0](t) = k |t - (a + b)/2| which is a convex function on [a, b], where k > 0, t [member of] [a, b]. Then

[f.sub.0-]'(b) = -k, [f.sub.0+]'(a) = k, [f.sub.0-]'([a + b]/2) = -k, [f.sub.0+]'([a + b]/2) = k, [f.sub.0]([a + b]/2) = 0 and [f.sub.0](a) = [f.sub.0](b) = [k(b - a)]/2

And [u.sub.0](x) = x, then [L.sub.1] = [L.sub.2] = r = 1. If in (4.4) we choose f = [f.sub.0], u = [u.sub.0] and x = (a + b)/2. According to the proof of the best possibility of the constant in Theorem 3 in (1), the sharpness of the constant 1 in the inequality (4.16) is the best possible.

5. Some applications

As applications, some bounds for the absolute value of the difference (1.4).

Proposition 5.1 If f: [a, b] [right arrow] R is of bounded variation and u: [a, b] [right arrow] R is r-H-Holder continuous on [a, b]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5.1)

where [p.sub.i] [greater than or equal to] 0, [[SIGMA].sub.i=1.sup.n] [p.sub.i ] = 1, the constant 1/2 is also the best possible in both branches of (5.1).

Proof. We use the third inequality in (2.5) to state:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5.2)

for i = 1, 2, ... , n.

If we multiply (5.2) by [p.sub.i] [greater than or equal to] 0, sum over i = 1 to n, we deduce the desired result (5.1).

The fact that 1/2 is the best possible follows from the fact that it is the best possible for n = 1.

In a similar manner, on utilizing the third inequality in (2.10), we can state the following result:

Proposition 5.2 If f: [a, b] [right arrow] R is L-Lipschitzian on [a, b] and u: [a, b] [right arrow] R is r-H-Holder continuous on [a, b]. Then

|[[PHI].sub.f](x)| [less than or equal to] HL(b - a)/|u(b) - u(a)|[[[1/2](b - a) + [n.summation over (i=1)][p.sub.i]|[x.sub.i] - [a + b]/2|].sup.r],

Where [p.sub.i] [greater than or equal to] 0, [[SIGMA].sub.i=1.sup.n] [p.sub.i] = 1.

Finally, on utilizing the inequality in (3.3), we can also state that:

Proposition 5.3 If f is absolutely continuous on [a, b], u: [a, b] [right arrow] R is r-H-Holder continuous on [a, b]. Then

|[[PHI].sub.f](x)| [less than or equal to] H/|u(b) - u(a)| [[n.summation over (i=1)] [[x.sub.i] - a].sup.r] || f' || [.sub.[a,[x.sub.i],1]] + [[n.summation over (i=1)] [b - [x.sub.i]].sup.r] || f' || [.sub.[[x.sub.i],b],1]], where [p.sub.i] [greater than or equal to] 0, [[SIGMA].sub.i=1.sup.n] [p.sub.i] = 1

Received October 4, 2010, Accepted March 6, 2013.

References

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(2.) S.S. Dragomir, On the Ostrowski's inequality for Riemann-Stieltjes integral, Korean J. Appl. Math., 7 (2000), 477-485.

(3.) D.S. Mitrinovic, J.E. Pecaric, and A.M. Fink, Inequalities for functions and Their integrals and derivatives, Kluwer Acadmic Publishers, Dordrecht, 1994.

(4.) S.S. Dragomir, C. Buse, M.V. Boldea, and L. Braescu, A generalisation of the trapezoid rule for the Riemann-Stieltjes integral and applications, Nonlinear Anal. Forum, 6 no. 2 (2001), 337-351.

(5.) S.S. Dragomir, An inequality improving the first Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math., 3 no. 2 (2002), Art. 31.

(6.) S.S. Dragomir, An inequality improving the second Hermite-Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products, J. Inequal. Pure Appl. Math., 3 no. 3 (2002), Art. 35.

* 2010 Mathematics Subject Classification. Primary 26D10, 26D15.

Wengui Yang [dagger]

Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia 472200, China

[dagger] E-mail: yangwg8088@163.com
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Author: Printer friendly Cite/link Email Feedback Yang, Wengui Tamsui Oxford Journal of Information and Mathematical Sciences Report 9CHIN May 1, 2013 4131 Some new modular equations and their applications. On [W.sub.2]-curvature tensor in a Kenmotsu manifold. Inequalities (Mathematics) Riemann integral

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