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A Generalzation of Ostrowski-gruss type inequality for first differentiable mappings.

1. Introduction

In 1938, A. Ostrowski (8) proved the following integral inequality.

Theorem 1. Let f: I[??] R [right arrow]R be a differentiable mapping in [I.sup.0] (interior of I), and let a, b [member of] [I.sup.0] with a < b. If [f.sup.']:(a,b)[right arrow]R is bounded on (a, b) with sup |[f.sup.'](t)| [less than or equal to] M, then we have

T[member of][a,b]

|f(x) - [1/[b - a]][[[integral].sub.a.sup.b] f(t)dt|[less than or equal to][1/4 + [[(x - [a + b]/2)].sup.2]/[(b - a).sup.2]](b - a)M, (1.1)

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

The constant 1/4 is sharp in the sense that it cannot be replaced by a smaller one.

This inequality provides an upper bound for the approximation of integral mean of a function f by the functional value f(x) at x[member of] [a, b] .In 1997, Dragomir and Wang (3), by the use of the Gruss inequality proved the following Ostrowski-Gruss type integral inequality.

Theorem 2. Let f: I[right arrow]R, where I [??] R is an interval, be a mapping differentiable in the interior [I.sup.0] of I, and let a, b [member of] [I.sup.0] with a < b. If [gamma] [less than or equal to] [f.sup.'](x) [less than or equal to] [GAMMA], x [member of] [a, b] for some constants [gamma], [GAMMA] [member of] R, then

|f(x) - [1/[b - a]][[b][integral][a]]f(t)dt - [[f(b) - f(a)]/[b - a]](x - [a + b]/2)|[less than or equal to][1/4](b - a)([GAMMA] - [Gamma]), (1.2)

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

This inequality provides a connection between Ostrowski inequality (8) and the Gruss inequality (5). In 2000, M. Matic, J. Pecaric and N. Ujevic (7), by the use of pre-Gruss inequality improved the factor of the right membership of (1.2) with [1/[4[square root of 3]]] as follows.

Theorem 3. Under the assumption of Theorem 2, we have

|f(x) - [1/[b - 1]][[b][integral][a]]f(t)dt - [[f(b) - f(a)]/[b - a]](x - [a + b]/2)| [less than or equal to][1/[4[square root of (3)]]]([GAMAM] - [gamma])(b - a), (1.3)

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

In 2000, Barnett et al. (1), by the use of Chebyshev's functional, improved the Matic-Pecaric-Ujevic result by providing first membership of the right side of (1.3) in terms of Euclidean norm as follows.

Theorem 4. Let f: [a,b] [right arrow] R be an absolutely continuous function whose derivative [f.sup.'][member of] [L.sub.2] [a, b] . Then we have the inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

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

We define for two mappings f, g: [a,b] [right arrow]R, the Chebyshev functional as

T(f,g) = [1/[b - a]][[integral].sub.a.sup.b]f(t)g(t)dt - (1/[b - a][[integral].sub.a.sup.b]f(t)dt)(1/[b - a][[integral].sub.a.sup.b]g(t)dt),

provided that f, g and fg are integrable on [a, b].

Also in (1) we can find the pre-Gruss inequality as

[T.sup.2] (f, g) [less than or equal to] T (f, f) T (g, g),

where f, g [member of] [L.sub.2] [a, b] and T (f, g) is the Chebyshev functional as defined above.

In this paper, we give a generalization of (1.4) and then apply it to probability density functions, generalized beta random variable and special means.

2. Main Results

Theorem 5. Let f: [a,b] [right arrow] R be an absolutely continuous function whose first derivative [f.sup.']

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

Proof. We consider the kernel p: [[a,b].sup.2] [right arrow] R as defined in (2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using Korkine's identity

T(f,g): = [1/[2[[(b - a)].sup.2]]][[b][integral][a]][[b][integral][a]](f(t) - f(s))(g(t) - g(s))dtds,

we obtain

[1/[b - a]][[b][integral][a]]p(x,t)[f.sup.'](t)dt - [1/[b - a]][[b][integral][a]]p(x,t)dt[1/[b - a]][[b][integral][a]][f.sup.'](t)dt

= [1/[2[[(b - a)].sup.2]]][[b][integral][a]][[b][integral][a]](p(x,t) - p(x,s))([f.sup.'](t) - [f.sup.'](s))dtds. (2.2)

Since,

[1/[b - a]][[b][integral][a]]p(x,t)[f.sup.'](t)dt = (1 - h)f(x) + h[[f(a) + f(b)]/2] - [1/[b - a]][[b][integral][a]]f(t)dt,

[1/[b - a]][[b][integral][a]]p(x,t)dt = (1 - h)(x - [a + b]/2),

and

[1/[b - a]][[b][integral][a]][f.sup.'](t)dt = [[f(b) - f(a)]/[b - a]],

then by (2.2) we get the following identity

(1 - h)[f(x) - [[f(b) - f(a)]/[b - a]](x - [a + b]/2)] + h[[f(a) + f(b)]/2] - [1/[b - a]][[b][integral][a]]f(t)dt

= [1/[2[[(b - a)].sup.2]]][[b][integral][a]][[b][integral][a]](p(x,t) - p(x,s))([f.sup.'](t) - [f.sup.'](s))dtds, (2.3)

for all x [member of] [a + h [b - a]/2, b - h [b - a]/2] and h [member of] [0,1]

Using the Cauchy-Bunaikowski-Schwartz inequality for double integrals, we may write

[1/[2[[(b - a)].sup.2]]]|[[b][integral][a]][[b][integral][a]](p(x,t) - p(x,s))([f.sup.'](t) - [f.sup.'](s))dtds|

[less than or equal to][[(1/[2[(b - a).sup.2]][[b][integral][a]][[b][integral][a]][[(p(x,t) - p(x,s))].sup.2]dtds)].sup.[1/2]] X [[(1/[2[(b - a).sup.2]][[b][integral][a]][[b][integral][a]][[([f.sup.'](t) - [f.sup.'](s))].sup.2]dtds)].sup.[1/2]]. (2.4)

However,

[1/[2[[(b - a)].sup.2]]][[b][integral][a]][[b][integral][a]][[(p(x,t) - p(x,s))].sup.2]dtds

= [1/[b - a]][[b][integral][a]][p.sup.2](x,t)dt - (1/[b - a][[b][integral][a]]p(x,t)dt[).sup.2]

= [1/[b - a]][[[[(x - (a + h[b - a]/2))].sup.3] + [[(b - h[[b - a]/2] - x)].sup.3]]/3 + [[h.sup.3][[(b - a)].sup.3]]/12] - [[(1 - h)].sup.2][[(x - [a + b]/2)].sup.2]. (2.5)

In addition simple calculations shows that

[[(x - (a + h[b - a]/2))].sup.3] + [[(b - h[[b - a]/2] - x)].sup.3]

= (b - a)(1 - h)[3[[(x - [a + b]/2)].sup.2] + [[[(1 - h)].sup.2][[(b - a)].sup.2]]/4], (2.6)

and

[1/[2[[(b - a)].sup.2]]][[b][integral][a]][[b][integral][a]][[([f.sup.'](t) - [f.sup.'](s))].sup.2]dtds

= [1/[b - a]][parallel][f.sup.'][parallel]sub.2.sup.2] - [[([f(b) - f(a)]/[b - a])].sup.2]. (2.7)

Using (2.3)-(2.7),we deduce the first inequality.

Moreover, if [gamma] [less than or equal to][f.sup.'](t) [less than or equal to] [GAMMA] almost everywhere t on (a, b), then, by using Gruss inequality, we have

0[less than equal or equal to][1/[b - a]][[b][integral][a]][([f.sup.'](t)).sup.2]dt - (1/[b - a][[b][integral][a]][f.sup.'](t)dt[).sup.2][less than or equal to][1/4][[([GAMMA] - [gamma])].sup.2],

which proves the last inequality of (2.1).

Remark 1. Since

[3h.sup.2]-3h + 1 [less than or equal to] 1, [for all] h [member of] [0, 1].

and is minimum for h = 1/2.

Thus, (2.1) shows an overall improvement in the inequality obtained by Barnett et al. (5).

The following corollary contains some special cases of (2.1).

Corollary 1. 1. For h = 1, i.e., x = [a + b]/2, (2.1) gives

|(b - a)[[f(a) + f(b)]/2] - [[b][integral][a]]f(t)dt|

[less than or equal to][[[(b - a)].sup.2]/[2[square root of 3]]][[1/[b - a][parallel][f.sup.'][parallel][.sub.2.sup.2] - [[([f(b) - f(a)]/[b - a])].sup.2]].sup.[1/2]],

[less than or equal to] [1/[4[square root of 3]]]([GAMMA] - [gamma])[[(b - a)].sup.2].

(if [gamma] [less than or equal to] [f.sup.'](t) [less than or equal to] [GAMMA] almost everywhere t on (a, b).) (2.8)

which is trapezoid inequality.

2. For h = 0 and x = [a + b]/2, (2.1) gives

|(b - a)f([a + b]/2) - [[b][integral][a]]f(t)dt|

[less than or equal to][[[(b - a)].sup.2]/[2[square root of 3]]][[[1/[b - a][parallel][f.sup.'][parallel][.sub.2.sup.2] - [[([f(b) - f(a)]/[b - a])].sup.2]]].sup.[1/2]],

[less than or equal to][1/[4[square root of 3]]]([GAMMA] - [gamma]][[(b - a)].sup.2].

(if [gamma] [less than or equal to] [f.sup.'](t) [less than or equal to] [GAMMA] almost everywhere t on (a, b).) (2.9)

which is mid-point inequality.

3. For h = 1/2 and x = [a + b]/2, (2.1) gives

|[[f(a) + 2f([a + b]/2) + f(b)]/4] - [1/[b - a]][[b][integral][a]]f(t)dt|

[less than or equal to][[[(b - a)].sup.2]/[4[square root of 3]]][[1/[b - a][parallel][f.sup.'][parallel[.sub.2.sup.2] - [[([f(b) - f(a)]/[b - a])].sup.2]].sup.[1/2]],

[less than or equal to][1/[8[square root of 3]]]([GAMMA] - [gamma])[[(b - a)].sup.2].

(if [gamma] [less than or equal to] [f.sup.'](t) [less than or equal to] [GAMMA] almost everywhere t on (a, b).) (2.10)

which is an averaged mid-point and trapezoid inequality.

4. For h =1/3 and x = [a + b]/2, (2.1) gives

|(b - a[[f(a) + 4f([a + b]/2) + f(b)]/6] - [[b][integral][a]]f(t)dt|

[less than or equal to][[[(b - a)].sup.2]/6][[1/[b - a][parallel][f.sup.'][parallel][.sub.2.sup.2] - [[([f(b) - f(a)]/[b - a])].sup.2]].sup.[1/2]],

[less than or equal to][1/12]([GAMMA] - [gamma])[[(b - a)].sup.2].

(if [gamma] [less than or equal to] [f.sup.'] (t) [less than or equal to] [GAMMA] almost everywhere t on (a, b).) (2.11)

which is a variant of Simpson's inequality for first differentiable function f, [f.sup.'] is integrable and there exist constants [gamma], [GAMMA] [member of] R such that [gamma] [less than or equal to] [f.sup.'](t) [less than or equal to] [GAMMA], t [member of] (a, b).

3. Applications

3.1 Application for P.D.F's

Let X be a random variable having the p.d.f f: [a, b] [right arrow] [R.sub.+] and the cumulative distribution function F: [a, b] [right arrow] [0, 1], i.e.,

F(x) = [[x][integral][a]]f(t)dt,x[member of][a + h[[b - a]/2],b - h[b - a]/2][subset][a, b],

and

E(X) = [[b][integral][a]]tf(t)dt,

is the expectation of the random variable X on the interval [a, b] . Then, we may have the following.

Theorem 6. Under the above assumptions and if the p.d.f belongs to [L.sub.2] [a, b],then, we have the inequality:

|(1 - h)[F(x) - [1/[b - a]](x - [a + b]/2)] + [h/2] - [[b - E(X)]/[b - a]]|

[less than or equal to][1/[b - a]][[1/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(x - [a + b]/2)].sup.2]].sup.[1/2]] X [[(b - a)[parallel]f[parallel][.sub.2.sup.2] - 1].sup.[1/2]],

[less than or equal to][[(M - m)]/[2(b - a)]][[1/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(x - [a + b]/2)].sup.2]].sup.[1/2]],

(if m [less than or equal to] f [less than or equal to] M almost everywhere on [a, b].) (3.1)

for all x [member of] [a + h [b - a]/2, b - h [b - a]/2].

Proof. Put in (2.1) F instead of f to get (3.1) and the details are omitted.

Corollary 2. Under the above assumptions, we have

|(1 - h)Pr(X[less than or equal to][a + b]/2) + [h/2] - [[b - E(X)]/[b - a]]|

[less than or equal to][1/[2[square root of 3]]][[(3[h.sup.2] - 3h + 1)].sup.[1/2]][[(b - a)[parallel]f[parallel][.sub.2.sup.2] - 1].sup.[1/2]],

[less than or equal to][1/[4[square root of 3]]][[(3[h.sup.2] - 3h + 1)].sup.[1/2]](M - m),m[less than or equal to]f[less than or equal to]M as above. (3.2)

3.2 Applications for generalized beta random variable

If X is a beta random variable with parameters [[beta].sub.3] > -1, [[beta].sub.4] > -1 and for [[beta].sub.2] > 0 and any [[beta].sub.1], the generalized beta random variable

Y = [[beta].sub.1]+[[beta].sub.2]X,

is said to have a generalized beta distribution (4) and the probability density function of the generalized beta distribution of beta random variable is,

f(x) = {[[[[(x - [beta].sub.1)].sup.[beta].sub.3][ [([[beta].sub.1] + [([[beta].sub.2] - x)].sup.[beta]4]]/[[beta]([[beta].sub.3] + 1,[[beta].sub.4] + 1)[[beta].sub.2.sup.[([beta].sub.3 + [beta].sub.4 + 1)]]]],for[beta].sub.1 < x < [beta].sub.1 + [beta].sub.2, 0, otherwise.

where [beta] (l,m) is the beta function with l, m > 0 and is defined as

[beta](l,m) = [[1][integral][0]][x.sup.[l - 1]][[(1 - x)].sup.[m - 1]]dx.

For p, q > 0 and h [member of] [0, 1), we choose,

[[beta].sub.1] = h/2,

[[beta].sub.2] = (1-h),

[[beta].sub.3] = p-1,

[[beta].sub.4] = q-1,

Then, the probability density function associated with generalized beta random variable

Y = h/2 + (1-h)X,

takes the form

f(x) = {[[[[(x - h/2)].sup.[p - 1]][[(1 - [h/2] - x)].sup.[q - 1]]]/[[beta](p, q)[[(1 - h)].sup.[p + q - 1]]]],[h/2] < x < 1 - [h/2] 0, otherwise.

Now,

E(Y) = [(1 - [h/2]).[integral](h/2)]]xf(x)dx = (1 - h)[p/[p + q]] + [h/2]. (3.3)

and

[parallel]f(.;p, q)[[parallel].sub.2.sup.2] = [1/[(1 - h)[[beta].sup.2](p, q)]][beta](2p - 1,2q - 1).

Then, by Theorem 6, we may state the following.

Proposition 1. Let X be a beta random variable with parameters (p, q). Then for generalized beta random variable

Y = h/2 + (1 - h) X,

we have the inequality

|[Pr(Y[less than or equal to]x) - x + 1/2] - [q/[p + q]]|

[less than or equal to][[1/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(x - 1/2)].sup.2]].sup.[1/2]] X [[[[beta](2p - 1,2q - 1) - (1 - h)[[beta].sup.2](p, q)].sup.[1/2]]/[[[(1 - h)].sup.[3/2]][beta](p, q)]]. (3.4)

for all x [member of] [h/2, 1 - h/2].

In particular, for x = 1/2 in (3.4), we have:

|Pr(Y[less than or equal to]1/2) - [q/[p + q]]|

[less than or equal to][1/[2[square root of 3]]][[(3[h.sup.2] - 3h + 1)].sup.[1/2]][[[[beta](2p - 1,2q - 1) - (1 - h)[[beta].sup.2](p, q)].sup.[1/2]]/[[[(1 - h)].sup.[3/2]][beta](p, q)]].

3.3 Applications for Special Means

Example 1. Consider the mapping f (x) = [x.sup.p], p [member of] R \ {-1,0}. Then

[1/[b - a]][[integral].sub.a.sup.b]f(t)dt = [L.sub.p.sup.p](a,b),

[[f(b) - f(a)]/[b - a]] = p[L.sub.[p - 1].sup.[p - 1]],

[[f(a) + f(b)]/2] = [[[a.sup.p] + [b.sup.p]]/2] = A([a.sup.p],[b.sup.p]),

and

[1/[b - a]][parallel][f.sup.'][parallel][.sub.2.sup.2] = [1/[b - a]][[integral].sub.a.sup.b]|[f.sup.'](t)[|.sup.2]dt = [p.sup.2][L.sub.[2(p - 1)].sup.[2(p - 1)]].

Therefore, (2.1) takes the form

|(1 - h)[[x.sup.p] - p[L.sub.[p - 1].sup.[p - 1]](x - A(a,b))] + hA([a.sup.p],[b.sup.p]) - [L.sub.p.sup.p]|

[less than or equal to]|p|[[[[(b - a)].sup.2]/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(x - A(a,b))].sup.2]].sup.[1/2]] X [[[L.sub.[2(p - 1)].sup.[2(p - 1)]] - [L.sub.[p - 1].sup.[2(p - 1)]]].sup.[1/2]] (3.5)

Choose x = A(a, b) in (3.5), to get

|(1 - h)[A.sup.p](a,b) + hA([a.sup.p],[b.sup.p]) - [L.sub.p.sup.p]|

[less than or equal to][[(b - a)]/[2[square root of 3]]][[(3[h.sup.2] - 3h + 1)].sup.[1/2]]|p|[[[L.sub.[2(p - 1)].sup.[2(p - 1)]] - [L.sub.[p - 1].sup.[2(p - 1)]]].sup.[1/2]].

which is minimum for h = 1/2. Moreover for h = 1,

|A([a.sup.p],[b.sup.p]) - [L.sub.p.sup.p]|

[less than or equal to][[(b - a)]/[2[square root of 3]]]|p|[[[L.sub.[2(p - 1)].sup.[2(p - 1)]] - [L.sub.[p - 1].sup.[2(p - 1)]]].sup.[1/2]].

Example 2. Consider the mapping f(x) = [1/x],(x [member of] [a + h [b - a]/2,b - h [b - a]/2][subset](0,[infinity])).

Then,

[1/[b - a]][[b][integral][a]]f(t)dt = [1/L],

[[f(b) - f(a)]/[b - a]] = - [1/[G.sup.2]],

[[f(b) + f(a)]/2] = [[A(a,b)]/[G.sup.2]],

[1/[b - a]][[b][integral][a]]|[f.sup.'](t)[|.sup.2]dt = [[[a.sup.2] + ab + [b.sup.2]]/[3[a.sup.3][b.sup.3]]],

and

[1/[b - a]][[b][integral][a]]|[f.sup.'](t)[|.sup.2]dt - [[([f(b) - f(a)]/[b - a])].sup.2] = [[[(b - a)].sup.2]/[3[a.sup.3][b.sup.3]]].

Therefore, (2.1) becomes,

|(1 - h)[1/x + [1/[G.sup.2]](x - A(a,b))] + h[[A(a,b)]/[G.sup.2]] - [1/L]|

[less than or equal to][[[[(b - a)].sup.2]/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(x - A(a,b))].sup.2]].sup.[1/2]] X [[(b - a)]/[[square root of 3][G.sup.3]]]. (3.6)

Choosing x = A(a, b) in (3.6),

|(1 - h)[1/A] + h[[A(a,b)]/[G.sup.2]] - [1/L]|

[less than or equal to][[[(b - a)].sup.2]/[6[G.sup.3]]][[(3[h.sup.2] - 3h + 1)].sup.[1/2]].

If we choose x = L in (3.6),we get

|(1 - h)[L/[G.sup.2]] + (2h - 1)[[A(a,b)]/[G.sup.2]] - h[1/L]|

[less than or equal to][[[[(b - a)].sup.2]/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(L - A(a,b))].sup.2]].sup.[1/2]] X [[(b - a)]/[[square root of 3][G.sup.3]]].

Example 3. Finally, let us consider the mapping

f(x) = ln x,(x [member of] [a + h [b - a]/2, b - h [b - a]/2][subset](0,[infinity])). Then

[1/[b - a]][[b][integral][a]]f(t)dt = ln I,

[[f(b) - f(a)]/[b - a]] = [1/L],

[[f(a) + f(b)]/2] = ln G,

[1/[b - a]][[b][integral][a]]|[f.sup.'](t)[|.sup.2]dt = [1/[G.sup.2]],

and

[1/[b - a]][[b][integral][a]]|[f.sup.'](t)[|.sup.2]dt - ([f(b) - f(a)]/[b - a][).sup.2] = [[[L.sup.2] - [G.sup.2]]/[[L.sup.2][G.sup.2]]].

Thus, (2.1) takes the form,

|ln[[[x.sup.[(1 - h)]][G.sup.h]]/I] - (1 - h)[[x - A(a,b)]/L]|

[less than or equal to][[[[(b - a)].sup.2]/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(x - A(a,b))].sup.2]].sup.[1/2]] X [[[([L.sup.2] - [G.sup.2])].sup.[1/2]]/[LG]].(3.7)

For x = A(a, b) in (3.7), we get:

|ln[[[(A(a,b)).sup.[(1 - h)]][G.sup.h]]/I]|

[less than or equal to][[(b - a)[[(3[h.sup.2] - 3h + 1)].sup.[1/2]][[([L.sup.2] - [G.sup.2])].sup.[1/2]]]/[LG]].

which for h = 1, takes the form

|ln[G/I]|

[less than or equal to][[(b - a)[[([L.sup.2] - [G.sup.2])].sup.[1/2]]]/[2[square root of 3]LG]].

Also, choosing x = I, we get

|ln[[G.sup.h]/[I.sup.h]] - (1 - h)[[I - A(a,b)]/L]|

[less than or equal to][[[[(b - a)].sup.2]/12(3[h.sup.2] - 3h + 1) + h(1 - h)[[(I - A(a,b))].sup.2]].sup.[1/2]] X [[[([L.sup.2] - [G.sup.2])].sup.[1/2]]/[LG]].

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(8) A. Ostrowski, Uber die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert, Comment. Math. Hel, 10 (1938), 226-227.

Fiza Zafar [dagger] CASPAM, Bahauddin Zakariya University, Multan 60800, Pakistan. and

Nazir Ahmad Mir [double dagger] Dept. of Mathematics, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan.

Received September 15, 2008, Accepted October 20, 2008.

* 2000 Mathematics Subject Classification. Primary 26D15; Secondary 41A55.

[dagger] Corresponding author. E-mail: fizazafar@gmail.com

[double dagger] E-mail: nazirahmad.mir@gmail.com
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Author:Zafar, Fiza; Mir, Nazir Ahmad
Publication:Tamsui Oxford Journal of Mathematical Sciences
Article Type:Report
Geographic Code:9PAKI
Date:May 1, 2010
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