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volume 5, issue 2, article 23, 2004.

Received 28 January, 2004;

accepted 28 January, 2004.

Communicated by:J.E. Peˇcari´c

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Journal of Inequalities in Pure and Applied Mathematics

HISTORY, GENERALIZATIONS AND UNIFIED TREATMENT OF TWO OSTROWSKI’S INEQUALITIES

SANJA VAROŠANEC

Department of Mathematics University of Zagreb, Zagreb, Croatia.

EMail:varosans@math.hr

c

2000Victoria University ISSN (electronic): 1443-5756 022-04

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History, Generalizations And Unified Treatment Of Two

Ostrowski’s Inequalities Sanja Varošanec

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J. Ineq. Pure and Appl. Math. 5(2) Art. 23, 2004

http://jipam.vu.edu.au

Abstract

In this paper we present a historical review of the investigation of two Ostrowski inequalities and describe several distinct streams for their generalizations. Also we point out some new methods to obtain known results and give a number of new results related to Ostrowski’s inequalities.

2000 Mathematics Subject Classification:Primary 26D15, Secondary 46C05.

Key words: Ostrowski’s inequalities, Cauchy-Buniakowski-Schwarz inequality, Gram’s determinant, Unitary vector space, Superadditive function, Interpolation.

1. History and Generalizations

In his book Vorlesungen über Differential und Integralrechnung II, A. Ostrowski presented the following interesting inequalities.

Theorem 1.1. [12, p. 289, problem 61], [10, pp. 92–93]. The minimum of the sumx²₁+· · ·+x²_nunder the conditions

(1.1)

n

X

i=1

a_ix_i = 0 and

n

X

i=1

b_ix_i = 1 is

(1.2)

Pn i=1a²_i P

i<j(aibj−ajbi)²,

n

X

i=1

a²_i +

n

X

i=1

b²_i >0

! .

Theorem 1.2. [12, p. 290, problem 63], [10, p. 94]. The maximum of the sum (Pn

i=1b_ix_i)²under the conditions (1.3)

n

X

i=1

a_ix_i = 0 and

n

X

i=1

x²_i = 1 is

(1.4)

P

i<j(a_ib_j −a_jb_i)² Pn

i=1a²_i ,

n

X

i=1

a²_i >0

! .

According to the Lagrange identity [10, p. 84], Theorem1.1can be rewritten in the following form.

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Theorem 1.3. Let a = (a₁, . . . , a_n) and b = (b₁, . . . , b_n) be two nonpropor- tional sequences of real numbers and letx= (x1, . . . , xn)be any real sequence which satisfies

(1.5)

n

X

i=1

a_ix_i = 0 and

n

X

i=1

b_ix_i = 1.

Then

(1.6)

n

X

i=1

x²_i ≥

Pn i=1a²_i (Pn

i=1a²_i)(Pn

i=1b²_i)−(Pn

i=1a_ib_i)².

The second Ostrowski problem can also be written in the analogue form. In the literature those forms are used more frequently than the original and have been extended, improved and generalized in different ways.

The aim of this paper is to give a brief historical review and to carry those ideas somewhat further.

K. Fan and J. Todd, [8], using Theorem 1.1, i.e. Theorem1.3, they established the following theorem.

Theorem 1.4. Let a = (a₁, . . . , a_n) and b = (b₁, . . . , b_n) (n ≥ 2) be two sequences of real numbers such thata_ib_j 6=a_jb_i fori6=j. Then

(1.7)

Pn i=1a²_i (Pn

i=1a²_i)(Pn

i=1b²_i)−(Pn

i=1a_ib_i)²

≤

2 n(n−1)

2 n

X

i=1 n

X

j=1,j6=i

a_j ajbi−aibj

!2

.

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They also generalized Theorem1.4using more than two vectors.

Another direction of generalization has arisen from the fact that the map (x₁, . . . , x_n) 7→ pPn

i=1x²_i is a Euclidean norm in Rⁿ generated by the inner producthx, yi = Pn

i=1x_iy_i. It is natural to consider an arbitrary inner product instead of the Euclidean inner product. The first generalization of that kind was done by Ž. Mitrovi´c, [11] and after that some similar results were given in [6], [7] and [15]. Here we quote Mitrovi´c’s result.

Theorem 1.5. Letaandbbe linearly independent vectors of a unitary complex vector spaceV and letxbe a vector inV such that

(1.8) hx, ai=α and hx, bi=β.

Then

(1.9) G(a, b)kxk² ≥ kαb−βak²,

whereG(a, b)is the Gram determinant of vectorsaandb. Equality holds if and only if

(1.10) x= 1

G(a, b)(ha, βa−αbib− hb, βa−αbia).

Here we present a rough outline of Mitrovi´c’s proof. Letybe a vector inV given by

y= 1

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If vector xsatisfies conditions (1.8), thenhy, yi = hx, yi = _G(a,b)¹ kαb−βak² andkx−yk² =kxk² − kyk². Sincekx−yk² ≥0, we obtain

kxk² ≥ kyk² = 1

G(a, b)kαb−βak² and inequality (1.9) holds.

Remark 1.1. Now, we point out another proof of Theorem1.5. It is well known that Gram’s determinant of the vectorsx₁, x₂, x₃ is nonnegative, i.e. inequality

G(x₁, x₂, x₃)≥0

holds with equality iff the vectors x₁, x₂, x₃ are linearly dependent. Putting x₁ =x,x₂ =a,x₃ =band using notationshx, ai=αandhx, bi=βwe have the following

0≤G(x, a, b)

=

hx, xi hx, ai hx, bi ha, xi ha, ai ha, bi hb, xi hb, ai hb, bi

=G(a, b)kxk²− hx, ai

ha, xi ha, bi hb, xi hb, bi

+hx, bi

ha, xi ha, ai hb, xi hb, ai

=G(a, b)kxk²−α(αhb, bi −βha, bi) +β(αhb, ai −βha, ai),

G(a, b)kxk² ≥ |α|²hb, bi −αβha, bi −βαhb, ai+|β|²ha, ai=kαb−βak².

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Equality holds iff vectors x, aand b are linearly dependent, i.e. there exist scalarsλandµsuch that

x=λa+µb.

Multiplying that identity byaandbrespectively, we obtainα=λha, ai+µhb, ai andβ =λha, bi+µhb, bifrom where we easily find that

λ= 1

G(a, b)(αhb, bi −βhb, ai), µ= 1

G(a, b)(βha, ai −αha, bi).

So,xis the vector given in (1.10).

In the same paper [11] a generalization of Fan-Todd’s result is given. Fur- thermore, in the paper [2] P.R. Beesack noticed that inequality (1.9) and a for- tiori also Ostrowski’s inequality (1.6) can be regarded as a special case of the Bessel inequality for non-orthonormal vectors.

Theorem 1.6. [2] Leta₁, . . . , a_k,(k ≥1)be linearly independent vectors of a Hilbert spaceHand letα₁, . . . , α_kbe given scalars. Ifx∈Hsatisfies

(1.11) hx, a_ii=α_i 1≤i≤k, then

(1.12) G(a1, . . . , ak)²kxk² ≥

k

X

i=1

γ_i^(k)ai

2

,

where G(a₁, . . . , a_k)is the Gram determinant of a₁, . . . , a_k andγ_i^(k) is the de- terminant obtained from G by replacing the elements of the i^th row of G by

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(α₁, . . . , α_k). Moreover, equality holds if and only if (1.13) G(a1, . . . , ak)x=

k

X

i=1

γ_i^(k)ai.

Finally, an analogue of Theorem1.5 and related generalizations in 2-inner andn-inner spaces are given in [4] and [5].

The second stream of generalization of Ostrowski’s inequality (1.6) was started by Madevski’s paper [9]. He used Theorem1.3to obtain inequalities be- tween certain statistical central moments. Also, he gave the followingp-version of Ostrowski’s inequality.

Theorem 1.7. [9] Leta= (a₁, . . . , a_n)andb = (b₁, . . . , b_n)be two nonpropor- tional sequences of real numbers and letx= (x₁, . . . , x_n)be any real sequence which satisfies

(1.14)

n

X

i=1

a_ix_i = 0 and

n

X

i=1

b_ix_i = 1.

Ifpis an integer, then

(1.15)

n

X

i=1

x²_i

!p

≥ (Pn

i=1a²_i)^p (Pn

i=1a²_i)^p(Pn

i=1b²_i)^p−(Pn

i=1a_ib_i)^2p.

In [1] M. Ali´c and J. Peˇcari´c proved that the integerpcan be substituted by an arbitrary real numberp ≥1. In the same paper a sequence of results involving moments of discrete distribution function has been given. An integral version

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of those results and some generalizations of known statistical inequalities given in [9], [14] and [16] are obtained in [13].

Recently, Theorems1.1and1.2have been the focus of investigation. In the papers [6] and [7] the authors have used elementary arguments and the Cauchy- Buniakowski-Schwarz inequality to obtain Ostrowski type inequalities in unitary space. Indeed, the following theorems are obtained.

Theorem 1.8. [7] Let a and b be linearly independent vectors of a real or complex unitary vector spaceV and letxbe a vector inV such that

(1.16) hx, ai= 0 and |hx, bi|= 1.

Then

(1.17) kxk² ≥ kak²

kak²kbk² − |ha, bi|². Equality holds if and only if

(1.18) x=µ

b−ha, bi kak²a

, whereµ∈K(K=R,C)is such that

|µ|= kak²

kak²kbk²− |ha, bi|².

Theorem 1.9. [6] Let a and b be linearly independent vectors of a real or complex unitary vector spaceV and letxbe a vector inV such that

(1.19) hx, ai= 0 and kxk= 1.

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Then

(1.20) |hx, bi|² ≤ kak²kbk²− |ha, bi|²

kak² .

Equality holds if and only if

(1.21) x=ν

b− hb, ai kak²a

, whereν ∈K(K=R,C)is so that

|ν|= kak

(kak²kbk²− |ha, bi|²)¹².

It is obvious that these results are special cases of Theorem1.5but we mentioned it because the method of proving is different from Mitrovi´c’s method and leads to another generalization which will be given in the next section. Proofs of the previous two theorems are based on the Cauchy-Buniakowski-Schwarz inequality:

kuk²kvk² ≥ |hu, vi|², u, v ∈V.

Applying it on vectorsu=z−^hz,ci_kck2candv =d−^hd,ci_kck2c, wherec6= 0and taking into account that

(1.22)

z− hz, ci kck² c

2

= kzk²kck² − |hz, ci|²

kck² ,

(1.23)

d− hd, ci kck² c

2

= kdk²kck²− |hd, ci|²

kck² ,

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and (1.24)

z− hz, ci

kck² c, d− hd, ci kck² c

= hz, dikck²− hz, cihc, di kck²

we have the following inequality (1.25) kzk²kck²− |hz, ci|²

kdk²kck²− |hd, ci|²

≥

hz, dikck²− hz, cihc, di

2. Putting in inequality (1.25)z = x, c = a andd = b wherea andx satisfy hx, ai = 0 and kxk = 1 we get inequality (1.20), while if a and x satisfy hx, ai= 0and|hx, bi|= 1inequality (1.17) is obtained.

Remark 1.2. Let us mention that inequality (1.9) also can be obtained by the above-mentioned method. In fact, putting in inequality (1.25)z =x,c=aand d =b,hx, ai=αandhx, bi=β we get

kxk²kak²− |hx, ai|²

kbk²kak²− |hb, ai|²

≥

βkak²−αha, bi

2, kxk²kak²G(a, b)≥

βkak²−αha, bi

2+|α|²G(a, b) =kak²kαb−βak² from where inequality (1.9) occurs. Using the fact that in the Cauchy-Buniakowski- Schwarz inequality, equality holds iff vectors are proportional, we get (1.10).

Remark 1.3. Inequality (1.25) is a special case of the more general result re- lated to Gram’s determinant given in [10, p. 599]. That result is as follows.

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Theorem 1.10. For vectorsx₁, . . . , x_nandy₁, . . . , y_nfrom unitary spaceV the following inequality holds

det







hx₁, y₁i . . . hx₁, y_ni

... ...

hx_n, y₁i . . . hx_n, z_ni







2

≤G(x₁, . . . , x_n)G(y₁, . . . , y_n),

with equality iff the vectors x1, . . . , xn span the same subspace as the vectors y₁, . . . , y_n.

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2. Further Generalizations of Ostrowski’s Inequalities

In this section we extend results from papers [1], [6], [7], [13] introducing su- per(sub)additive function.

Theorem 2.1. Letaandbbe linearly independent vectors of a unitary complex vector spaceV and letxbe a vector inV such that

Ifφ: [0,∞)→Ris a nondecreasing, superadditive function, then (2.2) φ(kak²kbk²)−φ(|ha, bi|²)≥φ

kαb−βak² kxk²

.

Ifφis a nonincreasing, subadditive function then a reverse in (2.2) holds.

Proof. Let us suppose that φ is a superadditive nondecreasing function. Then we have

(2.3) φ(u) =φ((u−v) +v)≥φ(u−v) +φ(v), i.e.φ(u)−φ(v)≥φ(u−v).

Taking into account the nondecreasing property of φ, results of Theorem 1.5 and inequality (2.3) we conclude

φ(kak²kbk²)−φ(|ha, bi|²)≥φ(kak²kbk²−(|ha, bi|²)

≥φ

kαb−βak² kxk²

.

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The case when φ is a nonincreasing and subadditive function has been done similarly.

In particular, inequality (2.2) holds for any nondecreasing convex function φ, while its reverse holds for any nonincreasing concave function. The result of Theorem2.1can be improved if functionφis a power function. In that case we have the following result.

Theorem 2.2. Suppose thata,bandxare as in Theorem2.1. Ifp≥1, then (2.4) kxk^2pkak^2p(kak^2pkbk^2p− |ha, bi|^2p)

≥max

kak^2pkαb−βak^2p,

|βkak² −αha, bi|^2p+|α|^2p(kak^2pkbk^2p− |ha, bi|^2p) . Proof. The functionφ(x) =x^p,p≥1is a nondecreasing superadditive function so, a direct consequence of the previous theorem is that for a, b and x which satisfy assumptions of Theorem2.1we have the following inequality

(2.5) kak^2pkbk^2p− |ha, bi|^2p ≥ kαb−βak^2p kxk^2p .

Applying the method of proving in Theorem2.1on inequality (1.25) we get (2.6) kzk^2pkck^2p− |hz, ci|^2p

kdk^2pkck^2p− |hd, ci|^2p

≥

hz, dikck²− hz, cihc, di

2p

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i.e. putting z = x, c = a, d = b and taking into account thathx, ai = α and hx, bi=β we have

(2.7) kxk^2pkak^2p− |α|^2p

kbk^2pkak^2p− |hb, ai|^2p

≥

βkak²−αha, bi

2p. After simple calculations, inequalities (2.6) and (2.7) give inequality (2.4).

Remark 2.1. If p = 1, then the two terms on the righthand side of inequality (2.4) are equal, but ifp > 1termskak^2pkαb−βak^2pand|βkak²−αha, bi|^2p+

|α|^2p(kak^2pkbk^2p− |ha, bi|^2p)are not comparable. For example, ifp= 2,kak= 1,kbk= 1,ha, bi= ¹₂ andα = 1,β ∈Rthe first term is equal to(β²−β+ 1)², while the second term is equal to(β²−β+¹₄)²+¹⁵₁₆. Ifβ ∈(0,1)the first term is less than the second term and ifβ >1the opposite inequality holds.

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3. Interpolation

Some results about refinements of the original first Ostrowski’s inequality are given in [3]. Here we give more general results in which we consider refinements of Ostrowski’s inequalities in arbitrary unitary complex vector spaces.

Theorem 3.1. Letaandbbe linearly independent vectors in a unitary complex vector spaceV and letxbe a vector inV such that

Letybe a vector defined by

(3.2) y= 1

Then the vectorF(x) = θx+ (1−θ)y,θ ∈[0,1], satisfies

(3.3) kxk² ≥ kF(x)k²

and

(3.4) G(a, b)kF(x)k² ≥ kαb−βak².

Proof. Let us note thatyis a vector for which equality in (1.9) holds, i.e.

(3.5) G(a, b)kyk² =kαb−βak².

So, without any calculation we conclude thathy, ai=αandhy, bi=β. Now, (3.6) hF(x), ai=hθx+ (1−θ)y, ai=θα+ (1−θ)α =α.

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Similarly, we obtain

(3.7) hF(x), bi=β.

According to Theorem1.5and in view of (3.6) and (3.7) we get G(a, b)kF(x)k² ≥ kαb−βak².

Let us calculate the producthy, xi.

G(a, b)hy, xi=hha, βa−αbib− hb, βa−αbia, xi

=ha, βa−αbihb, xi − hb, βa−αbiha, xi

=β(βha, ai −αha, bi)−α(βhb, ai −αhb, bi)

=|βk²kak²−αβha, bi −αβhb, ai+|α|²kbk²

=kαb−βak².

Comparing this result with (3.5) we havehy, xi=hy, yi =hx, yi. Using these equalities we obtain

kF(x)k² =hF(x), F(x)i

=θ²kxk²+θ(1−θ)hx, yi+ (1−θ)θhy, xi+ (1−θ)²hy, yi

=θ²kxk²+ (1−θ²)kyk².

kxk²− kF(x)k² = (1−θ²)(kxk²− kyk²) = (1−θ²)(kx−yk²)≥0 and inequality (3.3) has been established.

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Thus we obtain a sequence of succesive approximations x, F(x), F²(x), . . . , Fⁿ(x), . . . converging toyforθ <1which interpolate inequality (1.9)

kxk² ≥ kF(x)k² ≥ kF²(x)k² ≥ · · · ≥ kFⁿ(x)k² ≥ · · · ≥ kyk² = kαb−βak² G(a, b) . Ifα = 0, β = 1, θ = ¹₂ and kxk² = Pn

i=1x²_i, then we get a result of M.

Bjelica, [3].

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