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

http://jipam.vu.edu.au/

Volume 5, Issue 2, Article 23, 2004

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

SANJA VAROŠANEC DEPARTMENT OFMATHEMATICS

UNIVERSITY OFZAGREB, ZAGREB, CROATIA. varosans@math.hr

Received 28 January, 2004; accepted 28 January, 2004 Communicated by J. E. Peˇcari´c

ABSTRACT. In this paper we present a historical review of the investigation of two Ostrowski in- equalities 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.

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

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

1. HISTORY ANDGENERALIZATIONS

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_n under the conditions

(1.1)

n

X

i=1

aixi = 0 and

n

X

i=1

bixi = 1 is

(1.2)

Pn i=1a²_i P

i<j(a_ib_j−a_jb_i)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)² under the conditions

(1.3)

n

X

i=1

a_ix_i = 0 and

n

X

i=1

x²_i = 1

ISSN (electronic): 1443-5756

c 2004 Victoria University. All rights reserved.

022-04

is

(1.4)

P

i<j(aibj −ajbi)² Pn

i=1a²_i ,

n

X

i=1

a²_i >0

! .

According to the Lagrange identity [10, p. 84], Theorem 1.1 can be rewritten in the following form.

Theorem 1.3. Leta = (a₁, . . . , a_n)andb = (b₁, . . . , b_n)be two nonproportional sequences of real numbers and letx= (x₁, . . . , x_n)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. Theorem 1.3, 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 thataibj 6=ajbifori6=j. Then

(1.7)

Pn i=1a²_i (Pn

i=1a²_i)(Pn

i=1b²_i)−(Pn

i=1aibi)² ≤

2 n(n−1)

2 n

X

i=1 n

X

j=1,j6=i

a_j ajbi−aibj

!2

.

They also generalized Theorem 1.4 using more than two vectors.

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

pPn

i=1x²_i is a Euclidean norm inRⁿgenerated 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

G(a, b)(ha, βa−αbib− hb, βa−αbia).

If vectorxsatisfies 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 1.6. Now, we point out another proof of Theorem 1.5. It is well known that Gram’s determinant of the vectorsx₁, x₂, x₃is 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=β we 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.

Equality holds iff vectors x, aand b are linearly dependent, i.e. there exist scalarsλ and µ such that

x=λa+µb.

Multiplying that identity by a and b respectively, we obtain α = λha, ai +µhb, ai and β = 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. Furthermore, in the paper [2] P.R. Beesack noticed that inequality (1.9) and a fortiori also Ostrowski’s inequality (1.6) can be regarded as a special case of the Bessel inequality for non-orthonormal vectors.

Theorem 1.7. [2] Leta₁, . . . , a_k,(k≥1)be linearly independent vectors of a Hilbert spaceH and letα₁, . . . , α_kbe given scalars. Ifx∈H satisfies

(1.11) hx, aii=αi 1≤i≤k,

then

(1.12) G(a₁, . . . , a_k)²kxk² ≥

k

X

i=1

γ_i^(k)a_i

2

,

whereG(a₁, . . . , a_k)is the Gram determinant ofa₁, . . . , a_kandγ_i^(k)is the determinant obtained fromGby replacing the elements of thei^th row ofGby(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 Theorem 1.5 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 Made- vski’s paper [9]. He used Theorem 1.3 to obtain inequalities between certain statistical central moments. Also, he gave the followingp-version of Ostrowski’s inequality.

Theorem 1.8. [9] Leta= (a₁, . . . , a_n)andb = (b₁, . . . , b_n)be two nonproportional 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 distribu- tion function has been given. An integral version of those results and some generalizations of known statistical inequalities given in [9], [14] and [16] are obtained in [13].

Recently, Theorems 1.1 and 1.2 have been the focus of investigation. In the papers [6] and [7] the authors have used elementary arguments and the Cauchy-Buniakowski-Schwarz inequal- ity to obtain Ostrowski type inequalities in unitary space. Indeed, the following theorems are obtained.

Theorem 1.9. [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 1.10. [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.

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 Theorem 1.5 but 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 ,

and (1.24)

z− hz, ci

kck²c, 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 = aand d = b where aand xsatisfy hx, ai = 0and kxk= 1we get inequality (1.20), while ifaandxsatisfyhx, ai= 0and|hx, bi|= 1inequality (1.17) is obtained.

Remark 1.11. 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=aandd =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 1.12. Inequality (1.25) is a special case of the more general result related to Gram’s determinant given in [10, p. 599]. That result is as follows.

Theorem 1.13. For vectors x₁, . . . , x_n and y₁, . . . , y_n from unitary space V the following in- equality 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 vectorsx₁, . . . , x_nspan the same subspace as the vectorsy₁, . . . , y_n. 2. FURTHER GENERALIZATIONS OF OSTROWSKI’S INEQUALITIES

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

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

(2.1) hx, ai=α and hx, bi=β.

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 thatφis 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

.

The case whenφis a nonincreasing and subadditive function has been done similarly.

In particular, inequality (2.2) holds for any nondecreasing convex functionφ, while its reverse holds for any nonincreasing concave function. The result of Theorem 2.1 can be improved if functionφis a power function. In that case we have the following result.

Theorem 2.2. Suppose thata,bandxare as in Theorem 2.1. 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 function φ(x) = x^p, p ≥ 1 is a nondecreasing superadditive function so, a direct consequence of the previous theorem is that fora,bandxwhich satisfy assumptions of Theorem 2.1 we have the following inequality

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

Applying the method of proving in Theorem 2.1 on 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

i.e. puttingz =x,c=a,d=band taking into account thathx, ai=αandhx, bi=βwe 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 2.3. Ifp = 1, then the two terms on the righthand side of inequality (2.4) are equal, but ifp > 1termskak^2pkαb−βak^2p and|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. If β ∈(0,1)the first term is less than the second term and ifβ >1the opposite inequality holds.

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

(3.1) hx, ai=α and hx, bi=β.

Letybe a vector defined by

(3.2) y = 1

G(a, b)(ha, βa−αbib− hb, βa−αbia).

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.

Similarly, we obtain

(3.7) hF(x), bi=β.

According to Theorem 1.5 and 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.

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 andkxk² =Pn

i=1x²_i, then we get a result of M. Bjelica, [3].

REFERENCES

[1] M. ALI ´CANDJ. PE ˇCARI ´C, On some inequalities of Ž. Madevski and A.M. Ostrowski, Rad HAZU, Matematiˇcke znanosti, 472 (1997), 77–82.

[2] P.R. BEESACK, On Bessel’s inequality and Ostrowski’s, Univ. Beograd Publ.Elektrotehn. Fak. Ser.

Mat. Fiz., No 498-541 (1975), 69–71.

[3] M. BJELICA, Refinements of Ostrowski’s and Fan-Todd’s inequalities, in Recent Progress in In- equalities, G.V. Milovanovi´c (Ed.), Kluwer Academic Publishers, Dordrecht, 1998, 445–448.

[4] Y.J. CHO, M. MATI ´C AND J. PE ˇCARI ´C, On Gram’s determinant in 2-inner product spaces, J.

Korean Math. Soc., 38(6) (2001), 1125–1156.

[5] Y.J. CHO, M. MATI ´CANDJ. PE ˇCARI ´C, On Gram’s determinant in n-inner product spaces, Bull.

Korean Math. Soc.,to appear.

[6] S.S. DRAGOMIR AND A.C. GO ¸SA, A generalization of an Ostrowski inequality in inner prod- uct spaces, RGMIA Research Report Collection, 6(2) (2003), Article 20. [ONLINE] http:

//rgmia.vu.edu.au/v6n2.html.

[7] S.S. DRAGOMIR, Ostrowski’s inequality in complex inner product spaces, RGMIA Research Re- port Collection 6(Supp.) (2003), Article 6. [ONLINE]http://rgmia.vu.edu.au/v6(E) .html.

[8] K. FANANDJ. TODD, A determinantal inequality, J. London Math. Soc., 30 (1955), 58–64.

[9] Ž. MADEVSKI, Quelques conséquences d’une inégalité d’Ostrowski, Univ. Beograd Publ. Elek- trotehn. Fak. Ser. Mat. Fiz., No 412-460 (1973), 168–170.

[10] D.S. MITRINOVI ´C, J.E. PE ˇCARI ´C AND A.M. FINK, Classical and New Inequalities, Kluwer Academic Publishers, Dordrecht, 1993.

[11] Ž. MITROVI ´C, On a generalization of Fan-Todd’s inequality, Univ. Beograd Publ. Elektrotehn.

Fak. Ser. Mat. Fiz. No 412-460 (1973), 151–154.

[12] A. OSTROWSKI, Vorlesungen über Differential und Integralrechnung II, Verlag Birkhüser Basel, 1951.

[13] C.E.M. PEARCE, J. PE ˇCARI ´C AND S. VAROŠANEC, An integral analogue of the Ostrowski inequality, J. Ineq. Appl., 2 (1998), 275–283.

[14] K. PEARSON, Mathematical contributions to the theory of evolution. XIX: second supplement to a memoir on skew variation, Phil.Trans.Roy. Soc. A, 216 (1916), 432.

[15] H. ŠIKI ´CANDT. ŠIKI ´C, A note on Ostrowski’s inequality, Math. Ineq. Appl., 4(2) (2001), 297–

299.

[16] J.E. WILKINS, A note on skewness and kurtosis, Ann. Math. Statist., 15 (1944), 333–335.