A generalisation of inner product spaces of an inequality due to Ostrowski
and applications for sequences and integrals are given

Dragomir, Sever S.

Gosa, Anca C.

English

arXiv

A generalisation of inner product spaces of an inequality due to Ostrowski and applications for sequences and integrals are given

Dragomir, Sever S

Goşa, Anca C

Victoria University Institutional Repository (VUIR)

A Generalisation of an Ostrowski Inequality in Inner Product Spaces

Name not available

Victoria University Eprints Repository

A GENERALISATION OF AN OSTROWSKI INEQUALITY ININNER PRODUCT SPACESSEVER S. DRAGOMIR AND ANCA C. GOS¸AAbstract. A generalisation of inner product spaces of an inequality due toOstrowski and applications for sequences and integrals are given.1. IntroductionIn 1951, A.M. Ostrowski [2, p. 289] obtained the following result (see also [1, p.92]).Theorem 1. Suppose that a = (a1, . . . , an) , b = (b1, . . . , bn) and x = (x1, . . . , xn)are real n−tuples such that a 6= 0 and(1.1)n∑i=1aixi = 0 andn∑i=1bixi = 1.Then(1.2)n∑i=1x2i ≥∑ni=1 a2i∑ni=1 a2i∑ni=1 b2i − (∑ni=1 aibi)2 ,with equality if and only if(1.3) xk =bk∑ni=1 a2i − ak∑ni=1 aibi∑ni=1 a2i∑ni=1 b2i − (∑ni=1 aibi)2 , k = 1, . . . , n.Another similar result due to Ostrowski which is far less known and obtained inthe same work [2, p. 130] (see also [1, p. 94]), is the following one.Theorem 2. Let a, b and x be n−tuples of real numbers with a 6= 0 and(1.4)n∑i=1aixi = 0 andn∑i=1x2i = 1.Then(1.5)∑ni=1 a2i∑ni=1 b2i − (∑ni=1 aibi)2∑ni=1 a2i≥(n∑i=1bixi)2.If a and b are not proportional, then the equality holds in (1.5) iff(1.6) xk = q · bk∑ni=1 a2i − ak∑ni=1 aibi(∑nk=1 a2k)12[∑ni=1 a2i∑ni=1 b2i − (∑ni=1 aibi)2] 12, k ∈ {1, . . . , n} ,Date: May 19, 2003.1991 Mathematics Subject Classification. 26D15, 46C99.Key words and phrases. Ostrowski’s inequality, Inner products.12 SEVER S. DRAGOMIR AND ANCA C. GOS¸Awith q ∈ {−1, 1, } .The case of equality which was neither mentioned in [1] nor in [2] is consideredin Remark 1.In the present paper, by the use of an elementary argument based on Schwarz’sinequality, a natural generalisation in inner-product spaces of (1.5) is given. Thecase of equality is analyzed. Applications for sequences and integrals are also pro-vided.2. The ResultsThe following theorem holds.Theorem 3. Let (H, 〈·, ·〉) be a real or complex inner product space and a, b ∈ Htwo linearly independent vectors. If x ∈ H is such that((i)) 〈x, a〉 = 0 and ‖x‖ = 1,then(2.1)‖a‖2 ‖b‖2 − |〈a, b〉|2‖a‖2 ≥ |〈x, b〉|2.The equality holds in (2.1) iff(2.2) x = ν(b− 〈a, b〉‖a‖2 · a),where ν ∈ K (C,R) is such that(2.3) |ν| = ‖a‖[‖a‖2 ‖b‖2 − |〈a, b〉|2] 12.Proof. We use Schwarz’s inequality in the inner product space H, i.e.,(2.4) ‖u‖2 ‖v‖2 ≥ |〈u, v〉|2 , u, v ∈ Hwith equality iff there is a scalar α ∈ K such that(2.5) u = αv.If we apply (2.4) for u = z − 〈z,c〉‖c‖2 · c, v = d−〈d,c〉‖c‖2 · c, where c 6= 0 and c, d, z ∈ H,and taking into account that∥∥∥∥∥z − 〈z, c〉‖c‖2 · c∥∥∥∥∥2=‖z‖2 ‖c‖2 − |〈z, c〉|2‖c‖2 ,∥∥∥∥∥d− 〈d, c〉‖c‖2 · c∥∥∥∥∥2=‖d‖2 ‖c‖2 − |〈d, c〉|2‖c‖2and 〈z − 〈z, c〉‖c‖2 · c, d−〈d, c〉‖c‖2 · c〉=〈z, d〉 ‖c‖2 − 〈z, c〉 〈c, d〉‖c‖2 ,we deduce the inequality(2.6)[‖z‖2 ‖c‖2 − |〈z, c〉|2] [‖d‖2 ‖c‖2 − |〈d, c〉|2]≥∣∣∣〈z, d〉 ‖c‖2 − 〈z, c〉 〈c, d〉∣∣∣2OSTROWSKI INEQUALITY 3with equality iff there is a β ∈ K such that(2.7) z =〈z, c〉‖c‖2 · c+ β(d− 〈d, c〉‖c‖2 · c).If in (2.6) we choose z = x, c = a and d = b, where a and x statisfy (i), then wededuce‖a‖2[‖a‖2 ‖b‖2 − |〈a, b〉|2]≥[〈x, b〉 ‖a‖2]2which is clearly equivalent to (2.1).The equality holds in (2.1) iffx = ν(b− 〈a, b〉‖a‖2 · a),where ν ∈ K satisfies the condition(2.8) 1 = ‖x‖ = |ν|∥∥∥∥∥b− 〈a, b〉‖a‖2 · a∥∥∥∥∥ = |ν|[‖a‖2 ‖b‖2 − |〈a, b〉|2‖a‖2] 12,and the theorem is thus proved. The following particular cases hold.1. If a, b, x ∈ `2 (K) , K = C,R, where`2 (K) :={x = (xi)i∈N ,∞∑i=1|xi|2 <∞}with a, b linearly independent and(a)∞∑i=1xiai = 0,∞∑i=1|xi|2 = 1,then(2.9)∑∞i=1 |ai|2∑∞i=1 |bi|2 −∣∣∑∞i=1 aibi∣∣2∑∞i=1 |ai|2≥∣∣∣∣∣∞∑i=1xibi∣∣∣∣∣2.The equality holds in (2.9) iff(2.10) xi = ν[bi −∑∞k=1 akbk∑∞k=1 |ak|2· ai], i ∈ {1, 2, . . . }with ν ∈ K is such that(2.11) |ν| =(∑∞k=1 |ak|2) 12[∑∞k=1 |ak|2∑∞k=1 |bk|2 −∣∣∑∞k=1 akbk∣∣2] 12 .Remark 1. The case of equality in (1.5) is obviously a particular case of the above.We omit the details.4 SEVER S. DRAGOMIR AND ANCA C. GOS¸A2. If f, g, h ∈ L2 (Ω,m) , where Ω is an m−measurable space andL2 (Ω,m) :={f : Ω → K,∫Ω|f (x)|2 dm (x) <∞},with f, g being linearly independent and(2.12)∫Ωh (x) f (x)dm (x) = 0,∫Ω|h (x)|2 dm (x) = 1,then(2.13)∫Ω|f (x)|2 dm (x) ∫Ω|g (x)|2 dm (x)−∣∣∣∫Ω f (x) g (x)dm (x)∣∣∣2∫Ω|f (x)|2 dm (x)≥∣∣∣∣∫Ωh (x) g (x)dm (x)∣∣∣∣2 .The equality holds in (2.13) iffh (x) = ν[g (x)−∫Ωg (x) f (x)dm (x)∫Ω|f (x)|2 dm (x) f (x)]for a.e. x ∈ Ωand ν ∈ K with|ν| =(∫Ω|f (x)|2 dm (x)) 12[∫Ω|f (x)|2 dm (x) ∫Ω|g (x)|2 dm (x)−∣∣∣∫Ω f (x) g (x)dm (x)∣∣∣2] 12.References[1] D.S. MITRINOVIC´, J.E. PECˇARIC´ and A.M. FINK, Classical and New Inequalities in Anal-ysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993.[2] A.M. OSTROWSKI, Varlesungen u¨ber Differential und Integralrechnung II, Birkha¨user, Basel,1957.School of Computer Science and Mathematics, Victoria University of Technology,PO Box 14428, MCMC 8001, Victoria, Australia.E-mail address: sever@matilda.vu.edu.auURL: http://rgmia.vu.edu.au/SSDragomirWeb.htmlCollege No. 12, Res¸it¸a, Jud. Caras¸-Severin, R0-1700, Res¸it¸a, Romania.E-mail address: ancagosa@hotmail.com

Classical and New Inequalities in Analysis,

Varlesungen ¨ uber Diﬀerential und Integralrechnung II, Birkh¨ auser,

A Generalisation of an Ostrowski Inequality in Inner Product Spaces

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