Journal of Inequalities in Pure and Applied Mathematics

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1 Joural of Iequalities i Pure ad Applied Mathematics Volume 4, Issue 3, Article 63, 003 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES S.S. DRAGOMIR SCHOOL OF COMPUTER SCIENCE AND MATHEMATICS VICTORIA UNIVERSITY, PO BOX 448, MCMC 800, MELBOURNE, VICTORIA, AUSTRALIA. sever.dragomir@vu.edu.au Received Jauary, 003; accepted 4 May, 003 Commuicated by P.S. Bulle ABSTRACT. The mai purpose of this survey is to idetify ad highlight the discrete iequalities that are coected with CBS) iequality ad provide refiemets ad reverse results as well as to study some fuctioal properties of certai mappigs that ca be aturally associated with this iequality such as superadditivity, supermultiplicity, the strog versios of these ad the correspodig mootoicity properties. May compaio, reverse ad related results both for real ad complex umbers are also preseted. Key words ad phrases: Cauchy-Buyakovsky-Schwarz iequality, Discrete iequalities. 000 Mathematics Subject Classificatio. 6D5, 6D0. CONTENTS. Itroductio 4. CBS) Type Iequalities 5.. CBS) Iequality for Real Numbers 5.. CBS) Iequality for Complex Numbers 6.3. A Additive Geeralisatio 7.4. A Related Additive Iequality 8.5. A Parameter Additive Iequality 0.6. A Geeralisatio Provided by Youg s Iequality.7. Further Geeralisatios via Youg s Iequality.8. A Geeralisatio Ivolvig JCovex Fuctios 6.9. A Fuctioal Geeralisatio 8.0. A Geeralisatio for Power Series 9.. A Geeralisatio of Callebaut s Iequality.. Wager s Iequality for Real Numbers.3. Wager s iequality for Complex Numbers 4 ISSN electroic): c 003 Victoria Uiversity. All rights reserved

2 S.S. DRAGOMIR Refereces 6 3. Refiemets of the CBS) Iequality A Refiemet i Terms of Moduli A Refiemet for a Sequece Whose Norm is Oe A Secod Refiemet i Terms of Moduli A Refiemet for a Sequece Less tha the Weights A Coditioal Iequality Providig a Refiemet A Refiemet for No-Costat Sequeces De Bruij s Iequality McLaughli s Iequality A Refiemet due to Dayki-Eliezer-Carlitz A Refiemet via Dukl-Williams Iequality Some Refiemets due to Alzer ad Zheg 45 Refereces Fuctioal Properties A Mootoicity Property A Superadditivity Property i Terms of Weights The Superadditivity as a Idex Set Mappig Strog Superadditivity i Terms of Weights Strog Superadditivity as a Idex Set Mappig Aother Superadditivity Property The Case of Idex Set Mappig Supermultiplicity i Terms of Weights Supermultiplicity as a Idex Set Mappig 67 Refereces 7 5. Reverse Iequalities The Cassels Iequality The Pólya-Szegö Iequality The Greub-Rheiboldt Iequality A Cassels Type Iequality for Complex Numbers A Reverse Iequality for Real Numbers A Reverse Iequality for Complex Numbers Shisha-Mod Type Iequalities Zagier Type Iequalities A Reverse Iequality i Terms of the sup Norm A Reverse Iequality i Terms of the Norm A Reverse Iequality i Terms of the pnorm A Reverse Iequality Via a Adrica-Badea Result A Refiemet of Cassels Iequality Two Reverse Results Via Diaz-Metcalf Results Some Reverse Results Via the Čebyšev Fuctioal Aother Reverse Result via a Grüss Type Result 05 Refereces Related Iequalities Ostrowski s Iequality for Real Sequeces Ostrowski s Iequality for Complex Sequeces Aother Ostrowski s Iequality 6.4. Fa ad Todd Iequalities Some Results for Asychroous Sequeces A Iequality via A G H Mea Iequality A Related Result via Jese s Iequality for Power Fuctios Iequalities Derived from the Double Sums Case 7 J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

3 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES A Fuctioal Geeralisatio for Double Sums A CBS) Type Result for Lipschitzia Fuctios A Iequality via Jese s Discrete Iequality 6.. A Iequality via Lah-Ribarić Iequality 6.3. A Iequality via Dragomir-Ioescu Iequality A Iequality via a Refiemet of Jese s Iequality Aother Refiemet via Jese s Iequality A Iequality via Slater s Result A Iequality via a Adrica-Raşa Result A Iequality via Jese s Result for Double Sums Some Iequalities for the Čebyšev Fuctioal Other Iequalities for the Čebyšev Fuctioal Bouds for the Čebyšev Fuctioal 37 Refereces 40 Idex 4 J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

4 4 S.S. DRAGOMIR. INTRODUCTION The Cauchy-Buyakovsky-Schwarz iequality, or for short, the CBS) iequality, plays a importat role i differet braches of Moder Mathematics icludig Hilbert Spaces Theory, Probability & Statistics, Classical Real ad Complex Aalysis, Numerical Aalysis, Qualitative Theory of Differetial Equatios ad their applicatios. The mai purpose of this survey is to idetify ad highlight the discrete iequalities that are coected with CBS) iequality ad provide refiemets ad reverse results as well as to study some fuctioal properties of certai mappigs that ca be aturally associated with this iequality such as superadditivity, supermultiplicity, the strog versios of these ad the correspodig mootoicity properties. May compaios ad related results both for real ad complex umbers are also preseted. The first sectio is devoted to a umber of CBS) type iequalities that provides ot oly atural geeralizatios but also several extesios for differet classes of aalytic fuctios of a real variable. A geeralizatio of the Wager iequality for complex umbers is obtaied. Several results discovered by the author i the late eighties ad published i differet jourals of lesser circulatio are also surveyed. The secod sectio cotais differet refiemets of the CBS) iequality icludig de Bruij s iequality, McLaughli s iequality, the Dayki-Eliezer-Carlitz result i the versio preseted by Mitriović-Pečarić ad Fik as well as the refiemets of a particular versio obtaied by Alzer ad Zheg. A umber of ew results obtaied by the author, which are coected with the above oes, are also preseted. Sectio 4 is devoted to the study of fuctioal properties of differet mappigs aturally associated to the CBS) iequality. Properties such as superadditivity, strog superadditivity, mootoicity ad supermultiplicity ad the correspodig iequalities are metioed. I the ext sectio, Sectio 5, reverse results for the CBS) iequality are surveyed. The results of Cassels, Pólya-Szegö, Greub-Rheibold, Shisha-Mod ad Zagier are preseted with their origial proofs. New results ad versios for complex umbers are also obtaied. Reverse results i terms of porms of the forward differece recetly discovered by the author ad some refiemets of Cassels ad Pólya-Szegö results obtaied via Adrica-Badea iequality are metioed. Some ew facts derived from Grüss type iequalities are also poited out. Sectio 6 is devoted to various iequalities related to the CBS) iequality. The two iequalities obtaied by Ostrowski ad Fa-Todd results are preseted. New iequalities obtaied via Jese type iequality for covex fuctios are derived, some iequalities for the Čebyşev fuctioals are poited out. Versios for complex umbers that geeralize Ostrowski results are also emphasised. It was oe of the mai aims of the survey to provide complete proofs for the results cosidered. We also ote that i most cases oly the origial refereces are metioed. Each sectio cocludes with a list of the refereces utilized ad thus may be read idepedetly. Beig self cotaied, the survey may be used by both postgraduate studets ad researchers iterested i Theory of Iequalities & Applicatios as well as by Mathematicias ad other Scietists dealig with umerical computatios, bouds ad estimates where the CBS) iequality may be used as a powerful tool. The author iteds to cotiue this survey with aother oe devoted to the fuctioal ad itegral versios of the CBS) iequality. The correspodig results holdig i ier-product ad ormed spaces will be cosidered as well. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

5 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 5. CBS) TYPE INEQUALITIES.. CBS) Iequality for Real Numbers. The followig iequality is kow i the literature as Cauchy s or Cauchy-Schwarz s or Cauchy-Buyakovsky-Schwarz s iequality. For simplicity, we shall refer to it throughout this work as the CBS) iequality. Theorem.. If ā = a,..., a ) ad b = b,..., b ) are sequeces of real umbers, the.) ) a k b k a k b k with equality if ad oly if the sequeces ā ad b are proportioal, i.e., there is a r R such that a k = rb k for each k {,..., }. Proof. ) Cosider the quadratic polyomial P : R R,.) P t) = It is obvious that P t) = a k a k t b k ). ) ) t a k b k t + for ay t R. Sice P t) 0 for ay t R it follows that the discrimiat of P is egative, i.e., 0 4 = a k b k ) ad the iequality.) is proved. ) If we use Lagrage s idetity ).3) a i b i a i b i = a k b k b k a i b j a j b i ) i,j= = i<j a i b j a j b i ) the.) obviously holds. The equality holds i.) iff a i b j a j b i ) = 0 for ay i, j {,..., } which is equivalet with the fact that ā ad b are proportioal. Remark.. The iequality.) apparetly was firstly metioed i the work [] of A.L. Cauchy i 8. The itegral form was obtaied i 859 by V.Y. Buyakovsky []. The correspodig versio for ier-product spaces obtaied by H.A. Schwartz is maily kow as Schwarz s iequality. For a short history of this iequality see [3]. I what follows we use the spellig adopted i the paper [3]. For other spelligs of Buyakovsky s ame, see MathSciNet. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

6 6 S.S. DRAGOMIR.. CBS) Iequality for Complex Numbers. The followig versio of the CBS) iequality for complex umbers holds [4, p. 84]. Theorem.3. If ā = a,..., a ) ad b = b,..., b ) are sequeces of complex umbers, the.4) a k b k a k b k, with equality if ad oly if there is a complex umber c C such that a k k {,..., }. Proof..5) ) For ay complex umber λ C oe has the equality ak λ b k ) = ak λ b k āk λb ) k = a k + λ b k Re λ If i.5) we choose λ 0 C, λ 0 := the we get the idetity.6) 0 ak λ 0 bk = a kb k b k, b 0 ) a k b k. a k a kb k b k, = c b k for ay which proves.4). By virtue of.6), we coclude that equality holds i.4) if ad oly if a k = λ 0 bk for ay k {,..., }. ) Usig Biet-Cauchy s idetity for complex umbers.7) x i y i z i t i x i t i z i y i.8) = x i z j x j z i ) y i t j y j t i ) i,j= = i<j x i z j x j z i ) y i t j y j t i ) for the choices x i = ā i, z i = b i, y i = a i, t i = b i, i = {,..., }, we get a i b i a i b i = ā i b j ā j b i i,j= = ā i b j ā j b i. i<j Now the iequality.4) is a simple cosequece of.8). The case of equality is obvious by the idetity.8) as well. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

7 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 7 Remark.4. By the CBS) iequality for real umbers ad the geeralised triagle iequality for complex umbers z i z i, z i C, i {,..., } we also have a k b k a k b k ) a k b k. Remark.5. The Lagrage idetity for complex umbers stated i [4, p. 85] is wrog. It should be corrected as i.8)..3. A Additive Geeralisatio. The followig geeralisatio of the CBS) iequality was obtaied i [5, p. 5]. Theorem.6. If ā = a,..., a ), b = b,..., b ), c = c,..., c ) ad d = d,..., d ) are sequeces of real umbers ad p = p,..., p ), q = q,..., q ) are oegative, the.9) p i a i q i b i + p i c i q i d i p i a i c i q i b i d i. If p ad q are sequeces of positive umbers, the the equality holds i.9) iff a i b j = c i d j for ay i, j {,..., }. Proof. We will follow the proof from [5]. From the elemetary iequality.0) a + b ab for ay a, b R with equality iff a = b, we have.) a i b j + c i d j a i c i b j d j for ay i, j {,..., }. Multiplyig.) by p i q j 0, i, j {,..., } ad summig over i ad j from to, we deduce.9). If p i, q j > 0 i =,..., ), the the equality holds i.9) iff a i b j = c i d j for ay i, j {,..., }. Remark.7. The coditio a i b j = c i d j for c i 0, b j 0 i, j =,..., ) is equivalet with = d j b j i, j =,..., ), i.e., ā, c ad b, d are proportioal with the same costat k. a i c i Remark.8. If i.9) we choose p i = q i = i =,..., ), c i = b i, ad d i = a i i =,..., ), the we recapture the CBS) iequality. The followig corollary holds [5, p. 6]. Corollary.9. If ā, b, c ad d are oegative, the [ ].) a 3 i c i b 3 i d i + c 3 i a i d 3 i b i.3) [ a i b i d i b i a i c i + c i b i d i Aother result is embodied i the followig corollary [5, p. 6]. a i c i b i d i, ] ) d i a i c i a i b i c i d i. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

8 8 S.S. DRAGOMIR Corollary.0. If ā, b, c ad d are sequeces of positive ad real umbers, the: [ ] a 3 i b 3 i.4) + a i c i b i d i a i b i, c i d i.5) [ a i b i c i Fially, we also have [5, p. 6]. b i a i d i + b i c i Corollary.. If ā, ad b are positive, the a 3 ) i b 3 i a i b i b i a i ] ) a i d i a i b i. The followig versio for complex umbers also holds. a i ) b i a i b i 0. Theorem.. Let ā = a,..., a ), b = b,..., b ), c = c,..., c ) ad d = d,..., d ) be sequeces of complex umbers ad p = p,..., p ), q = q,..., q ) are oegative. The oe has the iequality [ ].6) p i a i q i b i + p i c i q i d i Re p i a i c i q i b i di. The case of equality for p, q positive holds iff a i b j = c i d j for ay i, j {,..., }. Proof. From the elemetary iequality for complex umbers a + b Re [ a b ], a, b C, with equality iff a = b, we have.7) a i b j + c i d j Re [ a i c i b j dj ] for ay i, j {,..., }. Multiplyig.7) by p i q j 0 ad summig over i ad j from to, we deduce.6). The case of equality is obvious ad we omit the details. Remark.3. Similar particular cases may be stated but we omit the details..4. A Related Additive Iequality. The followig iequality was obtaied i [5, Theorem.]. Theorem.4. If ā = a,..., a ), b = b,..., b ) are sequeces of real umbers ad c = c,..., c ), d = d,..., d ) are oegative, the.8) d i c i a i + c i d i b i c i a i d i b i. If c i ad d i i =,..., ) are positive, the equality holds i.8) iff ā = b = k where k = k, k,..., k) is a costat sequece. Proof. We will follow the proof from [5]. From the elemetary iequality.9) a + b ab for ay a, b R J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

9 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 9 with equality iff a = b; we have.0) a i + b j a i b j for ay i, j {,..., }. Multiplyig.0) by c i d j 0, i, j {,..., } ad summig over i from to ad over j from to, we deduce.8). If c i, d j > 0 i =,..., ), the the equality holds i.8) iff a i = b j for ay i, j {,..., } which is equivalet with the fact that a i = b i = k for ay i {,..., }. The followig corollary holds [5, p. 4]. Corollary.5. If ā ad b are oegative sequeces, the [ ].) a 3 i b i + a i b 3 i a i b i ;.) [ a i a i b i + b i ] ) b i a i a i b i. Aother corollary that may be obtaied is [5, p. 4 5]. Corollary.6. If ā ad b are sequeces of positive real umbers, the.3).4) ad.5) a i a i + b i a i b i b i + a i + b i a i b i a i a i b i, a i b i b i, The followig versio for complex umbers also holds. Theorem.7. If ā = a,..., a ), b = b,..., b ) are sequeces of complex umbers, the for p = p,..., p ) ad q = q,..., q ) two sequeces of oegative real umbers, oe has the iequality [ ].6) q i p i a i + p i q i b i Re p i a i q i bi. For p, q positive sequeces, the equality holds i.6) iff ā = b = k = k,..., k). The proof goes i a similar way with the oe i Theorem.4 o makig use of the followig elemetary iequality holdig for complex umbers a i b i..7) a + b Re [ a b ], a, b C; with equality iff a = b. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

10 0 S.S. DRAGOMIR.5. A Parameter Additive Iequality. The followig iequality was obtaied i [5, Theorem 4.]. Theorem.8. Let ā = a,..., a ), b = b,..., b ) be sequeces of real umbers ad c = c,..., c ), d = d,..., d ) be oegative. If α, β > 0 ad γ R such that γ αβ, the.8) α d i a i c i + β c i b i d i γ c i a i d i b i. Proof. We will follow the proof from [5]. Sice α, β > 0 ad γ αβ, it follows that for ay x, y R oe has.9) αx + βy γxy. Choosig i.9) x = a i, y = b j i, j =,..., ), we get.30) αa i + βb j γa i b j for ay i, j {,..., }. If we multiply.30) by c i d j 0 ad sum over i ad j from to, we deduce the desired iequality.8). The followig corollary holds. Corollary.9. If ā ad b are oegative sequeces ad α, β, γ are as i Theorem.8, the.3) α.3) α a i b i a 3 i + β a i b i + β a i b i b 3 i γ a i b i, ) b i a i γ a i b i. The followig particular case is importat [5, p. 8]. Theorem.0. Let ā, b be sequeces of real umbers. If p is a sequece of oegative real umbers with p i > 0, the:.33) p i a i p i b i p ia i b i p ia i p ib i p. i I particular,.34) a i b i a i b i a i b i. Proof. We will follow the proof from [5, p. 8]. If we choose i Theorem.8, c i = d i = p i i =,..., ) ad α = p ib i, β = p ia i, γ = p ia i b i, we observe, by the CBS) iequality with the weights p i i =,..., ) oe has γ αβ, ad the by.8) we deduce.33). Remark.. If we assume that ā ad b are asychroous, i.e., the by Čebyšev s iequality.35) a i a j ) b i b j ) 0 for ay i, j {,..., }, p i a i p i b i p i p i a i b i J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

11 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES respectively.36) a i b i a i b i, we have the followig refiemets of the CBS) iequality p i a i p i b i p ia i b i p ia i p ib i.37) p i ) p i a i b i provided p ia i b i 0, respectively.38) provided a ib i 0. a i b i a i b i a i ) b i a i b i.6. A Geeralisatio Provided by Youg s Iequality. The followig result was obtaied i [5, Theorem 5.]. Theorem.. Let ā = a,..., a ), b = b,..., b ), p = p,..., p ) ad q = q,..., q ) be sequeces of oegative real umbers ad α, β > with + =. The oe has the α β iequality.39) α q i p i b β i + β p i q i a α i αβ p i b i q i a i. If p ad q are sequeces of positive real umbers, the the equality holds i.39) iff there exists a costat k 0 such that a α i = b β i = k for each i {,..., }. Proof. It is, by the Arithmetic-Geometric iequality [6, p. 5], well kow that.40) α x + β y x α y β for x, y 0, with equality iff x = y. Applyig.40) for x = a α i, y = b β j α + β i, j =,..., ) we have =, α, β >.4) αb β j + βaα i αβa i b j for ay i, j {,..., } with equality iff a α i = b β j for ay i, j {,..., }. If we multiply.4) by q i p j 0 i, j {,..., }) ad sum over i ad j from to we deduce.39). The case of equality is obvious by the above cosideratios. The followig corollary is a atural cosequece of the above theorem. Corollary.3. Let ā, b, α ad β be as i Theorem.. The.4) b i a α+ i + a i b β+ i α β a i b i ; J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

12 S.S. DRAGOMIR.43) α a i b i a α i + β b i ) a i b β i a i b i. The followig result which provides a geeralisatio of the CBS) iequality may be obtaied by Theorem. as well [5, Theorem 5.]. Theorem.4. Let x ad ȳ be sequeces of positive real umbers. If α, β are as above, the ).44) x α i y α i + ) x β i α β yβ i yi x i y i. The equality holds iff x ad ȳ are proportioal. Proof. Follows by Theorem. o choosig p i = q i = y i, a i = x i y i, b i = x i y i, i {,..., }. Remark.5. For α = β =, we recapture the CBS) iequality. Remark.6. For a i = z i, b i = w i, with z i, w i C; i =,...,, we may obtai similar iequalities for complex umbers. We omit the details..7. Further Geeralisatios via Youg s Iequality. The followig iequality is kow i the literature as Youg s iequality.45) px q + qy p pqxy, x, y 0 ad p + q =, p > with equality iff x q = y p. The followig result geeralisig the CBS) iequality was obtaied i [7, Theorem.] see also [8, Theorem ]). Theorem.7. Let x = x,..., x ), ȳ = y,..., y ) be sequeces of complex umbers ad p = p,..., p ), q = q,..., q ) be two sequeces of oegative real umbers. If p >, + =, the p q.46) p k x k p q k y k p + q k x k q p k y k q p k x k y k q k x k y k. p q Proof. We shall follow the proof i [7]. Choosig x = x j y i, y = x i y j, i, j {,..., }, we get from.45).47) q x i p y j p + p x j q y i q pq x i y i x j y j for ay i, j {,..., }. Multiplyig with p i q j 0 ad summig over i ad j from to, we deduce the desired result.46). The followig corollary is a atural cosequece of the above theorem [7, Corollary.] see also [8, p. 05]). Corollary.8. If x ad ȳ are as i Theorem.7 ad m = m,..., m ) is a sequece of oegative real umbers, the.48) m k x k p m k y k p + m k x k q m k y k q m k x k y k ), p q where p >, p + q =. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

13 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 3 Remark.9. If i.48) we assume that m k =, k {,..., }, the we obtai [7, p. 7] see also [8, p. 05]).49) x k p y k p + x k q y k q x k y k ), p q which, i the particular case p = q = will provide the CBS) iequality. The secod geeralisatio of the CBS) iequality via Youg s iequality is icorporated i the followig theorem [7, Theorem.4] see also [8, Theorem ]). Theorem.30. Let x, ȳ, p, q ad p, q be as i Theorem.7. The oe has the iequality.50) p p k x k p q k y k q + q q k x k q p k y k p p k x k y k p Proof. We shall follow the proof i [7]. Choosig i.45), x = x j, y = x i y j y i, we get ) q ) p xj xi.5) p + q pq x i x j y j y i y i y j for ay y i 0, i, j {,..., }. It is easy to see that.5) is equivalet to.5) q x i p y j q + p y i p x j q pq x i y i p x j y j q q k x k y k q. for ay i, j {,..., }. Multiplyig.5) by p i q j 0 i, j {,..., }) ad summig over i ad j from to, we deduce the desired iequlality.50). The followig corollary holds [7, Corollary.5] see also [8, p. 06]). Corollary.3. Let x, ȳ, m ad p, q be as i Corollary.8. The.53) p m k x k p m k y k q + q m k x k q m k y k p m k x k y k p m k x k y k q. Remark.3. If i.53) we assume that m k =, k {,..., }, the we obtai [7, p. 8] see also [8, p. 06]).54) x k p y k q + x k q y k p x k y k p x k y k q, p q which, i the particular case p = q = will provide the CBS) iequality. The third result is embodied i the followig theorem [7, Theorem.7] see also [8, Theorem 3]). J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

14 4 S.S. DRAGOMIR Theorem.33. Let x, ȳ, p, q ad p, q be as i Theorem.7. The oe has the iequality.55) p p k x k p q k y k q + q q k x k p p k y k q p k x k y k Proof. We shall follow the proof i [7]. If we choose x = y i ad y = x i y j x j i.45) we get ) q ) p yi xi p + q pq x i y i y j x j x j y j, for ay x i, y j 0, i, j {,..., }, givig.56) q x i p y j q + p y i q x j p pq x i y i x j p y j q p k x k p y k q. for ay i, j {,..., }. Multiplyig.56) by p i q j 0 i, j {,..., }) ad summig over i ad j from to, we deduce the desired iequality.55). The followig corollary is a atural cosequece of the above theorem [8, p. 06]. Corollary.34. Let x, ȳ, m ad p, q be as i Corollary.8. The oe has the iequality:.57) m k x k p m k y k q m k x k y k m k x k p y k q. Remark.35. If i.57) we assume that m k =, k = {,..., }, the we obtai [7, p. 8] see also [8, p. 0]).58) x k p y k q x k y k x k p y k q, which, i the particular case p = q = will provide the CBS) iequality. The fourth geeralisatio of the CBS) iequality is embodied i the followig theorem [7, Theorem.9] see also [8, Theorem 4]). Theorem.36. Let x, ȳ, p, q ad p, q be as i Theorem.7. The oe has the iequality.59) q p k x k q k y k q + p p k y k q k x k p Proof. We shall follow the proof i [7]. Choosig i.45), x = x i q yj, y = x j y i p, we get q k x k y k.60) p x i y j q + q x j p y i pq x i q yi p xj y j p k x k q yk p. for ay i, j {,..., }. Multiply.60) by p i q j 0 i, j {,..., }) ad summig over i ad j from to, we deduce the desired iequality.60). J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

15 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 5 The followig corollary holds [7, Corollary.0] see also [8, p. 07]). Corollary.37. Let x, ȳ, m ad p, q be as i Corollary.8. The oe has the iequality:.6) q m k x k m k y k q + p m k y k m k x k p m k x k y k Remark.38. If i.6) we take m k =, k {,..., }, the we get.6) q x k y k q + p y k x k p x k y k which, i the particular case p = q = will provide the CBS) iequality. m k x k q yk p. x k q yk p, The fifth result geeralisig the CBS) iequality is embodied i the followig theorem [7, Theorem.] see also [8, Theorem 5]). Theorem.39. Let x, ȳ, p, q ad p, q be as i Theorem.7. The oe has the iequality.63) p p k x k q k y k q + q Proof. We will follow the proof i [7]. p k y k p q k x k p p k x k p yk q q k x k p y k q. Choosig i.45), x = y i q, y = x i, y y j x j i, x j 0, i, j {,..., }, we may write ) y i q ) q x i p p p + q pq y i q xi p, y j x j x j y j from where results.64) p y i x j p + q x i y j q pq x i p yi q xj p y j q for ay i, j {,..., }. Multiplyig.64) by p i q j 0 i, j {,..., }) ad summig over i ad j from to, we deduce the desired iequality.63). The followig corollary holds [7, Corollary.3] see also [8, p. 08]). Corollary.40. Let x, ȳ, m ad p, q be as i Corollary.8. The oe has the iequality:.65) p m k x k m k y k q + q m k y k m k x k p m k x k p yk q m k x k p y k q. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

16 6 S.S. DRAGOMIR Remark.4. If i.46) we choose m k =, k {,..., }, the we get [7, p. 0] see also [8, p. 08]).66) x k y k q + y k x k p x k p yk q x k p y k q, p q which i the particular case p = q = will provide the CBS) iequality. Fially, the followig result geeralisig the CBS) iequality holds [7, Theorem.5] see also [8, Theorem 6]). Theorem.4. Let x, ȳ, p, q ad p, q be as i Theorem.7. The oe has the iequality:.67) p p k x k q k y k p + q q k y k p k x k q p k x k p yk Proof. We shall follow the proof i [7]. From.45) oe has the iequality ).68) q x i p p yj + p x j q q yi ) pq xi p yi x j q yj q k x k q yk. for ay i, j {,..., }. Multiplyig.68) by p i q j 0 i, j {,..., }) ad summig over i ad j from to, we deduce the desired iequality.67). The followig corollary also holds [7, Corollary.6] see also [8, p. 08]). Corollary.43. With the assumptios i Corollary.8, oe has the iequality.69) m k x k m k p y k p + ) q y k q m k x k p yk m k x k q yk. Remark.44. If i.69) we choose m k = k {,..., }), the we get.70) x k p y k p + ) q y k q x k p yk x k q yk, which, i the particular case p = q =, provides the CBS) iequality..8. A Geeralisatio Ivolvig JCovex Fuctios. For a >, we deote by exp a the fuctio.7) exp a : R 0, ), exp a x) = a x. Defiitio.. A fuctio f : I R R is said to be Jcovex o a iterval I if ) x + y f x) + f y).7) f for ay x, y I. It is obvious that ay covex fuctio o I is a J covex fuctio o I, but the coverse does ot geerally hold. The followig lemma holds see [7, Lemma 4.3]). J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

17 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 7 Lemma.45. Let f : I R R be a Jcovex fuctio o I, a > ad x, y R\ {0} with log a x, log a y I. The log a xy I ad.73) {exp b [f log a xy )]} exp b [ f loga x )] exp b [ f loga y )] for ay b >. Proof. I, beig a iterval, is a covex set i R ad thus log a xy = [ loga x + log a y ] I. Sice f is Jcovex, oe has [.74) f log a xy ) = f loga x + log a y )] f log a x ) + f log a y ). Takig the exp b i both parts, we deduce exp b [f log a xy )] exp b [ f loga x ) + f log a y ) = { exp b [ f loga x )] exp b [ f loga y )]}, ] which is equivalet to.73). The followig geeralisatio of the CBS) iequality i terms of a J covex fuctio holds [7, Theorem 4.4]. Theorem.46. Let f : I R R be a Jcovex fuctio o I, a, b > ad ā = a,..., a ), b = b,..., b ) sequeces of ozero real umbers. If log a a k, log a b k I for all k {,..., }, the oe has the iequality: { [ )].75) exp b [f log a a k b k )]} [ exp b f loga a k exp b f loga bk)]. Proof. Usig Lemma.45 ad the CBS) iequality oe has exp b [f log a a k b k )] [ [ )] [ expb f loga a k expb f loga bk)]] { [expb [ )]] f loga a k which is clearly equivalet to.75). } ) { [expb [ )]] } f loga b k Remark.47. If i.75) we choose a = b > ad f x) = x, x R, the we recapture the CBS) iequality. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

18 8 S.S. DRAGOMIR.9. A Fuctioal Geeralisatio. The followig result was proved i [0, Theorem ]. Theorem.48. Let A be a subset of real umbers R, f : A R ad ā = a,..., a ), b = b,..., b ) sequeces of real umbers with the properties that i) a i b i, a i, b i A for ay i {,..., }, ii) f a i ), f b i ) 0 for ay i {,..., }, iii) f a i b i ) f a i ) f b i ) for ay i {,..., }. The oe has the iequality: [.76) f a i b i )] f ) a i f ) b i. Proof. We give here a simpler proof tha that foud i [0]. We have f a i b i ) f a i b i ) [ )] [ f a i f )] b i [ [f )] a i ) [ = f ) a i f ) ] b i ad the iequality.76) is proved. [f b i )] ) ] by the CBS)-iequality) Remark.49. It is obvious that for A = R ad f x) = x, we recapture the CBS) iequality. Assume that ϕ : N N is Euler s idicator. I 940, T. Popoviciu [] proved the followig iequality for ϕ.77) [ϕ ab)] ϕ a ) ϕ b ) for ay atural umber a, b; with equality iff a ad b have the same prime factors. A simple proof of this fact may be doe by usig the represetatio ) ) ϕ ) = p pk, where = p α p α p α k k [9, p. 09]. The followig geeralisatio of Popoviciu s result holds [0, Theorem ]. Theorem.50. Let a i, b i N i =,..., ). The oe has the iequality [.78) ϕ a i b i )] ϕ ) a i ϕ ) b i. Proof. Follows by Theorem.48 o takig ito accout that, by.77), [ϕ a i b i )] ϕ a i ) ϕ b i ) for ay i {,..., }. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

19 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 9 Further, let us deote by s ) the sum of all relatively prime umbers with ad less tha. The the followig result also holds [0, Theorem ]. Theorem.5. Let a i, b i N i =,..., ). The oe has the iequality [.79) s a i b i )] s ) a i s ) b i. Proof. It is kow see for example [9, p. 09]) that for ay N oe has.80) s ) = ϕ ). Thus.8) [s a i b i )] = 4 a i b i ϕ a i b i ) 4 a i b i ϕ a i ) ϕ b i ) = s a i ) s b i ) for each i {,..., }. Usig Theorem.48 we the deduce the desired iequality.79). The followig corollaries of Theorem.48 are also atural to be cosidered [0, p. 6]. Corollary.5. Let a i, b i R i =,..., ) ad a >. Deote exp a x = a x, x R. The oe has the iequality [ ).8) exp a a i b i )] ) exp a a i exp a b i. Corollary.53. Let a i, b i, ) i =,..., ) ad m > 0. The oe has the iequality: [ ].83) a i b i ) m. a i )m b i )m.0. A Geeralisatio for Power Series. The followig result holds [, Remark ]. Theorem.54. Let F : r, r) R, F x) = k=0 α kx k with α k 0, k N. If ā = a,..., a ), b = b,..., b ) are sequeces of real umbers such that.84) a i b i, a i, b i r, r) for ay i {,..., }, the oe has the iequality:.85) F ) a i F [ ) b i F a i b i )]. Proof. Firstly, let us observe that if x, y R such that xy, x, y r, r), the oe has the iequality.86) [F xy)] F x ) F y ). Ideed, by the CBS) iequality, we have [ ].87) α k x k y k α k x k k=0 k=0 α k y k, 0. k=0 Takig the limit as i.87), we deduce.86). J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

20 0 S.S. DRAGOMIR Usig the CBS) iequality ad.86) we have F a i b i ) F a i b i ) [ )] [ F a i F )] b i which is clearly equivalet to.85). { [F a i )] ) [ = F ) a i F ) ] b i, } [F )] ) b i The followig particular iequalities of CBS) type hold [, p. 64]. ) If ā, b are sequeces of real umbers, the oe has the iequality.88).89).90) exp ) a k exp [ ) b k exp a k b k )] ; sih ) a k sih [ ) b k sih a k b k )] ; cosh ) a k cosh [ ) b k cosh a k b k )]. ) If ā, b are such that a i, b i, ), i {,..., }, the oe has the iequalities.9).9) ta ) a k ta [ ) b k ta a k b k )] ; arcsi ) a k arcsi [ ) b k arcsi a k b k )] ; [ ) ] [ + a ) ] { [.93) l k + b ) ]} l k + ak b k l ; a k b k a k b k [ ) ] [ ) ] { [ ) ]}.94) l l l ; a k b k a k b k.95) a k )m [ ] b k )m a k b k ) m, m > 0. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

21 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES.. A Geeralisatio of Callebaut s Iequality. The followig result holds see also [, Theorem ] for a geeralisatio for positive liear fuctioals). Theorem.55. Let F : r, r) R, F x) = k=0 α kx k with α k 0, k N. If ā = a,..., a ), b = b,..., b ) are sequeces of oegative real umbers such that.96) a i b i, a α i b α i, a α i b α i 0, r) for ay i {,..., } ; α [0, ], the oe has the iequality.97) [ F a i b i )] F a α i b α i ) F ) a α i b α i. Proof. Firstly, we ote that for ay x, y > 0 such that xy, x α y α, x α y α 0, r) oe has.98) [F xy)] F x α y α) F x α y α). Ideed, usig Callebaut s iequality, i.e., we recall it [4].99) m ) α i x i y i m α i x α i y α i m α i x α i y α i, we may write, for m 0, that.00) m ) α i x i y i i=0 m α i x α y α) i i=0 m α i x α y α) i. i=0 Takig the limit as m, we deduce.98). Usig the CBS) iequality ad.98) we may write: F a i b i ) = F a i b i ) { [ )] F a α i b α [ i F a α i [ [F a α i b α i F a α i b α i )] ) ) b α i )] [F a α i b α i F ) ] a α i b α i } )] ) which is clearly equivalet to.97). The followig particular iequalities also hold [, pp ]. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

22 S.S. DRAGOMIR ) Let ā ad b be sequeces of oegative real umbers. The oe has the iequalities.0) [.0) [.03) [ exp a k b k )] sih a k b k )] cosh a k b k )] exp a α k b α k sih a α k b α k cosh a α k b α k ) ) ) exp a α k b α k ) ; sih a α k b α k ) ; cosh a α k b α k ). ) Let ā ad b be such that a k, b k 0, ) for ay k {,..., }. The oe has the iequalities:.04) [ ta a k b k )] ta a α k b α k ) ta a α k b α k ) ;.05) [ arcsi a k b k )] arcsi a α k b α k ) arcsi a α k b α k ) ;.06) { [ ) ]} [ + ak b k l l a k b k + a α k b α ) ] k l a α k bα k [ + a α ) ] k b α k a α ; k b α k.07) { [ ) ]} [ ) ] [ ) ] l l l a k b k a α k bα k a α. k b α k.. Wager s Iequality for Real Numbers. The followig geeralisatio of the CBS) iequality for sequeces of real umbers is kow i the literature as Wager s iequality [5], or [4] see also [4, p. 85]). Theorem.56. Let ā = a,..., a ) ad b = b,..., b ) be sequeces of real umbers. If 0 x, the oe has the iequality.08) a k b k + x ) a i b j i j [ a k + x b k + x ] b i b j. i<j a i a j] [ i<j Proof. We shall follow the proof i [3] see also [4, p. 85]). J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

23 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 3 For ay x [0, ], cosider the quadratic polyomial i y P y) := x) = x) [ = x) [ a k y b k ) + x a k y b k ) y a k y a k b k + ) + x y a k y b k ] ] a k) ) [ a k + x a k y y x) + x) ) b k + x b k ) = a k + x a k [ y a k b k + x + a k a k ) b k + x b k Sice, it is obvious that: ad we get P y) = a k + x ) a k a k b k ) b k i<j a i a j ) y y b k b k a k = a k b k = b k = y a k b k + x. b k ) + a k b k + x )] a k b k i<j i j i<j i j a i a j, a i b j b i b j, a i b j ) + ) b k b k + x a k ] b k i<j b i b j. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

24 4 S.S. DRAGOMIR Takig ito cosideratio, by the defiitio of P, that P y) 0 for ay y R, it follows that the discrimiat 0, i.e., 0 4 = a k b k + x ) a i b j i j a k + x b k + x ) b i b j i<j a i a j) i<j ad the iequality.08) is proved. Remark.57. If x = 0, the from.08) we recapture the CBS) iequality for real umbers..3. Wager s iequality for Complex Numbers. The followig iequality which provides a versio for complex umbers of Wager s result holds [6]. Theorem.58. Let ā = a,..., a ) ad b = b,..., b ) be sequeces of complex umbers. The for ay x [0, ] oe has the iequality.09) [ Re ) a k bk + x Re ) ] a i bj i j [ a k + x ] [ Re a i ā j ) b k + x Re ) ] b i bj. i<j i<j Proof. Start with the fuctio f : R R,.0) f t) = x) We have ta k b k + x ta k b k )..) f t) = x) ta k b k ) tā k b ) k = x) + x t [ t + x ) a k b k t a k t [ t b k ā k t a k t b k ) ā k bk a k bk + ā k t ] b k a k ] bk + b k J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

25 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 5 = x) a k + x a k t [ + x) + x) Re [ ) a k bk + x Re b k + x b k. a k ]] bk t Observe that.) a k = = = = a i ā j i,j= a i + i j a i + a i + i<j a i ā j a i ā j + a i ā j i<j j<i Re a i ā j ) ad, similarly,.3) b k = b i + Re ) b i bj. i<j Also ad thus a k bk = a i bi + a i bj i j.4) Re a k ) bk = Re ) a i bi + Re ) a i bj. i j Utilisig.).4), by.), we deduce [.5) f t) = a k + x [ + Re ) a k bk + x i<j Re a i ā j ) i j i<j ] t Re ) ] a i bj t + b k + x Re ) b i bj. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

26 6 S.S. DRAGOMIR Sice, by.0), f t) 0 for ay t R, it follows that the discrimiat of the quadratic fuctio give by.5) is egative, i.e., 0 4 [ = Re ) a k bk + x Re ) ] a i bj i j [ a k + x ] [ Re a i ā j ) b k + x Re ) ] b i bj i<j i<j ad the iequality.09) is proved. Remark.59. If x = 0, the we get the CBS) iequality [.6) Re ) ] a k bk a k b k. REFERENCES [] V.Y. BUNIAKOWSKI, Sur quelques iégalités cocerat les itégrales aux differeces fiies, Mem. Acad. St. Petersburg, 7) 859), No. 9, -8. [] A.L. CAUCHY, Cours d Aalyse de l École Royale Polytechique, I re Partie, Aalyse Algébrique, Paris, 8. [3] P. SCHREIDER, The Cauchy-Buyakovsky-Schwarz iequality, Herma Graßma Lieschow, 994), [4] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical ad New Iequalities i Aalysis, Kluwer Academic Publishers, Dordrecht/Bosto/Lodo, 993. [5] S.S. DRAGOMIR, O some iequalities Romaia), Caiete Metodico Ştiiţifice, No. 3, 984, pp. 0. Faculty of Mathematics, Timişoara Uiversity, Romaia. [6] E.F. BECKENBACH AND R. BELLMAN, Iequalities, Spriger-Verlag, Berli-Göttige- Heidelberg, 96. [7] S.S. DRAGOMIR, O Cauchy-Buiakowski-Schwartz s Iequality for Real Numbers Romaia), Caiete Metodico-Ştiiţifice, No. 57, pp. 4, 989. Faculty of Mathematics, Timişoara Uiversity, Romaia. [8] S.S. DRAGOMIR AND J. SÁNDOR, Some geeralisatios of Cauchy-Buiakowski-Schwartz s iequality Romaia), Gaz. Mat. Metod. Bucharest), 990), [9] I. CUCUREZEANU, Problems o Number Theory Romaia), Ed. Techicǎ, Bucharest, 976. [0] S.S. DRAGOMIR, O a iequality of Tiberiu Popoviciu Romaia), Gaz. Mat. Metod., Bucharest) 8 987), 4 8. ZBL No. 7:0A. [] T. POPOVICIU, Gazeta Matematicǎ, 6 940), p [] S.S. DRAGOMIR, Iequalities of Cauchy-Buiakowski-Schwartz s type for positive liear fuctioals Romaia), Gaz. Mat. Metod. Bucharest), 9 988), [3] T. ANDRESCU, D. ANDRICA AND M.O. DRÎMBE, The triomial priciple i obtaiig iequalities Romaia), Gaz. Mat. Bucharest), ), [4] P. FLOR, Über eie Uglichug vo S.S. Wager, Elemete Math., 0 965), 36. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

27 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 7 [5] S.S. WAGNER, Amer. Math. Soc., Notices, 965), 0. [6] S.S. DRAGOMIR, A versio of Wager s iequality for complex umbers, submitted. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

28 8 S.S. DRAGOMIR 3. REFINEMENTS OF THE CBS) INEQUALITY 3.. A Refiemet i Terms of Moduli. The followig result was proved i []. Theorem 3.. Let ā = a,..., a ) ad b = b,..., b ) be sequeces of real umbers. The oe has the iequality ) 3.) a k b k a k b k a k a k b k b k a k b k a k b k 0. Proof. We will follow the proof from []. For ay i, j {,..., } the ext elemetary iequality is true: 3.) a i b j a j b i a i b j a j b i. By multiplyig this iequality with a i b j a j b i 0 we get 3.3) a i b j a j b i ) a i b j a j b i ) a i b j a j b i ) = a i a i b j b j + b i b i a j a j a i b i a j b j a i b j a j b i. Summig 3.3) over i ad j from to, we deduce a i b j a j b i ) i,j= a i a i b j b j + b i b i a j a j a i b i a j b j a i b j a j b i i,j= a i a i b j b j + b i b i a j a j a i b i a j b j a i b j a j b i ), i,j= givig the desired iequality 3.). The followig corollary is a atural cosequece of 3.) [, Corollary 4]. Corollary 3.. Let ā be a sequece of real umbers. The ) 3.4) a k a k a k a k a k a k 0. There are some particular iequalities that may also be deduced from the above Theorem 3. see [, p. 80]). ) Suppose that for ā ad b sequeces of real umbers, oe has sg a k ) = sg b k ) = e k {, }. The oe has the iequality ) ) 3.5) a k b k a k b k e k a k e k b k e k a k b k 0. ) If ā = a,..., a ), the we have the iequality [ ] 3.6) a k ) k a k a k ) k a k 0. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

29 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 9 3) If ā = a,..., a + ), the we have the iequality + + ) + 3.7) + ) a k ) k + a k a k ) k a k 0. The followig versio for complex umbers is valid as well. Theorem 3.3. Let ā = a,..., a ) ad b = b,..., b ) be sequeces of complex umbers. The oe has the iequality 3.8) a i b i a i b i a i ā i b i b i a i b i b i ā i 0. Proof. We have for ay i, j {,..., } that Multiplyig by ā i b j ā j b i 0, we get ā i b j ā j b i a i b j a j b i. ā i b j ā j b i a i ā i b j b j + a j ā j b i b i a i b i b j ā j b i ā i a j b j. Summig over i ad j from to ad usig the Lagrage s idetity for complex umbers: a i b i a i b i = ā i b j ā j b i i,j= we deduce the desired iequality 3.8). Remark 3.4. Similar particular iequalities may be stated, but we omit the details. 3.. A Refiemet for a Sequece Whose Norm is Oe. The followig result holds [, Theorem 6]. Theorem 3.5. Let ā = a,..., a ), b = b,..., b ) be sequeces of real umbers ad ē = e,..., e ) be such that e i =. The the followig iequality holds [ ] a i b 3.9) i a k b k e k a k e k b k + e k a k e k b k a k b k ). Proof. We will follow the proof from []. From the CBS) iequality, oe has 3.0) [ ) ] a k e i a i e k [ ) ] b k e i b i e k { [ ) ] ) ]} a k e i a i e k [b k e i b i e k. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

30 30 S.S. DRAGOMIR Sice e k ad =, a simple calculatio shows that [ ) ] ) a k e i a i e k = a k e k a k, [ ) ] b k e i b i e k = ) b k e k b k, [ ) ] ) ] a k e i a i e k [b k e i b i e k = ad the the iequality 3.0) becomes ) ) 3.) a k e k a k b k e k b k a k b k a k b k e k a k e k a k Usig the elemetary iequality m l ) p q ) mp lq), m, l, p, q R e k b k ) e k b k 0. for the choices ) m = a k, l = e k a k, p = ad q = e k b k the above iequality 3.) provides the followig result 3.) Sice a k ) a k b k ) ) e k a k b k ) e k b k a k b k e k a k b k ) e k a k e k b k the, by takig the square root i 3.) we deduce the first part of 3.9). The secod part is obvious, ad the theorem is proved. e k b k. The followig corollary is a atural cosequece of the above theorem [, Corollary 7]. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

31 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 3 Corollary 3.6. Let ā, b, ē be as i Theorem 3.5. If a kb k = 0, the oe has the iequality: 3.3) a k ) ) b k 4 e k a k e k b k. The followig iequalities are iterestig as well [, p. 8]. 3.4) 3.5) ) For ay ā, b oe has the iequality a k ) If a kb k = 0, the [ b k a k b k ) a k b k. a k a k b k + ) b k 4 ) a k b k. a k ] b k I a similar maer, we may state ad prove the followig result for complex umbers. Theorem 3.7. Let ā = a,..., a ), b = b,..., b ) be sequeces of complex umbers ad ē = e,..., e ) a sequece of complex umbers satisfyig the coditio e i =. The the followig refiemet of the CBS) iequality holds [ ] a i b i 3.6) a k bk a k ē k e k bk + a k ē k e k bk a k bk. The proof is similar to the oe i Theorem 3.5 o usig the correspodig CBS) iequality for complex umbers. Remark 3.8. Similar particular iequalities may be stated, but we omit the details A Secod Refiemet i Terms of Moduli. The followig lemma holds. Lemma 3.9. Let ā = a,..., a ) be a sequece of real umbers ad p = p,..., p ) a sequece of positive real umbers with p i =. The oe has the iequality: ) 3.7) p i a i p i a i p i a i a i p i a i p i a i. Proof. By the properties of moduli we have a i a j ) = a i a j ) a i a j ) a i a j ) a i a j ) for ay i, j {,..., }. This is equivalet to 3.8) a i a i a j + a j a i a i + a j a j a i a j a j a i for ay i, j {,..., }. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

32 3 S.S. DRAGOMIR If we multiply 3.8) by p i p j 0 ad sum over i ad j from to we deduce p j p i a i p i a i p j a j + p i p j a j j= j= j= p i p j a i a i + a j a j a i a j a j a i i,j= p i p j a i a i + a j a j a i a j a j a i ), i,j= which is clearly equivalet to 3.7). Usig the above lemma, we may prove the followig refiemet of the CBS) -iequality. Theorem 3.0. Let ā = a,..., a ) ad b = b,..., b ) be two sequeces of real umbers. The oe has the iequality ) 3.9) a i b i a i b i a i sg a i ) b i b i a i b i a i b i 0. Proof. If we choose for a i 0, i {,..., }) i 3.7), that we get a i a k bi a i from where we get b i a k ) p i := a i, x i = b i, i {,..., }, a k a i a i a k a i a k a ib i ) a k ) which is clearly equivalet to 3.9). bi a i The case for complex umbers is as follows. b i a i ) bi a i a i a i a k a i a k b i a i a i a k b i b i a ib i a ib i a k ) Lemma 3.. Let z = z,..., z ) be a sequece of complex umbers ad p = p,..., p ) a sequece of positive real umbers with p i =. The oe has the iequality: 3.0) p i z i p i z i p i z i z i p i z i p i z i. Proof. By the properties of moduli for complex umbers we have z i z j z i z j ) z i z j ) for ay i, j {,..., }, which is clearly equivalet to for ay i, j {,..., }. z i Re z i z j ) + z j z i z i + z j z j z i z j z i z j bi a i J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

33 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 33 If we multiply with p i p j iequality 3.0). 0 ad sum over i ad j from to, we deduce the desired Now, i a similar maer to the oe i Theorem 3.0, we may state the followig result for complex umbers. Theorem 3.. Let ā = a,..., a ) a i 0, i =,..., ) ad b = b,..., b ) be two sequeces of complex umbers. The oe has the iequality: 3.) a i b i ā i b i a i a i b i b i a i b i ā i b i A Refiemet for a Sequece Less tha the Weights. The followig result was obtaied i [, Theorem 9] see also [, Theorem 3.0]). Theorem 3.3. Let ā = a,..., a ), b = b,..., b ) be sequeces of real umbers ad p = p,..., p ), q = q,..., q ) be sequeces of oegative real umbers such that p k q k for ay k {,..., }. The we have the iequality 3.) p k a k p k b k p k q k ) a k b k + q k a k q k b k [ ] p k q k ) a k b k + q k a k b k p k a k b k ). Proof. We shall follow the proof i []. Sice p k q k 0, the the CBS) iequality for the weights r k := p k q k will give 3.3) p k a k ) q k a k p k b k Usig the elemetary iequality ) ) [ ] q k b k p k q k ) a k b k. ac bd) a b ) c d ), a, b, c, d R for the choices ) a = p k a k, b = q k a k ), c = p k b k ) ad d = q k b k ) J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

34 34 S.S. DRAGOMIR we deduce by 3.3) that 3.4) Sice, obviously, p k a k ) ) p k b k q k a k ) q k b k ) [ ] p k q k ) a k b k. p k a k ) p k b k ) q k a k ) q k b k the, by 3.4), o takig the square root, we would get ) ) ) ) p k a k p k b k q k a k q k b k + p k q k ) a k b k, which provides the first iequality i 3.). The other iequalities are obvious ad we omit the details. ) The followig corollary is a atural cosequece of the above theorem [, Corollary 3.]. Corollary 3.4. Let ā, b be sequeces of real umbers ad s = s,..., s ) be such that 0 s k for ay k {,..., }. The oe has the iequalities a k b k ) 3.5) s k ) a k b k + s k a k s k b k [ ] s k ) a k b k + s k a k b k a k b k ). Remark 3.5. Assume that ā, b ad s are as i Corollary 3.4. The followig iequalities hold see [, p. 5]). 3.6) 3.7) a) If a kb k = 0, the b) If s ka k b k = 0, the a k a k ) b k 4 s k a k b k. b k a k b k + α k a k α k b k ). J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

35 A SURVEY ON CAUCHY-BUNYAKOVSKY-SCHWARZ TYPE DISCRETE INEQUALITIES 35 I particular, we may obtai the followig particular iequalities ivolvig trigoometric fuctios see [, p. 5]) a k b k ) 3.8) a k b k cos α k + a k si α k b k si α k [ ] a k b k cos α k + a k b k si α k a k b k ), where a k, b k, α k R, k =,...,. If oe would assume that a kb k = 0, the 3.9) b k 4 a k a k b k si α k ). If a kb k si α k = 0, the 3.30) a k b k a k b k + a k si α k ) b k si α k 3.5. A Coditioal Iequality Providig a Refiemet. The followig lemma holds [, Lemma 4.]. Lemma 3.6. Cosider the sequeces of real umbers x = x,..., x ), ȳ = y,..., y ) ad z = z,..., z ). If 3.3) y k x k z k for ay k {,..., }, the oe has the iequality: 3.3) y k ) x k z k. Proof. We will follow the proof i []. Usig the coditio 3.3) ad the CBS) iequality, we have y k x k zk which is clearly equivalet to 3.3). [ x k = x k The followig result holds [, Theorem 4.6]. ) z k ) ] ) z k. J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003

36 36 S.S. DRAGOMIR Theorem 3.7. Let ā = a,..., a ), b = b,..., b ) ad c = c,..., c ) be sequeces of real umbers such that i) b k + c k 0 k {,..., }); ii) a k b kc k b k + c k for ay k {,..., }. The oe has the iequality 3.33) a k b k c k b. k + c k ) Proof. We will follow the proof i []. By ii) we observe that a k b kc k b k b k + c k for ay k {,..., } c k ad thus 3.34) x k := b k a k 0 ad z k := c k a k 0 for ay k {,..., }. A simple calculatio also shows that the relatio ii) is equivalet to 3.35) a k b k a k ) c k a k ) for ay k {,..., }. If we cosider y k := a k ad take x k, z k k =,..., ) as defied by 3.34), the we get yk x kz k with x k, z k 0) for ay k {,..., }. Applyig Lemma 3.6 we deduce 3.36) a k ) which is clearly equivalet to 3.33). b k ) a k c k ) a k The followig corollary is a atural cosequece of the above theorem [, Corollary 4.7]. Corollary 3.8. For ay sequece x ad ȳ of real umbers, with x k + y k 0 k =,..., ), oe has: x k y k 3.37) x k + y k x k y k x. k + y k ) ad For two positive real umbers, let us recall the followig meas A a, b) := a + b G a, b) := ab the arithmetic mea) the geometric mea) H a, b) := + the harmoic mea). a b We remark that if ā = a,..., a ), b = b,..., b ) are sequeces of real umbers, the obviously ) 3.38) A a i, b i ) = A a i, b i, J. Iequal. Pure ad Appl. Math., 43) Art. 63, 003