ON SOME INEQUALITIES IN NORMED LINEAR SPACES

Transcription

1 ON SOME INEQUALITIES IN NORMED LINEAR SPACES S.S. DRAGOMIR Abstract. Upper ad lower bouds for the orm of a liear combiatio of vectors are give. Applicatios i obtaiig various iequalities for the quatities / y/ ad / y/, where ad y are ozero vectors, that are related to the Massera-Schäffer ad the Dukl-Williams iequalities are also provided. Some bouds for the uweighted Čebyšev fuctioal are give as well.. Itroductio I [9], L. Maligrada has obtaied the followig iterestig iequality for two ozero vectors, y i a real or comple ormed liear space (X, ) : (.) y y +. ma {, Notice that, this iequality provides a refiemet for the celebrated Massera-Schäffer iequality [0]: (.) y y ma {,, which, i its tur, is a refiemet of the Dukl-Williams iequality [7] (.3) y 4 y +. More recetly, i order to provide a lower boud for the quatity / y/, P.R. Mercer obtaied i [] the followig result as well: y (.4) mi {, y. I a effort to geeralise the above results for vectors, J. Pečarić ad R. Rajić have obtaied i [3] the followig double iequality: (.5) ma k {,..., k j j k j j mi k {,..., k j j k, Date: 8 May, Mathematics Subject Classificatio. Primary 6D5, 46B05. Key words ad phrases. Normed liear spaces, Triagle iequality, Massera-Schäffer iequality, Dukl-Williams iequality, Čebyšev fuctioal.

2 S.S. DRAGOMIR for ay j X\ {0, where j {,..., ad observed that, for =, = ad = y, (.5) reduces to the Maligrada & Mercer iequalities outlied above. They also remarked that, the followig refiemet of the geeralised triagle iequality obtaied by M. Kato et al. i [8] (.6) mi { k j k {,..., j j j ma { k j k {,..., j ca be deduced from (.5) as well. We should remark that (.6) ca be also obtaied as a particular case from the author s recet result established i [] (.7) ma { j j p p j j j p j p p j mi { j j p p j j j, where p ad. Notice that, i [], a more geeral result for cove fuctios has bee obtaied as well. Motivated by the above results, we establish i this paper some upper ad lower bouds for the more geeral quatity j where αj, j {,..., are scalars i K (K = C, R) ad j, j {,..., are vectors i the ormed liear space. For = j with j X\ {0, j {,..., we obtai a result which is similar to (.5). For the case of two vectors we recapture Maligrada s result (.) ad provide various iequalities for the dual epressio / y/ with, y X\ {0. Some bouds for the uweighted Čebyšev fuctioal are give as well. The followig result may be stated.. Iequalities for Vectors

3 INEQUALITIES IN NORMED LINEAR SPACES 3 Theorem. If k X ad α k K, k {,...,, the (.) ma k {,..., k k j j mi k {,..., k + k j. Proof. For ay k {,...,, we have (.) j = k + ( j k ). Takig the orm i (.) ad usig the triagle iequality we have successively: (.3) j k + ( j k ) k + k j for ay k {,...,. Takig the miimum over k i (.3) we deduce the secod iequality i (.). From (.) we also have j = k ( k j ). Takig i this equality the orm ad usig the cotiuity property of the orm, we have (.4) j k ( k j ) k ( k j ) k k j, for each k {,...,. Takig the maimum i (.4) we deduce the first part of (.). Remark. If there eists a r > 0 such that j k r k for each j, k {,...,, the we get from the secod iequality i (.) that (.5) j mi k k {,..., + r.

4 4 S.S. DRAGOMIR Moreover, if K, j {,..., are such that r (ad i this case r should be i (0, )) the the opposite iequality (.6) ma k k {,..., + r j also holds. Corollary. For ay ozero vectors k X, k {,...,, we have the iequalities: (.7) ma k {,..., k j k j j j j mi k {,..., k j + k j j ad (.8) ma k {,..., k j j k j j j mi k {,..., k j + j j k. 3. Iequalities for Two Vectors The case for two vectors is of iterest due to the fact that some similar iequalities obtaied i the past by several authors have bee applied i ivestigatig various problems i the Geometry of Baach spaces, icludig the characterizatio problem of strict coveity ad the characterizatio of ier product spaces i the larger class of ormed spaces. Lemma. For ay α, β K ad, y X we have (3.) [( + ) α + β ( α + β ) y ] + α + β ( ) + ( α β ) y α + βy [ α + β ( + ) + ( α + β ) y ] α + β ( ) ( α β ) y.

5 INEQUALITIES IN NORMED LINEAR SPACES 5 Proof. If we choose i Theorem, α = α, α = β, = ad = y, the we get (3.) ma { α + β β y, α + β α y α + βy mi { α + β + β y, α + β + α y. Utilisig the properties for real umbers ma {a, b = [a + b + a b ] ad ma {a, b = [a + b a b ], a, b R we have: ad ma { α + β β y, α + β α y = [( + ) α + β ( α + β ) y ] + α + β ( ) + ( α β ) y mi { α + β + β y, α + β + α y = [ α + β ( + ) + ( α + β ) y ] which, by (3.) produces the desired result (3.). α + β ( ) ( α β ) y, The followig particular cases are of iterest: Corollary. If α, β K with α = β =, the (3.3) α + βy ( + ) α + β y α + β, for ay, y X. Corollary 3. If, y X with = =, the (3.4) α + βy α + β ma { α, β y, for ay α, β K. Corollary 4. For ay two ozero vectors, y X we have: (3.5) y y +. ma {,

6 6 S.S. DRAGOMIR Proof. We choose i the secod iequality from (3.) α = ad β =, the we get (3.6) y { y y mi +, + { = [ y + ] mi, y + = ma {, ad the iequality (3.5) is proved. Remark. The iequality (3.5) has bee firstly obtaied by L. Maligrada i [9] o utilisig a differet approach. Corollary 5. For ay two ozero vectors, y X we have the reverse of the triagle iequality (3.7) (0 ) + + y y mi {,. Proof. If we write the first iequality i (3.) for y ad for α =, β =, the we get { ( + ) ma which is clearly equivalet to (3.7). + y ( + ), + y y, Remark 3. I [9], P.R. Mercer has obtaied the followig lower boud for the quatity y : y (3.8) mi {, y for ay, y X\ {0. I order to compare the lower bouds provided by (3.7) ad (3.8) cosider B, B : X R B (, y) := + + y ad Now, we observe that B (, y) := y. B (, y) B (, y) = y + + y [ + + ] = y + + y mi {, 0 for ay, y X. Therefore the Mercer result is better tha (3.7) i providig a lower boud for the quatity y.

7 INEQUALITIES IN NORMED LINEAR SPACES 7 I the followig we cosider the dual problem, amely the problem of fidig upper ad lower bouds for the quatity y where, y X\ {0. The first result that provides a lower boud is icorporated i Theorem. For ay two ozero vectors, y X we have: + (3.9) 0 mi {, + y ma {, y. Proof. Takig α = ad β = { ( + ) ma [ = ( + ) + = = + ( + ) ( ) ( + ) ( + ) + + y = ( + ) ma = i the left side of the iequality (3.) we have y ( + ), y ( + ) ( + ) + ( + + ) y ] y ( ) + y ( + ) y ( + ) {, y mi + mi {, y ma {,. The, by the first iequality i (3.) we get which clearly implies (3.9). {, + mi {, y ma {, + y, Theorem 3. For ay two ozero vectors, y X we have (3.0) y mi {, + y ma {,.

8 8 S.S. DRAGOMIR Proof. Takig α = ad β = i the right side of (3.) we have successively { mi + y, + y = [ ] ( + ) y ( + ) + ( ) y ( ) = ( + ) + y ( + ) [ y ] = ( + ) + + y ( + ) y = [ + + ] + y = ma = [ + ] {, + y mi mi {, + y ma {,, ad by the secod part of (3.) we get the desired result (3.0). 4. Bouds for the Čebyšev Fuctioal {, For β = (β,..., β ) K ad y = (y,..., y ) X, we cosider the uweighted Čebyšev fuctioal defied by C (β, y) := β j y j β j y j. We remark that this fuctioal has bee cosidered previously by the author ad some bouds have bee established. We recall here some simple results. With the above assumptios for X, α ad y, we have (4.) C (α, y) ( ) ma j {,..., ma j {,..., y j, [6]; ( ) y j, [3]; ( 6 p) /p ( y j q) /q, p >, p + q =, [],

9 INEQUALITIES IN NORMED LINEAR SPACES 9 where z j = z j+ z j is the forward differece. Here the costats, ad 6 are best possible i the sese that they caot be replaced by smaller quatities. I [5] we also have established that C (α, y) ( ma j {,..., det j j k= α k k= α k ) y j ; ( ( det j j k= α k k= α k ) q) /q ( y j p) /p for p >, p + q = ; ( det j j k= α k k= α k ) ma j {,..., y j. ad (4.) C (α, y) ma j {,..., k= α k j j k= α k j y j ; ( j k= α k j j k= α k for p >, p + q = ; q ) /q ( j y j p) /p j k= α k j j k= α k ma j {,..., y j. Fially, we recall the followig result from [4]: If there eists the comple umbers a, A C such that Re [(A ) ( a)] 0 for each j {,..., or, equivaletly, a + A A a for each j {,...,, the oe has the iequality: (4.3) C (β, y) A a y j The costat i the right had side of the iequality is best possible i the sese that it caot be replaced by a smaller costat. For may other results that hold for -tuples β ad y of real umbers we recommed the chapters devoted to Grüss ad Čebyšev iequalities from the books [] ad [4]. I the followig we provide other upper ad lower bouds for C (β, y) : y j.

10 0 S.S. DRAGOMIR Propositio. For ay β ad y as above, we have: (4.4) C (β, y) mi { k {,..., mi k {,..., mi k {,..., mi k {,..., Proof. We observe that β j β l y j y k β j l= ma { y j y k j {,..., [ { ] [ y j y k p y j y k C (β, y) = ( β j p β j { ma β j j {,..., ) β l y j. l= β l l= l= l= β l q ] q β l. Now, o applyig the secod iequality i Theorem for = β j l= β l ad j = y j, we deduce the first part of (4.4). The secod part is obvious by the Hölder iequality. The followig result ca be stated as well: Propositio. For ay β = (β,..., β ) K ad y = (y,..., y ) X we have the double iequality: (4.5) ma k {,..., β j γ y k y l l= βj γ yj y k mi k {,..., C (β, y) β j δ y k y l l= Proof. Follows from Theorem o otig that C (β, y) = for ay t K. ( βj t ) ( y j βj δ yj y k. ) y l l=

11 INEQUALITIES IN NORMED LINEAR SPACES Remark 4. As a particular case of iterest we ca state the followig result: ma k {,..., β j y k y l (4.6) βj yj y k (4.7) (4.8) C (β, y) mi k {,..., β j l= y k y l l= Refereces βj yj y k. [] S.S. DRAGOMIR, Bouds for the ormalised Jese fuctioal. Bull. Austral. Math. Soc. 74 (006), o. 3, [] S.S. DRAGOMIR, Aother Grüss type iequality for sequeces of vectors i ormed liear spaces ad applicatios. J. Comput. Aal. Appl. 4 (00), o., [3] S.S. DRAGOMIR, A Grüss type iequality for sequeces of vectors i ormed liear spaces ad applicatios. Tamsui Of. J. Math. Sci. 0 (004), o., [4] S.S. DRAGOMIR, Grüss type discrete iequalities i ormed liear spaces, revisited. Noliear Fuct. Aal. Appl. 9 (004), o. 4, [5] S.S. DRAGOMIR, Boudig the Cebysev fuctioal for sequeces of vectors i ormed liear spaces, Filomat, 8 (004), 5-6. [6] S.S. DRAGOMIR ad G. L. BOOTH, Grüss-Lupaş type iequality ad its applicatios for the estimatio of p-momets of guessig mappigs. Math. Commu. 5 (000), o., 7 6. [7] C.F. DUNKL ad K.S. WILLIAMS, Asimple orm iequality, Amer. Math. Mothly, 7(964), [8] M. KATO, K.-S. SAITO ad T. TAMURA, Sharp triagle iequality ad its reverse i Baach spaces, Math. Iequal. & Appl., 0(007), No., [9] L. MALIGRANDA, Simple orm iequalities, Amer. Math. Mothly, 3(006), [0] J.L. MASSERA ad J.J. SCHÄFFER, Liear differetial equatios ad fuctioal aalysis (I), A. of Math., 67(958), [] P.R. MERCER, The Dukl-Williams iequality i a ier product space, Math. Iequal. & Appl., 0(007), No., [] D. S.MITRINOVIĆ, J. PEČARIĆ ad A. M. FINK, Classical ad New Iequalities i Aalysis. Mathematics ad its Applicatios (East Europea Series), 6. Kluwer Academic Publishers Group, Dordrecht, 993. [3] J. PEČARIĆ ad R. RAJIĆ, The Dukl-Williams iequality with elemets i ormed liear spaces, Math. Iequal. & Appl., 0(007), No., [4] J. PEČARIĆ, F. PROSCHAN ad Y. L.TONG, Cove Fuctios, Partial Orderigs, ad Statistical Applicatios. Mathematics i Sciece ad Egieerig, 87. Academic Press, Ic., Bosto, MA, 99. School of Computer Sciece ad Mathematics, Victoria Uiversity, PO Bo 448, Melboure VIC 800, Australia. address: sever.dragomir@vu.edu.au URL: