KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE WITH TWO NEGATIVE PARAMETERS. Received April 13, 2010; revised August 18, 2010

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1 Scientiae Matheaticae Japonicae Online, e-200, KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE WITH TWO NEGATIVE PARAMETERS Young Ok Ki, Jun Ichi Fujii, Masatoshi Fujii + and Yuki Seo ++ Received Apil 3, 200; evised August 8, 200 Abstact In this pape, we shall show Kantoovich type ineualities fo the diffeence with two negative paaetes as follows: Let A and B be positive invetible opeatos on a Hilbet space H such that MI A I fo soe positive nubes M>>0 If the usual ode A B holds, then B + C, M, p, I B + C M,, p, I A p fo all p, <, whee C, M, p,, the Kantoovich constant fo the diffeence with two paaetes, is defined as ρ M p p ff + p M p if ρ M p p ff M As applications, we show soe chaacteizations of the chaotic ode Theeby, we obseve the diffeence between the usual ode and the chaotic one by vitue of Kantoovich constant fo the diffeence Intoduction Thoughout this pape, we conside bounded linea opeatos on a coplex Hilbet space H An opeato T is said to be positive denoted by T 0 if Tx,x 0 fo all x H The positivity defines the usual ode A B fo selfadjoint opeatos A and B Fo the sake of convenience, T > 0 eans T is positive and invetible The Löwne- Heinz ineuality assets that A B 0 ensues A α B α fo all 0 α Howeve A B 0 does not ensue A α B α fo α> in geneal As a copleentay esult to the Löwne-Heinz ineuality, Fuuta [4] fistly showed Kantoovich type ineualities fo the atio and aftewad Yaazaki [7] showed the following Kantoovich type ineuality fo the diffeence: A B, MI B I > 0 iply A p + C, M, pi B p fo all p, whee the Kantoovich constant fo the diffeence C, M, p is defined by M p p p p M p M p C, M, p =p + p fo all p R, see also [5] We note that li p 0 C, M, p = 0 and li p C, M, p = 0 Kantoovich type ineualities fo the diffeence povide a new view on the usual ode and the chaotic one The following theoe was obtained in [6] as a two positive paaetes vesion of 2000 Matheatics Subject Classification 47A63, 47A64 Key wods and phases Kantoovich type ineuality, Fuuta ineuality, positive opeato, Mond- Pečaić ethod

2 428 YO KIM, JI FUJII, M FUJII, Y SEO Theoe A Let A and B be positive opeatos on H such that MI B I fo soe positive nubes M>>0 Then the usual ode A B is euivalent to A + C, M, p, I B p fo all p, >, whee C, M, p,, the Kantoovich constant fo the diffeence with two paaetes is defined by M p M p M + M p p M 2 C, M, p, = p if if M p p M M p p M < M M p M if M< M p p M fo all p, R such that p > 0 In paticula, if we put = p in 2, then we see that C, M, p, p =C, M, p, M p p M because the condition M autoatically holds We also note that C, M, p, = p espm p M ifm p p espm p M In this note, as a continuation of [], we discuss Kantoovich type ineualities fo the diffeence with two negative paaetes: If A B>0 such that MI A I > 0, then B + C, M, p, I B + C M,, p, I A p fo all p, < As applications, we show soe chaacteizations of the chaotic ode by eans of Kantoovich type ineualities fo the diffeence Theeby, we obseve the diffeence between the usual ode and the chaotic one by vitue of the Kantoovich constant fo the diffeence 2 Kantoovich type ineualities fo diffeence At the beginning, we show two negative paaetes vesion of Theoe A Fo this, we claify the eaning of the Kantoovich constant fo the diffeence C, M, p, defined as 2 Lea 2 Fo given M>>0and p, R with p, > o p, < 0, { M p p C, M, p, = ax t + p M p } t : t [, M] Poof Put ht = M p p M t + p M M p M t Note that h t = 0 has the uniue solution and [0, ] iplies { M p p } t = > 0 { M h p p } 2 t = < 0

3 KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE 429 Thus, in the case of t M, we have the uppe bound ht on[, M], whee M p p p M p ht = + Next, if t <, then the uppe bound h on[, M] is given by h = + M p p + p M p = p Siilaly, if M<t, then the uppe bound hm on[, M] attains at t = M: hm =M p M By vitue of Lea 2, we have the following Kantoovich type ineuality fo the diffeence with two negative paaetes: Coollay 22 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 IfA B, then B + C, M, p, I A p fo all p, < Poof Fist of all, we note that t =Ax, x [, M] fo evey unit vecto x H by the assuption Fo p, <, it follows fo Lea 2 that and hence A p x, x Ax, x + C, M, p, fo evey unit vecto x H A p x, x Ax, x + C, M, p, Bx,x + C, M, p, by A B>0 B x, x+c, M, p, by the Hölde-McCathy ineuality fo evey unit vecto x H If we let = p in Coollay 22, then C, M, p, p M 2 M 0, so that B + C, M, p, I A p fo all p, < does not iply A B To discuss oe pecise estiation than Coollay 22, we pepae the following lea: Lea 23 Fo M>>0 2 fo all p, > such that MM p p M p C, M, p, M p p p M M

4 430 YO KIM, JI FUJII, M FUJII, Y SEO Poof Fist of all, we put h = M Then the condition M p p p M M is euivalent to h p /h h p Theefoe we have the following ineuality as a tool h p h p h h p h As a atte of fact, h h p has an euivalent expession 0 < hp h p h Moeove hp h hp iplies that h p h h p hp h hp h p h = h p hp h p h p h Now both sides of 2 ae witten as follows: MM p p M p = Mp M p + M p M p M p and by Lea 2 C, M, p, Mp M p M p p + So, aiing at the second tes, it suffices to show that M p M p M p M p p Actually the pepaed ineuality ensues that M p M p M p = h p = p p h p h h h p h hp h h p p h h p p p = h M p p =, p p h p h h as desied If we put p = in 2, then it follows that MM p p C, M, p, p > 0 fo all p> Hence we have li p C, M, p, p =C, M,, = 0

5 KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE 43 Now we show the following two Kantoovich type ineualities fo the diffeence with two negative paaetes, which chaacteize the usual ode A B: Theoe 24 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 Then the usual ode A B is euivalent to B + C M, 22, p, I A p fo all p, < Poof Since B A and I A M I, we have B + C M,,p, I A p fo p, > by Theoe A Put = >,p = p > fo p = Then it follows that B + C M,, p, I A p fo p, < Convesely, suppose 22 Since C M,, p, 0asp = by Lea 23, the ineuality 22 iplies B A, ie, A B In paticula, if we put = 2 in Theoe 24, then we have the following coollay which is utilized well late: Coollay 25 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 IfA B, then 23 B 2 + M 2 2 p M p 2 +4M p+ p+ 4 2 I A p fo all p< Theoe 26 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 Then the usual ode A B is euivalent to 24 B + p+ M p+ M p+ + I B + C M,, p, I Ap fo all p, < such that M M p M p p+ M Poof Since B A and I A M I, we have the following ineualities by Lea 23: p M p fo all p, > such that p M M M p M p p M C M,,p,

6 432 YO KIM, JI FUJII, M FUJII, Y SEO Taking = > and p = p > fo given p, <, it follows that p+ p+ M p+ M + = p+ p+ M p+ M + M = p p+ M + M p C Theefoe, we have 24 by Theoe 24 Convesely, suppose 24 If p =, then we have M,, p, M B p + p+ M p+ I A p fo all p< Since p+ M p+ 0asp, the ineuality above iplies B A, ie, A B Theoe 24 is bette than Coollay 22 by the following copaison: Theoe 27 If M>>0fo positive nubes M and, then C, M, p, C M,,p, fo p, > Poof Fo each s R, G s x is an affine function coesponding to x s on an inteval [a, b] We define G p x,g p x on[, M] and [ M, ], espectively Fo any x [, M], thee exist positive nubes α and β such that x = β + αm and α + β = Then G p β + α M can be defined Now the weighted aithetic-haonic ean ineuality says that β + α β + αm, M so that β + α M β + αm Since G p x is an inceasing function, we have G p β + α M G p Moeove it follows fo the affinity of G s x that β + αm G p x =G p β + αm =βg p +αg p M p p β = β + α = βg p + αg p = G p M M + α M Thus, G p x G p x

7 KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE 433 Hence we have the desied ineuality as follows: C M, {,p, = ax G p : x [, M]} by Lea 2 x x = G p fo soe x 0 [, M] x 0 x 0 G p x 0 x 0 ax{g p x x : x [, M]} = C, M, p, Reak By Theoe 27, it follows that A B>0 with MI A I > 0 iplies B + C, M, p, I B + C M,, p, I Ap fo p, < As a genealization of Theoe 24, we shall show the following theoe: Theoe 28 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 Then the usual ode A B is euivalent to B + M C M +, +, p +, 25 I A p fo all >0 and p, < + Poof By the Fuuta ineuality [3], [5, Chapte 7], it follows that A B ensues that A + A 2 B A fo all >, >0 Fo p, > and >0, put A = A +, B = A 2 B A + 2 +, p = p+ + <, and = + + < in Theoe 24 Since A B and M + I A + I, then we have B + C M +, +, + p +, + + I A p and hence A 2 B A 2 + C M +, +, + p +, + I A p + Multiply A 2 on both sides and eplace p and by p and espectively, we have 25 Convesely, suppose 25 If we put 0 and p = in 25, then we have B A Coollay 29 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 IfA B, then B + 4+ M + + M + M p+ p+ + +p+2 M +p+2 2 4M +2 + M + 2 I A p fo all p<, < 2 Poof If we put = 2 in 25 of Theoe 28, then we have the desied ineuality

8 434 YO KIM, JI FUJII, M FUJII, Y SEO 3 Chaacteizations of chaotic ode In this section, as an application of Kantoovich type ineualities fo the diffeence with two negative paaetes, we give chaacteizations of the chaotic ode Fo positive invetible opeatos A and B, the ode defined by log A log B is called the chaotic ode We shall show a chaotic ode vesion of Theoe 24 Theoe 3 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 Iflog A log B, then B + M C M, 3, p, I A p fo all p, < 0 Poof Put p, > 0 By the chaotic Fuuta ineuality [2], [5, Chapte 7], it follows that log A log B ensues that A A /2 B A /2 /+ Since +, p + <, we apply Theoe 24 to obtain and hence A 2 B A 2 + C M,,p+,+I A p B + C M,,p+,+A A p Replacing p and by p and espectively, we have B + M C M,, p, I A p fo all p, < 0 Though 22 in Theoe 24 chaacteizes the usual ode, it follows that 3 in Theoe 3 does not chaacteize the usual ode, because C/M, /, p/p 0as p 0 By using 23 in Coollay 25, we show the following chaacteization of the chaotic ode Theoe 32 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 Then the chaotic ode log A log B is euivalent to B + p+ M p+ 2 +4M M M p p 4M M 2 I A p fo all p, < 0 Poof By the chaotic Fuuta ineuality [2], [5, Chapte 7], it follows that log A log B ensues that A A /2 B A /2 /2 fo all >0 Put A = A, B =A /2 B A /2 /2, p = +p <, = 2, M = M and = in Coollay 25 Then we have A 2 B A 2 + M 2 2 p M p 2 +4M M p p 4M 2 I A p and hence B + M M 2 2 p M p 2 +4M M p p 4M 2 I A p

9 KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE 435 fo all p, > 0 Replacing p and by p and, we have B + M M 2 2 p+ M p+ 2 +4M M p p 4M 2 I A p fo all p, < 0 and hence we have the desied ineuality Convesely, if we put p =, then we have fo all p<0 and hence This iplies that B p + 2p M 2p 2 +4M p p p M p M p p 4M p p M p 2 I A p B p + p M p 2 4M p I A p fo all p<0 B p I p + p M p 2 4pM p I Ap I p fo all p<0 and as p 0 we have log B log A As a genealization of Theoe 3, we shall show the following theoe Copaing Theoe 28 with Theoe 33 in the below, we obseve that vaiables p, and in the Kantoovich constant change to p, and espectively This is explained by the t fact that li p p 0 p = log t fo t>0, as used in above, that is, the exponent of log t can be egaded as 0 Theoe 33 Let A and B be positive invetible opeatos on H such that MI A I fo soe positive nubes M>>0 Then the chaotic ode log A log B is euivalent to 32 B + M C M,, p, I A p fo all >0 and p, < 0 Poof By the chaotic Fuuta ineuality [2], [5, Chapte 7], it follows that log A log B ensues that A A 2 B A 2 + fo all, 0 Put A = A,B =A 2 B A 2 +, p = +p <, = + <,M = M and = in Theoe 24 Then we have A 2 B A 2 + C M,, + p, + I A p fo all p, > 0, so that B + C M, Since M I A I, we have B + M C M, + p, p, + A A p fo all p, > 0, I A p fo all p, < 0

10 436 YO KIM, JI FUJII, M FUJII, Y SEO Put p = = >0 in the assuing ineuality Then we have B + M C M,, 2 I A fo all >0 and M C M,, 2 = M M by the definition of C M,, 2 Since B I + M M I A I fo all >0 and M 2 li M M = li 0 we have log B log A, ie, log A log B M 4 2 M =0, Coollay 34 Let A and B be positive invetible opeatos such that MI A I I>0fo soe positive nubes M>>0 Iflog A log B, then B + M p M p M p M I B + M C M,, p, I A p fo all p, < 0 such that M M M p M p p Poof Put = M, M =, p = p >, and = > fo p, < 0in Lea 23 Then we have { p p } p M M C M,, p, M Moeove, since the left hand side of the above coincides with it follows that p M p p M M M p M p M p M C M, = p M p M p M,, p Theefoe we have B + M p M p M p M I B + M C M, by Theoe 33,, p, I A p Reak 2 Theoe 33 will give an altenative poof to the fist half of a poof of Theoe 32: As a atte of fact, if we put = in 32 of Theoe 33, then we have B + M CM,, p +, 2I A p This is just the euied ineuality

11 KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE Copleentay ineualities In this final section, we conside copleentay ineualities to esults with espect to the usual ode in 2 Fo this, we pepae soe notations Fo 0 <<M and p, R with p > 0,p, β, M, p, = ax{ p,m p M } and β 2, M, p, = p p p p p p if p p if p <, p M, M p M if M< p p We hee ecall the following esult poved in [6]: Lea 4 Let A and B be positive opeatos on H such that M I A I>0 and M 2 I B 2 I>0 fo soe scalas M > > 0 and M 2 > 2 > 0 If A B, then the following ineualities hold: 0 <<p< A + β,m,p,i B p, 2 p>, 0 << A + β 2,M 2,p,I B p, 3 0 <p<, p< A + β 2,M,p,I B p Based on Lea 4, we give copleentay ineualities: Theoe 42 Let A and B be as in above Then the following ineualities hold: <p<<0 B + β 2,M 2,p,I A p, <<p<0 B p + β 2,M 2,,pI A, 2 <<0,p< B + β,m,p,i A p, 2 <p<0, < B p + β,m,,pi A, 3 <p<0, <p B + β 2 2,M 2,p,I A p, 3 <<0,p< B p + β 2 2,M 2,,pI A Poof Since B A > 0 and 2 I B M 2 I, it follows fo of Lea 4 that B + β,,p, I A p fo 0 < <p < M 2 2 Putting = and p = p, we have B + β,, p, I A p fo <p<<0 M 2 2 Since β M 2, { p p }, p, = ax, 2 M 2 M = ax{m p 2 M 2, p 2 2 } = β 2,M 2,p,,

12 438 YO KIM, JI FUJII, M FUJII, Y SEO we have the desied ineuality as follows: B + β 2,M 2,p,I A p fo <p<<0 2 Since B A > 0 and I A M I, it follows fo 2 of Lea 4 that B + β,,p, I A p fo p >, 0 < < M Putting =, p = p, then B + β,, p, I A p M fo p<, <<0 Hence we have B + β,m,p,i A p fo p<, <<0 3 Since B A > 0 and 2 I B M 2 I, it follows fo 3 of Lea 4 that B + β 2,,p, I A p fo 0 <p <, p < M 2 2 Putting p = p, =, then B + β 2,, p, I A p fo <p<0, <p M 2 2 Since β 2 M 2, 2, p, = β 2 2,M 2,p, by the definition of β 2, we have B + β 2 2,M 2,p,I A p fo <p<0, <p Poofs of, 2 and 3 ae given by eplacing p and by and p in, 2 and 3, espectively Acknowledgeent his couteous eview The authos would like to expess thei thanks to the efeee fo Refeences [] JI Fujii and YO Ki, A Kantoovich type ineuality with a negative paaete, Sci Math Japon, , [2] M Fujii, T Fuuta and E Kaei, Fuuta s ineuality and it s application to Ando s theoe, Linea Alg Appl, , 6-69 [3] T Fuuta, A B 0 assues B A p B / B p+2/ fo 0, p 0, with + 2 p +2, Poc Ae Math Soc, 0 987, [4] T Fuuta, Opeato ineualities associated with Hölde-McCathy and Kantoovich ineualities, J Ineual Appl, 2 998, 37 48

13 KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE 439 [5] T Fuuta, J Mićić, J Pe caić and Y Seo, Mond-Pe caić Method in Opeato Ineualities, Monogaphs in Ineualities,, Eleent, Zageb, 2005 [6] Y O Ki, A diffeence vesion of Fuuta-Giga theoe on Kantoovich type ineuality and its application, Sci Math Japon, , [7] T Yaazaki, An extension of Specht s theoe via Kantoovich ineuality and elated esults, Math Ineual Appl, , Depatent of Matheatics, Suwon Univesity, Bongdaoup, Whasungsi, Kyungkido , Koea E-ail addess : yoki@skkuedu Depatent of At and SciencesInfoation Science, Osaka Kyoiku Univesity, Asahigaoka, Kashiwaa, Osaka , Japan E-ail addess : fujii@ccosaka-kyoikuacjp + Depatent of Matheatics, Osaka Kyoiku Univesity, Asahigaoka, Kashiwaa, Osaka , Japan E-ail addess : fujii@ccosaka-kyoikuacjp ++ Faculty of Engineeing, Shibaua Institute of Technology, 307 Fukasaku, Minuaku, Saitaa-city, Saitaa , Japan E-ail addess : yukis@sicshibaua-itacjp

Banach Journal of Mathematical Analysis ISSN: (electronic)

Banach Journal of Mathematical Analysis ISSN: (electronic) Banach J. Math. Anal. 4 200), no., 87 9 Banach Jounal of Mathematical Analysis ISSN: 75-8787 electonic) www.emis.de/jounals/bjma/ ON A REVERSE OF ANDO HIAI INEQUALITY YUKI SEO This pape is dedicated to