A GRÜSS-TYPE INEQUALITY AND ITS APPLICATIONS

Transcription

1 A GRÜSS-TYPE INEQUALITY AND ITS APPLICATIONS SAICHI IZUMINO, JOSIP E. PEČARIĆ, AND BOŽIDARTEPEŠ Received 7 August 2003 We give a Grüss-type iequality which is a refiemet of a result due to Dragomir ad Agarwal. We also give its applicatios for the momets of radom variables, guessig mappigs, ad Ozeki s iequality.. Itroductio Let a = a,...,a ), b = b,...,b )be-tuples sequeces) of real umbers ad let p = p,..., p )bea-tuple of positive umbers. The discrete) Grüss iequality is a estimatio of the differece Ia,b; p):= P p i a i b i p i a i p i b i,.) where P = p i. If both a ad b are assumed to be odecreasig or oicreasig), that is, a a,b b or a a,b b ),.2) thethe abovedifferece Ia,b; p) isoegative, thatis, P p i a i b i p i a i p i b i..3) This is well kow as Čebyšev s iequality [0, page 240]. As a complemet of this iequality, oe of the authors proved the followig theorem. Theorem. [, Theorem8],[0, page 302]. Let a, b be odecreasig or oicreasig) -tuples of real umbers ad let p be a -tuple of positive umbers. The where P j = j k= p k. Ia,b; p) a a b b P ) j P P j,.4) j Copyright 2005 Hidawi Publishig Corporatio Joural of Iequalities ad Applicatios 2005:3 2005) DOI: 0.55/JIA

2 278 A Grüss-type iequality ad its applicatios Without ay assumptio of mootoicity o -tuples a ad b, the followig extesio of Theorem. was give by D. Adrica ad C. Badea. Theorem.2 [,Theorem2]. Let a, b be -tuples of real umbers satisfyig m a i M, m 2 b i M 2 i =,...,),.5) ad let p be a -tuple of positive umbers. The Ia,b; p) M m ) M2 m 2 ) J I PJ) P PJ) ),.6) where I ={,...,} ad PJ) = i J p i for J I cf.p = PI )). There are a umber of further results cocerig discrete or itegral type) Grüss iequalities with or without mootoicity coditios o -tuples a ad b [2, 5, 7, 8, 9, 2], ad so forth. See [0,ChapterX].) Recetly, Dragomir ad Agarwal [3] or Dragomir ad Diamod [4]) preseted a geeral Grüss-type iequalityfor complex-tuples i terms of the first differeces of them. Theorem.3 [3, Theorems 22 ad 23], [4, Theorems 4 ad 5]. Let a, b be -tuples of complex umbers ad let p be a -tuple of positive umbers such that p i =, that is, p is a probability distributio. The Ia,b; p) /s Ia,b; p) ) a k+ a k b i i p i p j i j, i, ) /s a k+ a k b i s a k+ a k b i p i p j i j, i,j a k+ a k i p i b i p q i p q j i j q s, q>, s + ), q = i, p i i j, i, ) /s a k+ a k p i b i s p i p j i j q s, q>, s + ), q = i, a k+ a k p i b i..7)

3 Saichi Izumio et al. 279 I [3], the followig iequality was basically used as a key fact to obtai Theorem.3: Ia,b; p) where a i = a i+ a i i =,..., ). I this paper, with the otatios P j = j k= a k p i p j i j b i,.8) i, p k, P j+ = P j j =,..., ).9) for a probability distributio p, further, coveietly puttig P 0 = P + = 0ad a 0 = a = 0), we prove a fudametal idetity Ia,b; p) = i which brigs a strogeriequality tha.8), cf. Remark 2.4) Ia,b; p) i P j a j P j+ a j )p i b i,.0) P j a j + P j+ a j )p i b i..) Usig the above idetity ad iequality, we give our mai results Theorems 2.3 ad 2.6) as refiemets of Theorem.3. We also give some applicatios for the momets of discrete radom variables, for guessig mappigs cosidered i [3, 4], ad for Ozeki s iequality related to [6, 7, 8]. 2. Grüss-type iequalities We prepare two useful facts before we give the mai results. Recall that for -tuples a, b ad a probability distributio p,we writecf..)) For this differece we have the followig lemma. Lemma 2.. Proof. First, by defiitio, Ia,b; p) = p i a i b i p i a i p i b i. 2.) Ia,b; p) = i P j a j P j+ a j )p i b i. 2.2) ) ) Ia,b; p) = p i a i b i p j a j p i b i = p j ai a j p i b i. 2.3)

4 280 A Grüss-type iequality ad its applicatios Next, ote the followig fact which is a extesio of Abel s idetity for a sequece c,...,c with c j = c j+ c j for j =,..., ad coveietly puttig c 0 = Γc = 0): i p j c j = c i P j c j + Puttig c j = a i a j i the above idetity, we have P j+ c j i =,...,). 2.4) ) ) i p j ai a j = ai a i P j ) a i a j + P j+ ) a i a j i = P j a j P j+ a j. Hece from 2.3)ad2.5), we obtai the desired idetity 2.2). Lemma 2.2 [0, page 4]. If a ad p are positive -tuples, the the fuctio ) /r p i a r i if r 0, P Mr) = a pi i mi { } a,...,a { } a,...,a ) /P if r = 0, if r =, if r = 2.5) 2.6) is mootoe icreasig o [, ]. Now we give the followig result as a refiemet of Theorem.3. Theorem 2.3. Let a ad b be -tuples of complex umbers ad let p be a probability distributio. The Ia,b; p ) i P j a j + P j+ a j )p i b i, ) /q i i j p j P j a j q + a k i j p j )p i b i, P j+ a j q p i b i q>), 2.7) Moreover, if R, R 2,adR 3 deote the right-had sides of 2.7), successively from above to below, the R R 2 R )

5 Saichi Izumio et al. 28 Proof. By Lemma 2., we immediately obtai Ia,b; p) i P j a j + P j+ a j )p i b i, 2.9) which is the first iequality i 2.7). To obtai the other iequalities, put i Q i = P j + P j+ i =,...,). 2.0) The from Lemma 2.2,wehave,forq>, i ) P j a j + P j+ a j Q i { Q i i j P j a j q + a j, )} /q P j+ a j q 2.) so that i P j a j + P j+ a j i Q /q i P j a j q + a j Q i. j P j+ a j q 2.2) For Q i, we have, by a elemetary computatio, Q i = i j p j. 2.3) Hece from 2.9), 2.2), ad 2.3), we obtai 2.8)adatthesametimethesecodad the third iequalities i 2.7). Remark 2.4. The iequality i P j a j + P j+ a j j a j i j p j 2.4) immediately obtaied from 2.2) ad2.3)) says that our basic iequality 2.9) is stroger tha.8) which was used i [3,Theorem C].

6 282 A Grüss-type iequality ad its applicatios Remark 2.5. If we defie by Fq) the right-had side of the secod iequality i 2.7), the we see, by Lemma 2.2,thatFq) is a icreasig fuctio ad lim q Fq) Fq) lim q Fq), which implies 2.8). For positive -tuples a ad b with a assumptio of mootoicity, we have the followig theorem. Theorem 2.6. Let a ad b be -tuples of positive umbers ad let p be a probability distributio. Assume thata is odecreasig. The Ia,b; p) i P j a j )p i b i, i ) /q i i j)p j a k i i j)p j )p i b i, P j a j ) q p i b i q>), 2.5) where coveietly p 0 = 0)ad P j+ a j )p i b i, ) /q Ia,b; p) j i)p j a k j i)p j )p i b i. P j+ aj ) q p i b i q>), 2.6) Moreover, as i Theorem 2.3,ifR, R 2,adR 3 deote resp., R, R 2,adR 3 ) the right-had sides of 2.5)resp.,2.6)), successively from above to below, the R R 2 R 3 resp., R R 2 R 3 ). 2.7) Proof. Sice a j 0, we have i i P j+ a j P j a j P j+ a j P j a j, 2.8) so that from Lemma 2. we obtai the first iequalities i 2.5) ad2.6). To see the remaiig iequalities, ote that the followig idetities i i P j = i j)p j, P j+ = j i)p j 2.9) hold. From these ad the first iequalities i 2.5) ad2.6), we ca obtai all other desired iequalities by the similar argumet as i the proof of Theorem 2.3.

7 Saichi Izumio et al Applicatios for the momets of radom variables, guessig mappigs, ad Ozeki s iequality Cosider a discrete radom variable ) x,...,x X :, 3.) p,..., p which takes positive values x,...,x with a probability distributio p = p,..., p ). The the γ-momet γ>0) of X is defied as the expectatio EX γ ), that is, M γ X) = EX γ ) = p i x γ i. 3.2) A approximatio result which compares M α+β X)α,β>0) with the product of M α X) ad M β X) was show i [3, 4]. We here give a improvemet of the result by applyig Theorem 2.3. Propositio 3.. Uder the above assumptios, M α+β X) M α X)M β X) i ) P j x α j + P j+ x α j p i x β i, ) /q i i j p j P j x a q j + xk α i j p j )p i x β i. Proof. Writig x γ = x γ,...,x γ )forγ>0, we see that P j+ x α q j p i x β i q>), 3.3) M α+β X) M α X)M β X) = I x α,x β ; p ). 3.4) Hece we obtai the desired iequalities i 3.3)fromTheorem 2.3. Now i coectiowith the radom variable X, cosider a uiformly distributed radom variable x,...,x U :,..., 3.5) ad its α-momet M α U) = /) x α i. The we have the followig propositio as a improvemet of a result i [3, 4].

8 284 A Grüss-type iequality ad its applicatios Propositio 3.2. Uder the above assumptios, M α X) M α U) i j p j + j) p j )xi α, ) /q i S i + S i j p j q + j) p j q ) p k Si + S i x α i, M α X) M α U) i j x α j + j) ) x α j p i, ) /q i S i + S i j x α q j + j) x α q j p i x α ) k Si + S i pi, x α i q>), 3.6) q>), 3.7) where S k = kk +)/2. Proof. Let p = /,...,/). The we see that M α X) M α U) = I p,x α ; p ), M α X) M α U) = I x α, p; p ). 3.8) Hece correspodig to the first or the secod idetity i the above 3.8), the first iequality i 3.6)or3.7)is obtaied immediately by Theorem 2.3. For the other iequalities i 3.6)ad3.7), otice that Q i := i j + j) = S i + S i. 3.9) Hece agai from Theorem 2.3, we ca obtai all desired iequalities. A discrete guessig mappig G is defied [3, 4] as a special radom variable Sice g γ = γ,..., γ )γ>0) is odecreasig ad ),..., G :. 3.0) p,..., p M α+β G) M α G)M β G) = I g α,g β ; p ) α,β>0), 3.)

9 Saichi Izumio et al. 285 we ca obtai the followig propositio as a refiemet of a result i [3, 4] by usig Theorem 2.6. Propositio 3.3. Uder the above assumptios, M α+β G) M α G)M β G) i P j j )p α i i β, i ) /q i i j)p j kα i P j j α ) q p i i β ) i j)p j p i i β = δ α ) M α+β G) M α G)M β G) P j+ j )p α i i β, where kα ) /q j i)p j ) j i)p j i q>), ) i j)p j p i i ), β P j+ j α ) q p i i β p i i = δ β α ) q>), ) j i)p j p i i ), β 3.2) δ α ) = k +) α k α α ) α if α, = 2 α if 0 <α<. 3.3) Let,..., V :,..., 3.4) be the size- guessig mappig with the uiform probability distributio. The, similarly as 3.8), we see that M α G) M α V) = I p,g α ; p ) = I g α, p; p )), M α V) = i ). α 3.5) Hece we ca obtai the followig result by the similar argumet as i Propositio 3.3.

10 286 A Grüss-type iequality ad its applicatios Propositio 3.4. Uder the above assumptios, M α G) i α = M a G) M a V) ) i j j )p a i, ii ) 2 δ α) M α G) i α ) /q i ii ) p i = 2 δ α) j j α) q p i q>), 2 EG2 ) )) 2 EG), j) j )p a i, ) i) i +) /q j) j α) q p i q>), 2 δ i) i+) α) p i = 2 δ α) 2 E G 2) 2 + EG)+ )) ) 3.6) Ozeki s iequality [6, 7, 8] cf.[0, page 2]) is a complemet of Cauchy-Schwarz iequality, which estimates the differece ) 2 I 2 a,b; p):= p i a 2 i p i bi 2 p i a i b i 0) 3.7) for positive -tuples a ad b with a probability distributio p.notethat I a 2,b 2 ; p ) = p i a 2 i bi 2 p i a 2 i p i bi 2, Iab,ab; p) = ) 2 p i a 2 i bi 2 p i a i b i, 3.8) where ab = a b,...,a b ). Hece, we see that I 2 a,b; p) = I a 2,b 2 ; p ) + Iab,ab; p). 3.9) From this fact, we obtai a ew estimatio of the differece I 2 a,b; p)itermsofthefirst differeces of a 2 ad ab.

11 Saichi Izumio et al. 287 Propositio 3.5. Uder the above assumptios, I 2 a,b; p) i ) P j a 2 j + P j+ a 2 j p i bi 2 + ) /q i i j p j P j a 2 q j + ) /q i + i j p j P j ab) j q + i P j ab) j + P j+ ab) j )p i a i b i, P j+ a 2 q j p i bi 2 P j+ ab) j q p i a i b i q>), a 2 k i j p j )p i a 2 i + ab) k i j p j )p i a i b i. 3.20) Proof. By 3.9)weseethat I 2 a,b; p) I a 2,b 2 ; p ) + Iab,ab; p). 3.2) Hece, we ca obtai the desired iequalities i 3.20)fromTheorem 2.3. For -tuples a, b with a assumptio of mootoicity, we have the followig propositio. Propositio 3.6. Let a, b be positive -tuples ad let p be a probability distribitio. Assume that The, 0 <)a a, 0<)b b. 3.22) I 2 a,b; p) i P j ab) j )p i a i b i, i ) /q i i j)p j ab) k i P j ab)j ) q p i a i b i i j)p j )p i a i b i. q>), 3.23) Proof. Sice all the -tuples a 2, b 2, ab are odecreasig, we see, by Čebyšev s iequality, that I a 2,b 2 ; p ) 0, Iab,ab; p) )

12 288 A Grüss-type iequality ad its applicatios Hece from 3.9) I 2 a,b; p) Iab,ab; p), 3.25) so that from Theorem 2.6, we obtai the desired iequalities i 3.23). Refereces [] D. Adrica ad C.Badea, Grüss iequality for positive liear fuctioals, Period. Math. Hugar ), o. 2, [2] M. Bieracki, H. Pidek, ad C. Ryll-Nardzewski, Sur ue iégalité etre des itégrales défiies, A. Uiv. Mariae Curie-Skłodowska Sect. A ), 4 Frech). [3] S. S. Dragomir ad R. P. Agarwal, Some iequalities ad their applicatio for estimatig the momets of guessig mappigs, Math. Comput. Modellig ), o. 3-4, [4] S. S. Dragomir ad N. T. Diamod, A discrete Grüss type iequality ad applicatios for the momets of radom variables ad guessig mappigs, Stochastic Aalysis ad Applicatios. Vol. 3, Nova Sciece Publishers, New York, 2003, pp [5] G. Grüss, Über des Maximum des absolute Betrages vo / b a)) b a f x)gx)dx / b a) 2 ) b a f x)dx b a gx)dx, Math. Z ), Germa). [6] S. Izumio, H. Mori, ad Y. Seo, O Ozeki s iequality, J.Iequal.Appl.2 998), o. 3, [7] S. IzumioadJ. E. Pečarić, A weighted versio of Ozeki s iequality, Sci. Math. Jp ), o. 3, [8], Some extesios of Grüss iequality ad its applicatios, Nihokai Math. J ), o. 2, [9] A. Lupaş, O a iequality, Uiv. Beograd. Publ. Elektroteh. Fak. Ser. Mat. Fiz. 98), o , [0] D. S. Mitriović, J. E. Pečarić, ada. M. Fik, Classical ad New Iequalities i Aalysis, Mathematics ad Its Applicatios East Europea Series), vol. 6, Kluwer Academic Publishers, Dordrecht, 993. [] J. E. Pečarić, O a iequality of T. Popoviciu. I, II,Bul.Şti. Teh. Ist. Politeh. Traia Vuia Timişoara 2438) 979), o. 2, 9 5. [2], O some iequalities aalogous to Grüss iequality, Mat. Vesik47)32) 980), o. 2, Saichi Izumio: Departmet of Mathematics, Faculty of Educatio, Toyama Uiversity, 390 Gofuku, Toyama , Japa address: izumio@edu.toyama-u.ac.jp Josip E. Pečarić: Faculty of Textile Techology, Uiversity of Zagreb, Pierottijeva 6,0 000 Zagreb, Croatia address: pecaric@mahazu.hazu.hr Božidar Tepeš: Faculty of Philosophy, Uiversity of Zagreb, I. Lučića 3, 0000 Zagreb, Croatia address: btepes@ffzg.hr