Matrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality

Transcription

1 Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5] of he sandard arihmeicgeomeric-harmonic mean inequaliy for scalars.. Inroducion The classical arihmeic-geomeric mean inequaliy assers ha if w,..., w k are posiive numbers summing o and x,..., x k are posiive numbers, hen he arihmeic mean A w = A w x,..., x k := w x + + w k x k is a leas as grea as he geomeric mean G w = G w x,..., x k := x w x w k k. As his is a consequence of he relaively simple propery of convexiy of he logarihm funcion, i is naural o expec more complex and precise relaionships o exis beween A w and G w. Indeed, many auhors for example, Alzer ], Mercer 5] have refined he inequaliy A w G w 0. A paricularly ineresing resul, due o Carwrigh and Field 4], gives boh upper and lower bounds for A w G w in erms of he variance associaed wih he arihmeic mean. Theorem Carwrigh-Field Le w i i k be posiive numbers summing o. If x i i k are posiive numbers in he inerval a, b], where a > 0, hen b w j x j A w A w G w a w j x j A w. Carwrigh and Field noed ha heir inequaliy is sharp, in he sense ha here may be equaliy on boh sides. Key words and phrases: Arihmeic mean, geomeric mean, harmonic mean, posiive marix. 000 Mahemaical Subjec Classificaion: 5A7, 5A45, 5A48, 5A60, 47A63.

2 Recenly, Mercer 5] discovered an exensive collecion of inequaliies wih he same general flavor as hose of Carwrigh and Field. For now, our emphasis is on exending he Carwrigh-Field inequaliy o posiive marices. I is reasonable o hope ha such an exension exiss, since Ando ] and Sagae and Tanabe 7] succeeded in esablishing an arihmeic-geomeric mean inequaliy for posiive marices. We recall ha he n n marix M is posiive if he inner produc < Mx, x > is posiive for all non-zero complex n-vecors x.. The geomeric mean of posiive marices One problem inheren in working wih he geomeric mean of posiive n n marices a leas in he case of non-commuing marices is finding an appropriae definiion. A major sumbling block is ha if M, M are posiive n n marices, heir produc M M need no be posiive, and fracional powers canno be defined adequaely. Ando s definiion ] bu see also 6] of he geomeric mean of wo equally weighed, i.e. w = w = / posiive marices M, M was GM, M := M M M M M, where all square roos are posiive square roos. By design, GM, M > 0. Remarkably, in spie of he apparen asymmery of he definiion, GM, M = GM, M. This commuaiviy propery is a consequence of an imporan exremal propery, 6]: GM, M is he leas posiive n n marix M wih he propery ha M M 0. M M I is ineresing, however, o give a simple direc proof, ha does no seen o have been noed before. Observe ha GM, M = GM, M is equivalen o M M M M M M M = M M M. As posiive marices are equal if and only if heir squares are equal, his is in urn equivalen o M M M M M ] M M M M M M M M = M M M., Since he erm in square brackes is jus M M M he lef hand side of he expression above does indeed reduce o he righ hand side.

3 Building on Ando s success in showing ha GM, M AM, M := M + M, Sagae and Tanabe 7] inroduced more general geomeric means of an arbirary number of posiive n n marices. For posiive numbers w,..., w k summing o and posiive n n marices M,..., M k, hey defined he geomeric mean G w = G w M,..., M k o be M k M k M k M 3 M M M M u M M 3 u M k M k u k M k, where u i = w i+ / i+ w j for i =,..., k. All powers are o be inerpreed as posiive powers, so G w is easily seen o be posiive. If n =, hen G w M, M = M M M M w M, which is consisen wih Ando s definiion of he geomeric mean of wo marices in he case w = w = /. Sagae and Tanabe showed ha wih he naural definiion of he weighed marix arihmeic mean as A w M,..., M k := w M + + w k M k, he marix analog of he arihmeic-geomeric mean inequaliy is rue. In oher words, G w M,..., M k A w M,..., M k. Neverheless, heir marix geomeric mean has a drawback ha is poenially problemaic in he search for more refined inequaliies: he marix geomeric mean of more han wo marices is order dependen. Since his does no appear o have been observed before, we give a simple example generaed by Mahemaica. We work wih he case k = 3 and equal weighs w = w = w 3 = /3. Two possible geomeric means of posiive marices M, M and M 3 are and / /3 GM 3, M, M = M / M / M / M / M 3 M / / M M / M / / /3 GM, M, M 3 = M / 3 M / 3 M / M / M M / / M M / 3 M / 3. Bu, if M := 3 0, M := 0 3 and M 3 :=,

4 hen, afer compuing wih Mahemaica, we find GM 3, M, M = , GM, M, M 3 = In oher words, GM, M, M 3 GM 3, M, M. 3. Marix versions of he carwrigh-field inequaliy Our firs objecive is o esablish a version of he Carwrigh-Field inequaliy for wo posiive marices. The proof will depend on a resul equivalen o special case of he original inequaliy wih x =, x =. We give a proof ha is differen from hose published previously. Lemma Le w > 0, w > 0 saisfy w + w =. If 0, ], hen w + w w + w w w + w w. Equaliy holds if and only if =. Proof We firs prove he righ hand inequaliy. Consider Since f = w w w + w w. f = w w w + w w = w w w 0 for all 0, ], he firs derivaive f is decreasing on 0, ], and so f f = 0 for all 0, ]. This implies ha f is increasing on 0, ], and so f f = 0 for all 0, ]. We have hus esablished he righ hand inequaliy. I is obvious ha equaliy holds if and only if =. hen The lef hand inequaliy is proved similarly. Consider and hence g = w + w w + w w, g = w + w w w + w w, g = w w + w w + w w, g = w + w w w < 0 for all 0, ], he second derivaive g is decreasing on 0, ], and so g g = 0 for all 0, ]. This implies ha g is increasing on 0, ], and so g g = 0 for 4

5 all 0, ]. Hence, g is decreasing on 0, ], and g g = 0 for all 0, ]. We have herefore proved he lef hand inequaliy. Again i is obvious ha equaliy holds if and only if =. Wih he aid of his lemma, we can refine a mehod of Ando ] o prove a marix version of he Carwrigh-Field inequaliy for wo posiive marices. Theorem Le w > 0, w > 0 saisfy w + w =. Le M, M be posiive n n marices wih M M. Wrie A w = A w M, M and G w = G w M, M. Then w j M j A w M M j A w A w G w w j M j A w M M j A w. Equaliy holds if and only if M = M. Proof We sar wih he lef hand inequaliy. The fac ha M A w and M A w are proporional will allow us o work wih a single variable N := M M M. Observe ha w j M j A w M M j A w = w w M w M M w M w M + w w M w M M w M w M = w w w + w M M M M M = w w M I N M, where I is he n n ideniy marix. Noe ha A w = M w N + w IM and Thus, o prove ha G w = M N w M. w j M j A w M M j A w A w G w, i is enough o esablish w w M I N M M w N + w IM M N w M. 5

6 This is rue if and only if w w I N w N + w I N w. Noe ha, since M M, N is a posiive marix wih N I. Now N = U DU, where D is a diagonal marix d,, d n ] and U is a uniary marix. Since 0 < N I, i follows ha 0 < D I and so 0 < d i i n. By Lemma, we have w w d i w d i + w d w i i n. So and his implies ha Hence w w I D w D + w I D w, w w I N w N + w I N w. w j M j A w M M j A w A w G w. Equaliy holds if and only if D = I, ha is, N = I and M = M. Nex, we prove he righ hand inequaliy, using similar echniques. A compuaion shows ha Hence, o esablish w j M j A w M M j A w = w w M I N N M. A w G w w j M j A w M M j A w, we need o show M w N + w IM M N w M w w M I N N M. Bu his is rue if and only if w N + w N N w + w w I N, and his follows from Lemma via he diagonalizaion echnique above. Again, equaliy holds if and only if N = I, ha is M = M. I is naural o ask wheher our marix exension of he Carwrigh-Field inequaliy can be carried over o he case of hree or more marices. If he marices commue, his is 6

7 no problem. Indeed, no ordering hypohesis is necessary in he presence of commuaiviy. Commuing posiive marices can be viewed as elemens of a commuaive C algebra, which, by Gelfand s heorem 3], is isomerically -isomorphic o he algebra CK of coninuous funcions on an appropriae compac Hausdorff space K. The posiive marices hen correspond o posiive funcions, and he scalar inequaliy of Carwrigh and Field can be applied a each poin of K. When we drop he commuaiviy hypohesis, we are unable o prove analogs of Theorem for more han wo marices. This is possibly relaed o general problems wih he geomeric mean of more han wo marices ha we illusraed above. 4. Marix refinemens of he arihmeic-geomeric-harmonic mean inequaliy Recall ha if w,..., w k are posiive number summing o, he weighed harmonic mean of he posiive numbers x,..., x k is The classical resul ha H w := H w x,..., x k = w x + + w k x k. H w G w A w has been exensively refined. Building on a resul of Alzer ], Mercer 5] found an exensive collecion of inequaliies ha he proved using a global echnique. We summarize some of his resuls. Theorem 3 Mercer Le x i i k be real numbers, ordered so ha 0 < x x k, and le w i i k be posiive numbers wih k =. Then, if we w i i= wrie A w = A w x,..., x k, G w = G w x,..., x k, and H w = H w x,..., x k, we have x k w j x j G w A w G w x w j x j G w ; x k 3 G w x k w j x j H w A w H w ; w j x j G w G w H w G w x w j x j G w ; 4 G w w x j x j H w G w H w G w w k x j x j H w. These inequaliies are sric unless all he x i are equal. The righ hand side of had been esablished by Alzer ]. A naural quesion is wheher a version of Mercer s heorem holds for posiive marices. We are able o provide a posiive answer in he case of wo marices? 7

8 Lemma Le w > 0, w > 0 saisfy w + w =, and 0, ], hen w + w w + w w + w w ] w + w w. Equaliy holds if and only if =. Proof We sar o prove he righ hand inequaliy of. Firs of all, we have w w + w w ] w + w w w w + w + w w w w + + w 0. Le f = w w + w + w w w w + + w. Then f = 0 and f = w + w + w w w w + w + w w. Doing more we have ha f = 0, f = w + w w w w w + w + w w w, and f = 0. Taking he hird derivaive of f wih simplificaion, we have f = w w w 3 w + w w + w w + w w + w w ]. Noice ha he righ hand inequaliy of Lemma implies w + w w > 0, so we have f > 0, which implies ha f is increasing on 0, ] and f f = 0, ha means f is decreasing on 0, ] and f f = 0. Hence f is increasing on 0, ] and f f = 0, which means he righ hand inequaliy of holds. Similarly, using he same way we prove he lef hand inequaliy of. I is easy o check ha w + w w + w w + w w ] w + w + w + w w w w + w 0. Le g = w + w + w + w w w w + w, 8

9 hen g = 0 and Doing more we have g = 0, g = w + w + w w w w w + w w w. g = w w w w w w w + w w w w, and g = 0. Coninuously aking he hird derivaive of g, we have g = w w w 3 w w + w + w w + w w + w w ]. By he righ hand inequaliy of Lemma, we have g > 0 on 0, ]. Hence he lef hand inequaliy of holds. The generalizaion of inequaliy in Theorem 3 o he case of wo marices is given as follows. Theorem 4 Le w > 0, w > 0 saisfy w + w =. Le M, M be posiive n n marices wih M M. Wrie A w = A w M, M and G w = G w M, M. Then w j M j G w M M j G w A w G w Equaliy holds if and only if M = M. w j M j G w M M j G w. Proof We sar o prove he lef side of he inequaliy. Firs of all, we have which imply ha M G w = M N N w M and M G w = M I N w M, w j M j G w M M j G w = w M N N w M + w M I N w M. = M w N N w + w I N w ] M. Hence w j M j G w M M j G w A w G w M w N N w + w I N w ] M M w N + w I N w M w N N w + w I N w ] w N + w I N w, 9

10 which holds since he righ hand inequaliy of. Equaliy holds if and only if N = I, ha is M = M. Nex, we going o prove he righ side of he inequaliy. We have w j M j G w M M j G w = w M N N w N N N w M + w M I N w N I N w M = M w N N w + w I N w ] N M. Hence A w G w w j M j G w M M j G w w N + w I N w w N N w + w I N w ] N w N + w N N w + w N N w + w I N w ], which holds since he lef hand inequaliy of. Equaliy holds if and only if N = I, ha is M = M. For he generalizaion of inequaliy in Theorem 3, we need he following lemma. Lemma 3 Le w > 0, w > 0 saisfy w + w =, and 0, ], hen ] w + w w + w w + w w + w w + w. Equaliy holds if and only if =. Proof Afer algebraic simplificaion, we have w + w = w w w + w w + w w + w w + w. 0

11 Hence w + w w + w w w w + w w + w w w w + w w + w w + w w, ] w + w w + w w + w which is obvious. The marix version of inequaliy in Theorem 3 o he case of wo marices as follows. Theorem 5 Le w > 0, w > 0 saisfy w + w =. Le M, M be posiive n n marices wih M M. Wrie A w = A w M, M and H w = H w M, M. Then w j M j H w M M j H w A w H w. Equaliy holds if and only if M = M. Proof As before, i is convenien o use N = M M M. In addiion, we se C = w I + w N. We have already seen ha 0 < N I, when M M, and i is clear ha N and C commue. By he definiion of H w, we have H w = w M N M + w M IM = M Nw I + w N M = M NCM. Hence M H w = M N NCM, M H w = M I NCM.

12 Thus, w j M j H w M M j H w A w H w w N NC + w I NC ] w N + w I NC, which holds since inequaliy. Equaliy holds if and only if N = I, ha is M = M. We give he following lemma for generalizing he inequaliy 3 in Theorem 3. Lemma 4 Le w > 0, w > 0 saisfy w + w =, and 0, ], hen w + w + w Equaliy holds if and only if =. w w + w w ] w w + w. 3 Proof I is difficul ha giving immediaely proof for inequaliies 3 wih he same way of Lemma. Wih he help of inequaliies iii in Theorem 3, we give he following proof of inequaliies 3. Le k = in he inequaliies iii of Theorem 3, we have G w x which is equivalen o x w x w x w j x j G w G w H w G w x w j x j G w, w j x j x w x w x w x w w x +w x xw x w x w w x + w x = x w x w x w j x j x w x w. Since x x, le = x /x, hen 0 <, and afer a few seps of verifying we have ], and x w x w x x w x w x w w + w w j x j x w x w = x w w w + w w ], w j x j x w x w = x w w w + w w ]. From hese we imply ha he inequaliies 3 holds. The marix version corresponding he inequaliy 3 in Theorem 3 is he following.

13 Theorem 6 Le w > 0, w > 0 saisfy w + w =. Le M, M be posiive n n marices wih M M. Wrie G w = G w M, M and H w = H w M, M. Then G wm w j M j G w M M j G w G w H w G wm Equaliy holds if and only if M = M. w j M j G w M M j G w. Proof Firs we prove he lef side of he inequaliy. Using he noaion in Theorem 5 we used, we have G wm w j M j G w M M j G w G w H w M N w w N N w + w I N w ] M M N w NCM N w w N N w + w I N w ] N w NC w N N w + w I N w ] I N w C, which holds since he righ hand inequaliy of 3. Equaliy holds if and only if N = I, ha is M = M. Now we prove he righ side of he inequaliy. We have G w H w G wm w j M j G w M M j G w N w NC w N N w + w I N w ] N w N N w + C w N N w + w I N w ], which holds since he lef hand inequaliy of 3. Equaliy holds if and only if N = I, ha is M = M. We urn o he las inequaliy 4 in Theorem 3. Similarly we give he following lemma. Lemma 5 Le w > 0, w > 0 saisfy w + w =, and 0, ], hen w + w + w ] w + w w + w w + w Equaliy holds if and only if =. 3 w w + w. 4

14 Proof have Firs we prove he righ hand inequaliy of 4. From he proof of Lemma 3 we w + w w + w ] w + w w w w + w w + w w + w w + w w w + w w w + w w w w + w w + w w + w w. If =, his is cerainly rue, and in fac becomes an equaliy. If, by he righ hand inequaliy of Lemma, we have 0 < w w w + w w. Consequenly, we can achieve our goal by showing ha w + w < w + w, when. However his follows he facs ha 0, and w > 0. Nex we prove he lef hand inequaliy of 4. We know ha w w + w + w + w w + w ] w + w w + w w + w + w w + w w w w + w w + w w + w w + w w + w w w + w. If =, his is cerainly rue, and in fac becomes an equaliy. If, by he lef hand inequaliy of Lemma, we have 0 < w + w w + w w. Since 0 < <, holds, he proof is compleed. w + w < w + w Here is he marix version of inequaliy 4 in Theorem 3 o he case of wo marices. 4

15 Theorem 7 Le w > 0, w > 0 saisfy w + w =. Le M, M be posiive n n marices wih M M. Wrie G w = G w M, M and H w = H w M, M. Then G wm G wm w M j H w M M j H w G w H w Equaliy holds if and only if M = M. w M j H w M M j H w. Proof I is similar wih Theorem 5, we have G wm w M j H w M M j H w G w H w N w w N NC + w I NC ] N w NC w N NC + w I NC ] I N w C, which holds since he righ hand inequaliy of 4. Equaliy holds if and only if N = I, ha is M = M. For he righ hand inequaliy, we have G w H w G wm w M j H w M M j H w N w NC N w w N NC + w I NC ] N N w + C w N NC + w I NC ], which holds since he lef hand inequaliy of 4. Equaliy holds if and only if N = I, ha is M = M. As wih he Carwrigh-Field inequaliy, Theorem 4 o Theorem 7 can be exended wihou he need for an ordering hypohesis o he case of several commuing posiive n n marices. I would be ineresing o know wheher he commuaiviy or ordering hypohesis can be dropped. 5

16 References ] H. Alzer, A new refinemen of he arihmaeic mean-geomeric mean inequaliy. Rocky Mounain J. Mah. 7, No. 3997, MR 99a: 605. ] T. Ando, Concaviy of cerain maps on posiive definie marices and applicaions o Hadamard producs. Linear Algebra Appl., MR 80f: ] F. F. Bonsall and J. Duncan, Complee normed algebras. Berlin, New York, Springer- Verlag, ] D. I. Carwrigh and M. J. Field, A refinemen of he arihmeic - geomeric mean inequaliy. Proc. Amer. Mah. Soc. 7978, MR 57: # ] A. McD. Mercer, Bounds for A-G, A-H, G-H and a family of inequaliies of Ky Fan s ype, using a general mehod. J. Mah. Anal. Appl., MR 00a: 60. 6] W. Pusz and S. L. Woronowicz, Funcional calculus for sesquilinear forms and he purificaion map. Repors on Mahemaical Physics 8975, MR 54: # ] M. Sagae and K. Tanabe, Upper and lower bounds for he arihmeic-geomeric-harmonic means of posiive definie marices. Linear and Mulilinear Algebra, 37994, MR 95m: Bao Qi Feng Deparmen of Mahemaical Sciences Ken Sae Universiy New Philadelphia, OH , USA bfeng@usc.ken.edu Andrew Tonge Deparmen of Mahemaical Sciences Ken Sae Universiy Ken, OH 4440, USA onge@mah.ken.edu 6