OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2

Transcription

1 OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2 YUKI SEO Abstract. For 1 t 1, Lm-Pálfa defned a new famly of operator power means of postve defnte matrces and subsequently by Lawson-Lm ther noton and most of ther results extend to the settng of postve nvertble operators on a Hlbert space. Each of these means except t 0 arses as a unque postve nvertble soluton of a non-lnear operator equaton and satsfes all desrable propertes of power arthmetc means of postve real numbers. The purpose of ths paper s to extend the range n whch operator power means due to Lawson-Lm-Pálfa are defned. We nvestgate some propertes of operator power means for t ( 2, 2)\[ 1, 1]. 1. Introducton Let B(H) be the C -algebra of all bounded lnear operators on a Hlbert space H equpped wth the operator norm, S(H) the set of all bounded self-adjont operators, and P = P(H) the open convex cone of all postve nvertble operators. For X, Y S(H), we wrte X Y f Y X s postve, and X < Y f Y X s postve nvertble. For postve real numbers x 1,..., x n R and a weght vector ω = (ω 1,..., ω n ) such as ω 0 for = 1,..., n and n ω = 1, the power arthmetc means M t (ω; x 1,..., x n ) = ( ω 1 x t ω n x t n) 1/t for t R make a path of means from the harmonc one at t = 1 to the arthmetc one at t = 1 va the geometrc one at t 0. The followng s a noncommutatve verson of the power arthmetc mean: For postve nvertble operators A 1,..., A n P and a weght vector ω ( 1/t M t (ω; A 1,..., A n ) = ω A) t for t R. Bhagwat and Subramanan [2] showed that the power arthmetc mean has the followng monotoncty: 1 t s = M t (ω; A 1,..., A n ) M s (ω; A 1,..., A n ) and the lmt M 0 (ω; A 1,..., A n ) = u- lm t 0+ M t (ω; A 1,..., A n ) exsts and s equal to the chaotc geometrc mean exp( n ω log A ), also see [7, 15], whch reduced to the usual geometrc mean n the case of commutng operators. However, M t (ω; A 1,..., A n ) does not have the monotoncty for 1 < t < 1 n general and they are not operator means except for the case of t = ±1. Recently, for 1 t 1, Lm-Pálfa [13] defned a new famly of operator power means of postve defnte matrces and subsequently by Lawson-Lm [12] ther noton and most of ther results extend to the settng of postve nvertble operators on a Hlbert space Mathematcs Subject Classfcaton. 47A63, 47A64. Key words and phrases. Power mean, operator geometrc mean, postve operator, Thompson metrc. 1

2 2 Y. SEO We denote by {P t (ω; A)}, where ω s a weght vector and A s an n-tuple of postve nvertble operators on a Hlbert space. Each of these means except t 0 arses as a unque postve nvertble soluton P t (ω; A) of a non-lnear operator equaton X = ω (X t A ) (t [ 1, 1]\{0}) and satsfes desrable propertes of power arthmetc means of postve real numbers and nterpolates between the weghted harmonc and arthmetc means. Moreover, Lawson- Lm showed that the Karcher mean of postve nvertble operators concdes wth the lmt of operator power means as t 0. For more detals on the Karcher mean; see [4, 5, 17]. In fact, f A mutually commute for = 1,..., n, then t follows that P t (ω; A) = ( n ω A t ) 1/t. Moreover, they showed that the power means P t (ω; A) have a monotone ncreasng property for 1 < t < 1: and an nformaton monotoncty: 1 < t s < 1 = P t (ω; A) P s (ω; A) Φ(P t (ω; A)) P t (ω; Φ(A)) (t (0, 1]) for any untal postve lmear map Φ. However, the range n whch the operator power means are defned, s lmted to [ 1, 1]. The purpose of ths paper s to extend the range of the defnton of power means P t (ω; A). Moreover, we nvestgate some propertes of P t (ω; A) for t ( 2, 2)\[ 1, 1]. 2. Prelmnares For A, B P and t [0, 1], the t-geometrc operator mean s defned as A t B = A 1/2 ( A 1/2 BA 1/2) t A 1/2. For convenence, we use the notaton t for the bnary operaton whose formula s the same as t. s [0, 1] s not. A t B = A 1/2 ( A 1/2 BA 1/2) t A 1/2 for t [0, 1], Though A t B for t [0, 1] s monotonc, A s B for Lemma 2.1. Let A, B, X, Y P and 1 < t 2. Then () If X Y, then Y t A X t A. () If A B wth m 1 A M 1, m 2 B M 2 and m X M for some scalars 0 < m M ( = 1, 2) and 0 < m M, then X t A K(m /M, M /m, t)x t B for = 1, 2, where the generalzed Kantorovch constant K(m, M, t) s defned by (1) K(m, M, t) = mm ( t Mm t t 1 M t m t (t 1)(M m) t mm t Mm t for any real number t R, see [11, Theorem 2.53]. () If m A M for some scalars 0 < m M, then X 1 t m t X t A X 1 (1 t) M t. ) t

3 OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2 3 Proof. (): For 1 < t 2 Y t A = A 1 t Y = A 1/2 ( A 1/2 Y A 1/2) 1 t A 1/2 = A 1/2 ( A 1/2 Y 1 A 1/2) t 1 A 1/2 A 1/2 ( A 1/2 X 1 A 1/2) t 1 A 1/2 = X t A. by 0 < t 1 < 1 and Y 1 X 1 (): Snce A B, we have X 1/2 AX 1/2 X 1/2 BX 1/2 and m 1 /M X 1/2 AX 1/2 M 1 /m and m 2 /M X 1/2 BX 1/2 M 2 /m. By the generalzed Kantorovch nequalty [11, Theorem 8.3], t follows from 1 < t 2 that (X 1/2 AX 1/2 ) t K( m M, M m, t)(x 1/2 BX 1/2 ) t for = 1, 2, and we have the desred nequalty. (): It follows from X 1 1 X X and (). Remark 2.2. Let X 0 and 0 < A B. Then the nequalty AXA BXB doe not hold n general. If we put t = 2 n (2) of Lemma 2.1, then we have { (nm1 + NM 1 ) 2 AXA mn, (nm } 2 + NM 2 ) 2 BXB 4nNm 1 M 1 4nNm 2 M 2 where n X N and m 1 A M 1, m 2 B M 2 for some scalars 0 < n N and 0 < m M ( = 1, 2). If B = I, then we have AXA (n+n)2 X n [10, Lemma 4]. 4nN The Thompson metrc on P s defned by 3. Thompson metrc d(a, B) = log max{m(a/b), M(B/A)} where M(A/B) = nf{λ > 0 : A λb} = B 1/2 AB 1/2 = r(b 1 A). It s known that d s a complete metrc on P and d(a, B) = log B 1/2 AB 1/2 = log A 1/2 BA 1/2, see [16]. We lst some basc propertes of the Thompson metrc: Lemma 3.1 (see [3, 6]). For A, B, C, D P () d(a, B) = d(a 1, B 1 ) = d(t AT, T BT ) for nvertble T B(H); () d(a + B, C + D) max{d(a, C), d(b, D)}; () d(a t, B t ) td(a, B) for t [0, 1]; (v) d(αa, αb) = d(a, B) for postve real number α > 0; (v) d(a t B, C t D) (1 t)d(a, C) + td(b, D) for t [0, 1]. For A, B P, a map γ A,B : R P defned by γ A,B (t) = A t B for t R s a path jonng A and B. Then we have the followng: Theorem 3.2. Let A, B P. Then d(a s B, A t B) = s t d(a, B) for all s, t R.

4 4 Y. SEO Proof. By defnton of the Thompson metrc and Lemma 3.1 d(a s B, A t B) = d((a 1/2 BA 1/2 ) s, (A 1/2 BA 1/2 ) t ) = d((a 1/2 BA 1/2 ) s t, I) = log(a 1/2 BA 1/2 ) s t = s t log A 1/2 BA 1/2 = s t d(a, B). We have the followng estmate n the case of 1 < t < 2, whch corresponds to (v) of Lemma 3.1: Theorem 3.3. Let A, B, C, D P such that m 1 A C M 1 A and m 2 B D M 2 B for some scalars 0 < m 1 M 1 and 0 < m 2 M 2. For each 1 < t < 2 d(a t B, C t D) (t 1)d(A, C) + td(b, D) + log K(t) where K(t) = max{k(m 1, M 1, t), K(m 2, M 2, t)} and the generalzed Kantorovch constant K(m, M, t) s defned by (1). Proof. Snce A 1/2 C 1 A 1/2 1 A 1/2 CA 1/2, t follows from Lemma 2.1 that C t D = A 1/2 [ (A 1/2 CA 1/2 ) t (A 1/2 DA 1/2 ) ] A 1/2 A 1/2 [ A 1/2 C 1 A 1/2 1 t (A 1/2 DA 1/2 ) ] A 1/2 by () of Lemma 2.1 = A 1/2 C 1 A 1/2 (1 t) A t D = A 1/2 C 1 A 1/2 (1 t) B 1/2 [ (B 1/2 AB 1/2 ) t (B 1/2 DB 1/2 ) ] B 1/2 A 1/2 C 1 A 1/2 (1 t) B 1/2 DB 1/2 t K(m 2, M 2, t)b 1 t A by () of Lemma 2.1 = A 1/2 C 1 A 1/2 (1 t) B 1/2 DB 1/2 t K(m 2, M 2, t)a t B. Smlarly, t follows that Therefore, we have A t B C 1/2 A 1 C 1/2 t 1 D 1/2 BD 1/2 t K(m 1, M 1, t) C t D. (A t B) 1/2 (C t D)(A t B) 1/2 A 1/2 C 1 A 1/2 (1 t) B 1/2 DB 1/2 t K(m 2, M 2, t) and (C t D) 1/2 (A t B)(C t D) 1/2 C 1/2 A 1 C 1/2 t 1 D 1/2 BD 1/2 t K(m 1, M 1, t) and ths mples the desred nequalty. 4. Operator power means In ths secton, we extend the range n whch the power means due to Lawson-Lm-Pálfa are defned. For ths, we need the followng Lemma: Lemma 4.1. Let X, Y, A P and 1 < t 2. Then d(x t A, Y t A) (t 1)d(X, Y ).

5 Proof. For 1 < t 2, OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2 5 d(x t A, Y t A) = d(a 1 t X, A 1 t Y ) = d((a 1/2 X 1 A 1/2 ) t 1, (A 1/2 Y 1 A 1/2 ) t 1 ) by () of Lemma 3.1 (t 1)d(A 1/2 XA 1/2, A 1/2 Y A 1/2 ) by () of Lemma 3.1 = (t 1)d(X, Y ) by () of Lemma 3.1. Theorem 4.2. Let A 1, A 2,..., A n P and a weght vector ω = (ω 1,..., ω n ). Then for each 1 < t < 2, the followng equaton has a unque postve nvertble soluton: X = ω (X t A ). Proof. We wll show that the map f : P P defned by f(x) = n ω (X t A ) s a strct contracton wth respect to the Thompson metrc. Let X, Y > 0. d(f(x), f(y )) max 1 n {d(ω (X t A ), ω (Y t A ))} by () of Lemma 3.1 = max 1 n {d(x t A, Y t A )} by (v) of Lemma 3.1 (t 1)d(X, Y ) by Lemma 4.1. Snce 1 < t < 2, t follows that f s a strct contracton and hence f has a unque fxed pont. Defnton 4.3. Let A = (A 1,..., A n ) P n and a weght vector ω = (ω 1,..., ω n ). For t (1, 2), we denote by P t (ω; A) the unque postve nvertble soluton of X = ω (X t A ). For t ( 2, 1), we defne P t (ω; A) = P t (ω; A 1 ) 1, where A 1 = (A 1 1,..., A 1 n ). In fact, X = P t (ω; A) s the unque postve nvertble soluton of X = ( n ω (X t A ) 1 ) 1 and X 1 = n ω (X 1 t A 1 ) f and only f X 1 = P t (ω; A 1 ). Remark 4.4. Let t (1, 2). Put f : P P defned by f(x) = n ω (X t A ). By Theorem 4.2, f s a strct contracton for the Thompson metrc and by the Banach fxed pont theorem lm k f k (X) = P t (ω; A) for any X P. Smlarly, the map g(x) = ( n ω (X t A ) 1 ) 1 s a strct contracton for the Thompson metrc and lm k g k (X) = P t (ω; A) for any X P. For A = (A 1,..., A n ) P n, M B(H), ω = (ω 1,..., ω n ) and for a permutaton σ on n-letters, we set MAM = (MA 1 M,..., MA n M ), A σ = (A σ(1),..., A σ(n) ) 1 ˆω = (ω 1,..., ω n 1 ). 1 ω n We lst some basc propertes of P t (ω; A) for t ( 2, 2)\[ 1, 1].

6 6 Y. SEO Proposton 4.5. Let A = (A 1,..., A n ) P n, a weght vector ω = (ω 1,..., ω n ) and let t ( 2, 2)\[ 1, 1]. () P t (ω; A) = ( n ω A t ) 1/t f the A s commute; () P t (ω σ ; A σ ) = P t (ω; A) for any permutaton σ; () P t (ω; MAM ) = MP t (ω; A)M for any nvertble M; (v) P t (ω; A 1 ) 1 = P t (ω; A); (v) n ω A P t (ω; A) for t (1, 2); (v) P t (ω; A) ( n ω A 1 ) 1 for t ( 2, 1); (v) If m A M for = 1,..., n and some scalars 0 < m M, then m P t (ω; A) m 1 t M t for t (1, 2) and m t M 1+t P t (ω; A) M for t ( 2, 1); (v) For t (1, 2), P t (ω; A 1,..., A n 1, X) = X f and only f X = P t (ˆω; A 1,..., A n 1 ). Proof. Proofs from () to (v) are smlar to those of [13]. (v): Put X = P t (ω; A) for t (1, 2). Snce αt + (1 α)i T α for all α > 1 and postve nvertble T, we have (1 t)a + tb A t B for 1 < t < 2 puttng T = A 1/2 BA 1/2 and multplyng A 1/2 on both sdes, see [9, pp.123]. Therefore t follows that X = ω (X t A ) ω ((1 t)x + ta ) = (1 t)x + t ω A and hence X n ω A. (v): Put X = P t (ω; A) for t ( 2, 1). Snce X = ( n ω (X 1 t A 1 ) ) 1, t follows that X 1 = ω (X 1 t A 1 ) ω ((1 + t)x 1 + ( t)a 1 ) = (1 + t)x 1 t ω A 1 and hence X ( n ω A 1 ) 1 for t ( 2, 1). (v): Put X = P t (ω; A) for t (1, 2), and X n ω A m. Hence we have X = ω (X t A ) ω (m t A ) = ω (m 1 t A t ) m 1 t M t by () of Lemma 2.1. Smlarly, put X = P t (ω; A) for t ( 2, 1), and X ( n ω ) A 1 1 M. Hence we have X 1 = ω (X 1 t A 1 ) ω (M 1 t A 1 ) = ω (M 1 t A t ) M 1 t m t and m t M 1+t X. (v): For 1 < t < 2, P t (ω; A 1,..., A n 1, X) = X X = n 1 ω (X t A ) + ω n X X = n 1 ω 1 ω n (X t A ) X = P t (ˆω; A 1,..., A n 1 ). Theorem 4.6. Let A = (A 1,..., A n ) P n such that 0 < m A M for some scalars 0 < m M and a weght vector ω = (ω 1,..., ω n ). Let 1 < t s < 2. Then s t [ ( d(p t (ω; A), P s (ω; A)) t (A) + log K m/m, (M/m) t, t )], (2 s)(2 t) where the generalzed Kantorovch constant K(m, M, t) s defned by (1) and (A) = max 1,j n {d(a, A j )} denotes the d-dameter of A = (A 1,..., A n ).

7 OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2 7 Proof. Put X = P t (ω; A) and Y = P s (ω; A). Snce m A M for = 1,..., n and m X m 1 t M t, we have m/ma X (M/m) t A and m/ma A j M/mA for, j = 1,..., n. It follows from [18, Proposton 4] that K(m/M, M/m, t) K(m/M, (M/m) t, t) for 1 < t < 2. By Theorem 3.3, t follows that d(x, A j ) = d( ω (X t A ), ω A j ) max {d(x ta, A j t A j )} 1 n { max (t 1)d(X, Aj ) + td(a, A j ) + log K(m/M, (M/m) t, t) } 1 n = (t 1)d(X, A j ) + t (A) + log K(m/M, (M/m) t, t) for j = 1,..., n and hence we have d(x, A j ) By Lemma 4.1, we have d(x, Y ) = d(y, X) = d( t 2 t (A) t log K(m/M, (M/m)t, t). ω (Y s A ), ω (X t A )) max {d(y sa, X t A )} 1 n max {d(y sa, X s A ) + d(x s A, X t A )} 1 n max {(s 1)d(Y, X) + (s t)d(x, A )} 1 n [ t (s 1)d(X, Y ) + (s t) 2 t (A) t log K ( m/m, (M/m) t, t ) ] [ and hence we have d(x, Y ) s t t (A) + 1 log K (m/m, 2 s 2 t 2 t (M/m)t, t) ]. Theorem 4.7. Let A = (A 1,..., A n ) and B = (B 1,..., B n ) such that 0 < m 1 A M 1 and 0 < m 2 B M 2 for = 1,..., n for some scalars 0 < m 1 M 1 and 0 < m 2 M 2. Then for each 1 < t < 2 d(p t (ω; A), P t (ω; B)) t 2 t max 1 n {d(a, B )} t log K 1(t), where K 1 (t) = max{k(m 2 /m 1 t 1 M t 1, m 1 t 2 M t 2/m 1, t), K(m 2 /M 1, M 2 /m 1, t)}. Proof. For fxed t (1, 2), put X = P t (ω; A) and Y = P t (ω; B). Snce m 1 X m 1 t 1 M1 t and m 2 Y m 1 t 2 M2, t we have m 2 /m 1 t 1 M1X t Y m 1 t 2 M2/m t 1 X and m 2 /M 1 A

8 8 Y. SEO B M 2 /m 1 A for = 1,..., n. Then t follows from Theorem 3.3 that d(x, Y ) = d( ω (X t A ), ω (Y t B )) max {d(x ta, Y t B )} 1 n max {(t 1)d(X, Y ) + td(a, B ) + log K 1 (t)} 1 n = (t 1)d(X, Y ) + t max {d(a, B )} + log K 1 (t). 1 n Remark 4.8. As t 1 n Theorem 4.7, we have K 1 (t) 1. Snce P 1 (ω; A) = n ω A and P 1 (ω; B) = n ω B, ths corresponds to d( n ω A, n ω B ) max 1 n {d(a, B )}, see [12, Lemma 2.4]. 5. Monotoncty of power means In ths secton, we consder the monotoncty of P t (ω; A) for 1 < t < 2. Before ths, n the case of n = 2, we consder an explct form of P t (1 α, α; A, B) for 1 < t < 2 and moreover a postve soluton of (2) X = (1 α)(x t A) + α(x t B) for t R. The soluton of (2) s X = Am t,α B = A 1/2 ( (1 α)i + α(a 1/2 BA 1/2 ) t) 1/t A 1/2. Indeed, put C = A 1/2 BA 1/2 and Y = ((1 α)i + αc t ) 1/t. Then t follows that (1 α)(x t A) + α(x t B) = A 1/2 ((1 α)(y t I) + α(y t C)) A 1/2 = A 1/2 ( (1 α)y 1 t + αy 1 t C t) A 1/2 = A 1/2 ( Y 1 t ((1 α)i + αc t ) ) A 1/2 = A 1/2 Y A 1/2 = X. It s known that the soluton X = Am t,α B for α [0, 1] s nondecreasng for t R, see [11, Theorem 5.21]. However, n the case of n 3, we have the followng order relaton: Theorem 5.1. Let A = (A 1,..., A n ) P n such that m A M for = 1,..., n and some scalars 0 < m M. Then for each 1 < t s < 2 (3) (m/m) t(s 1) 2 s P s (ω; A) P t (ω; A) (M/m) s(t 1) 2 t P s (ω; A). To prove Theorem 5.1, we need the followng two lemmas. The followng lemma s the complement of the Löwner-Henz nequalty: Lemma 5.2. ([14, Lemma 2.2]) If Y X, then X p X p/2 Y p X p/2 Y p for 0 < p < 1. Lemma 5.3. For 1 < t < 2, defne f(x) = n ω (X t A ) where A are postve nvertble operators and ω s a weght vector. Let X, Y be postve nvertble operators such that m 1 Y M 1 and m 2 X M 2 for some scalars 0 < m M ( = 1, 2). Then Y X = f(x) f(y ) (M 2 /m 1 ) t 1 f(x). In partcular, f 2 s monotone, that s, f Y X, then f 2 (Y ) f 2 (X).

9 OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2 9 Proof. The assumpton Y X mples f(y ) = n (Y ta ) n (X ta ) = f(x) by () of Lemma 2.1. Put X = A 1/2 X 1 A 1/2, Y = A 1/2 Y 1 A 1/2 for = 1,..., n. Snce X 1 Y 1, we have X Y and t follows from Lemma 5.2 and 0 < t 1 < 1 that for = 1,..., n and hence Y t 1 Y (t 1)/2 Y t A Y (t 1)/2 X (t 1) Y (t 1)/2 X (t 1) X t 1 Y (t 1)/2 X t A. Also, by the Arak-Cordes nequalty [11, pp.67], we have Y (t 1)/2 X (t 1) Y (t 1)/2 = r((a 1/2 Y 1 A 1/2 )(A 1/2 Y 1/2 XA 1/2 )) t 1 X 1 Y 1/2 t 1 = r(xy 1 ) t 1 = X 1/2 Y 1 X 1/2 t 1 (M 2 /m 1 ) t 1. Therefore, we have f(y ) = ω (Y t A ) (M 2 /m 1 ) t 1 ω (X t A ) = (M 2 /m 1 ) t 1 f(x). Proof of Theorem 5.1 Put f(x) = n ω (X t A ) and then P t (ω; A) = lm k f k (X) for any X P. Snce X t A = X t/s (X s A ), t follows from 0 < t/s 1 that [ f(x) = ω (X t A ) = ω X t/s (X s A ) ] ω [(1 t s )X + t ] s (X sa ) = (1 t s )X + t s ω (X s A ). If we put X 0 = P s (ω; A), then we have f(x 0 ) (1 t s )X 0 + t ω (X 0 s A ) = (1 t s s )X 0 + t s X 0 = X 0. Moreover, we have m X 0 m 1 s M s and (m 1 s M s ) 1 t m t f(x 0 ) m 1 t M t. By Lemma 5.3, t follows that ( ) f 2 m 1 s M s t 1 (X 0 ) f(x (m 1 s M s ) 1 t m t 0 ) (M/m) st(t 1) X 0. Snce f 2 s monotonc, we have Inductvely we have f 4 (X 0 ) f 2 ( (M/m) st(t 1) X 0 ) = (M/m) st(t 1)(1 t) 2 f 2 (X 0 ) (M/m) st(t 1)(1 t)2 +st(t 1) X 0. f 2k (X 0 ) [ (M/m) 1 (1 t) 2k st(t 1)] 1 (1 t) 2 X 0.

10 10 Y. SEO As k, we have the desred nequalty P t (ω; A) (M/m) s(t 1) 2 t P s (ω; A). Smlarly, f we put Y 0 = P s (ω; A) and g(x) = n ω (X s A ), then we have the LHS of (3) n Theorem Postve lnear maps In ths secton, we consder an nformaton monotoncty of P t (ω; A) for 1 < t < 2. Though the Ando nequalty Φ(A t B) Φ(A) t Φ(B) holds for any postve lnear map and t [0, 1] (see [1]), the reverse holds n the case of t (1, 2): Lemma 6.1. Let Φ be a postve lnear map on B(H) and A, B P such that ma B MA for some scalars 0 < m M. Then Φ(A t B) Φ(A) t Φ(B) K(m, M, t) 1 Φ(A t B) for 1 < t < 2, where the generalzed Kantorovch constant K(m, M, t) s defned by (1). Proof. We may assume that Φ s strctly postve. For A, B > 0, put Ψ(X) = Φ(A) 1/2 Φ(A 1/2 XA 1/2 )Φ(A) 1/2 and hence Ψ s a untal postve lnear map. By the Jensen nequalty [11, Corollary 1.22, Corollary 2.12] (4) K(m, M, t)ψ(x) t Ψ(X t ) Ψ(X) t for 1 < t < 2. Therefore, t follows that Φ(A) 1/2 Φ(A t B)Φ(A) 1/2 = Φ(A) 1/2 Φ(A 1/2 (A 1/2 BA 1/2 ) t A 1/2 )Φ(A) 1/2 and we have Φ(A t B) Φ(A) t Φ(B). Lemma 6.1. = Ψ((A 1/2 BA 1/2 ) t ) Ψ(A 1/2 BA 1/2 ) t = ( Φ(A) 1/2 Φ(B)Φ(A) 1/2) t Smlarly, by usng (4), we have the RHS of The operator power means P t (ω; A) for t (0, 1] have an nformaton monotoncty, see [12, Proposton 3.6]. In the case of t (1, 2), we have the followng slghtly modfcaton: Theorem 6.2. Let A = (A 1,..., A n ) and a weght vector ω. Let Φ be a untal postve lnear map. For each 1 < t < 2 K ( (m/m) t, M/m, t ) 1 t(2 t) (m/m) t 1 2 t Φ(Pt (ω; A)) P t (ω; Φ(A)) (M/m) t(t 1) 2 t Φ(P t (ω; A)), where Φ(A) = (Φ(A 1 ),..., Φ(A n )) and the generalzed Kantorovch constant K(m, M, t) s defned by (1). Proof. Put X = P t (ω; A) and f(y ) = n ω (Y t Φ(A )) for any Y P. Snce Φ s untal, we have m Φ(X) m 1 t M t. Snce f(φ(x)) = n ω (Φ(X) t Φ(A )), we have (m 1 t M t ) 1 t m t f(φ(x)) m 1 t M t. It follows from f(φ(x)) Φ(X) that f 2 (Φ(X)) (M/m) t2 (t 1) f(φ(x)). Hence we have f 3 (Φ(X)) f 2 (Φ(X)) (M/m) t2 (t 1) Φ(X).

11 OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2 11 Inductvely t follows that f 2k 1 (Φ(X)) [ ] 1 (1 t) 2k (M/m) t2 (t 1) 1 (1 t) 2 Φ(X) and as k we have [ ] 1 P t (ω; Φ(A)) (M/m) t2 (t 1) 1 (1 t) 2 Φ(P t (ω; A)) = (M/m) t(t 1) 2 t Φ(P t (ω; A)). Conversely, snce K((m/M) t, M/m, t) 1 Φ(X) f(φ(x)), we have K((m/M) t, M/m, t) 1 (M/m) t(t 1) Φ(X) f 2 (Φ(X)). Snce f 2 s monotone, t follows that f 3 (Φ(X)) f 2 (K((m/M) t, M/m, t) 1 Φ(X)) = K((m/M) t, M/m, t) (1 t)2 f 2 (Φ(X)) Inductvely, we have K((m/M) t, M/m, t) 1 (1 t)2 (m/m) t(t 1) Φ(X). K((m/M) t, M/m, t) 1 (1 t) 2(k 1) [ ] 1 1 (1 t) 2 (m/m) t(t 1) 1 (1 t) 2 Φ(X) f 2k 1 (Φ(X)) for k = 1, 2 and as k K((m/M) t, M/m, t) 1 t(2 t) (m/m) t 1 2 t Φ(Pt (ω; A) P t (ω; Φ(A)). Remark 6.3. As t 1 n Theorem 6.2, we have K((m/M) t, M/m, t) 1 t 1 t(2 t) (m/m) 2 t 1 and (M/m) t2 (t 1) 1. Ths corresponds to P 1 (ω; Φ(A)) = n ω Φ(A ) = Φ( n ω A ) = Φ(P 1 (ω; A)). Acknowledgement. The authors would lke to express ther cordal thanks to the referee for hs/her valuable suggestons. References [1] T. Ando, Concavty of certan maps on postve defnte matrces and applcatons to Hadamard products, Lnear Algebra Appl., 26 (1979), [2] K.V. Bhagwat and R. Subramanan, Inequaltes between means of postve operators, Math. Proc. Camb. Phl. Soc., 83 (1978), [3] R. Bhata, On the exponental metrc ncreasng property, Lnear Algebra Appl., 375 (2003), [4] R. Bhata, Postve Defnte Matrces, Prnceton Ser. Appl. Math., Prnceton Unversty Press, Prnceton, NJ, [5] R. Bhata and J. Holbrook, Remannan geometry and matrx means, Lnear Algebra Appl., 413 (2006), [6] G. Corach, H. Porta and L. Recht, Convexty of the geodesc dstance on spaces of postve operators, Illnos J. Math. 38(1994), [7] M. Fuj and R. Nakamoto, A geometrc mean n the Furuta nequalty, Sc. Math. Jpn., 55 (2002), [8] M. Fuj, J. Mćć Hot, J. Pečarć and Y. Seo, Recent developments of Mond-Pečarć method n operator nequaltes, Monographs n Inequaltes 4, Element, Zagreb, [9] T. Furuta, Invtaton to Lnear Operators, Taylor&Francs, London, [10] T. Furuta and Y. Seo, An applcaton of generalzed Furuta nequalty to Kantorovch type nequaltes, Sc. Math., 2, (1999), [11] T. Furuta, J. Mćć Hot, J. Pečarć and Y. Seo, Mond-Pečarć Method n Operator Inequaltes, Monographs n Inequaltes 1, Element, Zagreb, 2005.

12 12 Y. SEO [12] J. Lawson and Y. Lm, Karcher means and Karcher equatons of postve defnte operators, Trans. Amer. Math. Soc., Seres B, 1 (2014), [13] Y. Lm and M. Pálfa, Matrx power means and the Karcher mean, J. Funct. Anal., 262 (2012), [14] Y. Seo, On a reverse of Ando-Ha nequalty, Banach J. Math. Anal., 4(2010), [15] Y. Seo, Generalzed Pólya-Szegö type nequaltes for some non-commutatve geometrc means, Lnear Algebra Appl., 438 (2013), [16] A. C. Thompson, On certan contracton mappngs n a partally ordered vector space, Proc. Amer. Math. Soc., 14(1963), [17] T. Yamazak, Remannan mean and matrx nequaltes related to the Ando-Ha nequalty and chaotc order, Oper. Matrces., 6 (2012), [18] T. Yamazak and M. Yanagda, Characterzatons of chaotc order assocated wth Kantorovch nequalty, Sc. Math., 2 (1999), Department of Mathematcs Educaton, Osaka Kyoku Unversty, Asahgaoka, Kashwara, Osaka , Japan E-mal address: yuks@cc.osaka-kyoku.ac.jp