Some operator monotone functions related to Petz-Hasegawa s functions

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1 Some operaor monoone funcions relaed o Pez-Hasegawa s funcions Masao Kawasaki and Masaru Nagisa Absrac Le f be an operaor monoone funcion on [, ) wih f() and f(). If f() is neiher he consan funcion nor he ideniy funcion, hen (f() f(a))(f () f (b)) is also operaor monoone on [, ), where a, b and Inroducion f () f(). We call a real coninuous funcion f() on an inerval I operaor monoone on I (in shor, f P(I) ), if A B implies f(a) f(b) for any self-adjoin marices A, B wih heir specrum conaind in I. In his paper, we consider only he case I [, ) or I (, ). We denoe f P + (I) if f P(I) saisfies f() for any I. Le H + be he upper half-plane of C, ha is, H + {z C Imz > } {z C z >, < arg z < π}, where Imz (resp. arg z) means he imaginary par (resp. he argumen) of z. As Loewner s heorem, i is known ha f is operaor monoone on I if and only if f has an analyic coninuaion o ha maps H + ino iself and also has an analyic coninuaion o he lower half-plane H ( H + ), obained by he reflecion across I (see [],[] ). D. Pez [5] proved ha an operaor monoone funcion f : [, ) [, ) saisfying he funcional equaion f() f( ) is relaed o a Morozova-Chensov funcion which gives a monoone meric on he manifold of n n densiy marices. In he work [6], he concree funcions (Pez-Hasegawa s funcions) ( ) f a () a( a) ( a )( a ) ( < a < ) appeared and heir operaor monooniciy was proved. V.E.S. Szabo inroduced an ineresing idea for checking heir operaor monooniciy in [7]. We use a

2 similar idea as Szabo s in our argumen. M. Uchiyama [8] proved he operaor monooniciy of he following exended funcions: ( p a p )( p b p ) for < p < and a, b >. I is well known ha he funcion p ( p ) is operaor monoone as Loewner-Heinz s inequaliy. The main resul of his paper is as follows: Theorem. Le a and b be non-negaive real. If f P + [, ) and boh f and f are no consan, hen (f() f(a))(f () f (b)) is operaor monoone on [, ), where f () f(). Proof of Main resul The following saemen was proved by M. Uchiyama [8]. Here we prove i based on he fac ha any operaor monoone funcion is a Pick funcion, bu his is essenially same as his proof. Proposiion. Le f P(, ) be no consan and a be posiive real. Then we have ha a g() f() f(a) is operaor monoone on [, ). Proof. Since f(h + ) H +, Im(f(z) f(a)) Im(f(z)) > for all z H +. z a Therefore g(z) f(z) f(a) is holomorphic on H +. Since f is no consan, f (a). We also have a lim g() (> ) if f() exiss f(a) f(). + oherwise This means g(z) is coninuous on H + [, ) and we have g([, )) [, ). By Loewner s heorem, we have he following inegral represenaion of f: for z H + (, ) f(z) α + βz + where ν is a posiive measure on (, ) such ha ( x + z + x x )dν(x) (α R, β ), + x dν(x) <. +

3 Using his relaion, we have g(z). β + (x + z)(x + a) dν(x) If we show ha Im(g(z)) for any z H +, hen g P + [, ). We remark ha So we have Im(β + z H + < argz < π < arg(x + z) < π (x [, )) π < arg < π (x [, )) (x + z)(x + a) Im < (x [, )). (x + z)(x + a) (x + z)(x + a) dν(x)) This shows ha Im(g(z)). Im dν(x) <. (x + z)(x + a) For f P[, ), we have he following inegral represenaion: where β and f(z) f() + βz + λz z + λ dw(λ) λ dw(λ) <. + λ (z H + [, )), When f() (i.e., f P + [, )), f(z) can be approximaed by where each f i (z) saisfies ha n f i (z), i < arg f i (z) arg z whenever < arg z < π. So we have < arg f(z) arg z whenever < arg z < π. By using elemenary geomery, i easily holds ha arg(z z ) π + arg z for any z H +. So we can ge he following saemen: Lemma 3. For any z H + and l >, we have arg z < arg(z l) < π + arg z if z > l. 3

4 Now we can prove he following heorem and remark ha Theorem easily follows from his: Theorem 4. Le f, g P + [, ) and boh f and g be non-consan. If f()g() is operaor monoone on [, ), hen (f() f(a))(g() g(b)) is also operaor monoone on [, ) for any a, b. Proof. By he assumpion we can consider he funcion h(z) (z a)(z b) (f(z) f(a))(g(z) g(b)) z H +. I is clear ha h(z) is holomorphic on H +. We may consider values, by aking he limi, h(a) a b f (a)(g(a) g(b)) and h(b) b a g (b)(f(b) f(a)). So we have h([, )) [, ). We assume ha f(z) and g(z) are coninuous on he closure H + of H + and f() f(a) and g() g(b) for any (, ). Then h(z) is coninuous on H +. z a Since f(z) f(a) and z b g(z) g(b) belong o P +[. ) by Proposiion, i is clear ha arg h(z) if arg z π. In he case z (, ), i.e., z > and arg z π, we have arg h(z) arg(z a) arg(f(z) f(a)) + arg(z b) arg(g(z) g(b)) π arg f(z) + π arg g(z) π arg z π (since arg f(z) + arg g(z) arg z ). So arg h(z) π. In he case z H + saisifying z > max{a, b}, i holds ha by above lemma. Since arg z < arg(z a), arg(z b) < π + arg z arg h(z) arg(z a) arg(f(z) f(a)) + arg(z b) arg(g(z) g(b)) π + arg z arg f(z) + π + arg z arg g(z) π + arg z arg f(z) arg g(z) π, we have < arg h(z) < π. 4

5 For r >, we define H(r) {z C z r, Imz }. r > l max{a, b}, we can ge Whenever arg h(z) π on he boundary of H(r). Since h(z) is holomorphic on H(r), Imh(z) is harmonic on H(r). Because Imh(z) on he boundary of H(r), we have h(h(r)) H + by he minimum principle of he harmonic funcion. This implies h(h + ) h( r>l H(r)) r>l h(h(r)) H +, and h P + [, ). In general case, we se f()g() F () and f() f() F () ( ). By he relaion f()g(), we have f P + [, ). We define he funcion f p, f p and g p ( < p < ) as follows: f p (z) f(z p ), fp (z) f(z p ), and g p (z) ( f p ) (z) for z H +. Then we have f p, g p P + [, ) and h p (z) z f p (z) zf (zp ) f(z p ) z p g(z p ) (z a)(z b) (f p (z) f p (a))(g p (z) g p (b)) is holomorphic on H + and coninuous on H +. By he fac f p()g p () F ( p ) is operaor monoone on [, ), h p () becomes operaor monoone on [, ). Since we have h p () (f p () f p (a))(g p () g p (b)) (f( p ) f(a p ))( p g( p ) b p g(b p )) lim h p() h(). p So we can ge he operaor monooniciy of h(). for, We can generalize his resul for many operaor monoone funcions under some assumpion([4]). We remark ha, for f P[, ), f belongs o P + [, ), where we pu f () f() f(). Le g P + [, ) and f ()g() be operaor monoone on 5

6 [, ). Under he assumpion ha f and g are no consan, we have (f () f (a))(g() g(b)) (f() f(a))(g() g(b)) P +[, ) for any a, b. Corollary 5. Le f P + (, ) and boh f and f be no consan. For any a >, we define h a () ( a)( a ) (f() f(a))(f () f (a )) (, ). Then we have () h a is operaor monoone on (, ). () f() f( ) implies h a () h a ( ). (3) a and f( ) f() implies h () h ( ). Proof. We can direcly prove () from heorem 3. Because we can compue h a ( ) ( a)( a ) (f( ) f(a))(f ( ) f (a )) ( a)( a ) (f( ) f(a))(f ( ) f (a )), (f( ) f(a))(f ( ) f (a )) (f() f(a))(f () f (a )) (f( ) f(a))(/f( ) /af(a )) (f() f(a))(/f() /af(a )) if i holds f() f( ) or a, f( ) f(). So we have () and (3). Example 6. Using his corollary, we can repeaedly consruc an operaor monoone funcion h() on [, ) saisfying he relaion h( ) >. (*) If we choose p ( < p < ) as f() in Corollary 5(3), ( ) ( p )( p ). 6

7 If we choose ( ) ( p )( p ) as f() in Corollary 5(), a ( ) ( p )( p ) (a ) (a p )(a p ) ( p )( p ) ( ) ) a a(a p )(a p ) (a ) for a >. If we choose p + p ( < p < ) as f() in Corollary 5(), a p + p a p a p a (a > ) p + p a p + a p (cosh(log ) cosh(log a)) cosh(log ) cosh(log + log( p + p ) log(a p + a p )). These funcions, h P + [, ), saisfy he relaion (*). References [] R. Bhaia, Marix Analysis, Springer, 996. [] F. Hiai, Marix Analysis: Marix monoone funcions, marix means, and majorizaion, Inerdecip. Inform. Sci. 6 () [3] E.A. Morozova, N.N. Chensov, Markov invarian geomery on sae manifolds, Iogi Nauki i Techniki 36 (99) 69, Translaed in J. Sovie Mah. 56(99) [4] M. Kawasaki, M. Nagisa, Transforms on operaor monoone funcions, in preparaion. [5] D. Pez, Monoone meric on marix spaces, Linear Algebra Appl. 44 (996) [6] D. Pez, H. Hasegawa, On he Riemannian meric of α-enropies of densiy marices, Le. Mah. Phys. 38 (996) 5. [7] V.E.S. Szabo, A class of marix monoone funcions, Linear Algebra Appl. 4 (7) [8] M. Uchiyama, Majorizaion and some operaor monoone funcions, Linear Algebra Appl. 43() Graduae School of Science, Chiba Universiy, Chiba 63-85, Japan address: kawasaki 64abc@yahoo.co.jp address: nagisa@mah.s.chiba-u.ac.jp 7