Some Quantum f-divergence Inequalities for Convex Functions of Self-adjoint Operators

Transcription

1 Bol. Soc. Paran. Mat. (3s.) v (019): c SPM ISSN on line ISSN in press SPM: doi:10.569/bsp.v37i Soe Quantu f-divergence Inequalities for Convex Functions of Self-adjoint Operators A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir abstract: In this paper, soe new inequalities for convex functions of self-adjoint operators are obtained. As applications, we present soe inequalities for quantu f-divergence of trace class operators in Hilbert Spaces. Key Words: Self-adjoint bounded linear operator, Trace inequality, Convex function, Quantu f-divergence. Contents 1 Introduction The Class of χ α -Divergences Dichotoy Class Divergences of Arioto-type Soe inequalities for convex functions of self-adjoint operators 18 3 Soe quantu f-divergence inequalities for convex functions Introduction Let (X,A) be a easurable space satisfying A > and µ be a σ-finite easure on (X,A). Let P be the set of all probability easures on (X,A) which are absolutely continuous with respect to µ. For P, Q P, let p = dp dq dµ and q = dµ denote the Radon-Nikody derivatives of P and Q with respect to µ. Two probability easures P, Q P are said to be orthogonal and we denote this by Q P if P ({q = 0}) = Q({p = 0}) = 1. Let f : [0, ) (, ] be a convex function that is continuous at 0, i.e., f (0) = li u 0 f (u). In 1963, I. Csiszár [6] introduced the concept of f-divergence as follows. Definition 1.1. Let P, Q P. Then I f (Q,P) = p(x)f X [ ] q(x) dµ(x), (1.1) p(x) is called the f-divergence of the probability distributions Q and P. 010 Matheatics Subject Classification: 47A63, 47A99. Subitted March 6, 016. Published March 10, Typeset by B S P M style. c Soc. Paran. de Mat.

2 16 A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir Reark 1.. Observe that, the integrand in the forula (1.1) is undefined when p(x) = 0. The way to overcoe this proble is to postulate for f as above that [ ] q(x) 0f 0 [ = q(x)li uf u 0 ( )] 1, x X. (1.) u We now give soe exaples of f-divergences that are well-known and often used in the literature (see also [3]) The Class of χ α -Divergences The f-divergences of this class, which is generated by the function χ α, α [1, ), defined by χ α (u) = u 1 α, u [0, ) have the for I f (Q,P) = X p q p 1 α dµ = X p 1 α q p α dµ. (1.3) Fro this class only the paraeter α = 1 provides a distance in the topological sense, naely the total variation distance V (Q,P) = X q p dµ. The ost proinent special case of this class is, however, Karl Pearson s χ -divergence χ (Q,P) = X q p dµ 1 that is obtained for α =. 1.. Dichotoy Class Fro this class, generated by the function f α : [0, ) R u 1 lnu for α = 0; 1 f α (u) = α(1 α) [αu+1 α uα ] for α R\{0,1}; 1 u+ulnu for α = 1; ( only the paraeter α = 1 f 1 (u) = ( u 1) ) provides a distance, naely, the Hellinger distance [ H(Q,P) = ( q ]1 p) dµ. X Another iportant divergence is the Kullback-Leibler divergence obtained for α = 1, ( ) q KL(Q,P) = qln dµ. p X

3 Quantu f-divergence Inequalities for Self-adjoint Operators Divergences of Arioto-type This class is generated by the functions [ ] α α 1 (1+u α ) 1 α 1 α 1 (1+u) for α (0, )\{1}; Ψ α (u) := (1+u)ln+ulnu (1+u)ln(1+u) for α = 1; 1 1 u for α =. It has been shown in [19] that this class provides the distances [I Ψα (Q,P)] in(α, α) 1 for α (0, ) and 1 V (Q,P) for α =. For f continuous convex on [0, ) we obtain the -conjugate function of f by ( ) 1 f (u) = uf, u (0, ) u and f (0) = li u 0 f (u). It is also known that if f is continuous convex on [0, ) then so is f. The following two theores contain the ost basic properties of f-divergences. For their proofs we refer the reader to Chapter 1 of [17] (see also [3]). Theore 1.3 (Uniqueness and Syetry Theore). Let f,f 1 be continuous convex on [0, ). We have I f1 (Q,P) = I f (Q,P), for all P,Q P if and only if there exists a constant c R such that for any u [0, ). f 1 (u) = f (u)+c(u 1), Theore 1.4 (Range of Values Theore). Let f : [0, ) R be a continuous convex function on [0, ). For any P,Q P, we have the double inequality f (1) I f (Q,P) f (0)+f (0). (1.4) (i) If P = Q, then the equality holds in the first part of (1.4). If f is strictly convex at 1, then the equality holds in the first part of (1.4) if and only if P = Q; (ii) If Q P, then the equality holds in the second part of (1.4). If f (0)+f (0) <, then equality holds in the second part of (1.4) if and only if Q P.

4 18 A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir The following result is a refineent of the second inequality in Theore 1.4 (see [3, Theore 3]). Theore 1.5. Let f be a continuous convex function on [0, ) with f (1) = 0 (f is noralised) and f (0)+f (0) <. Then for any Q,P P. 0 I f (Q,P) 1 [f (0)+f (0)]V (Q,P) (1.5) For other inequalities for f-divergence see [],[7]-[1] and [13] Motivated by the above results, in this paper we obtain soe new inequalities for quantu f- divergence of trace class operators in Hilbert spaces. It is shown that for noralised convex functions it is nonnegative. Soe upper bounds for quantu f-divergence in ters of variational and χ -distance are provided. Applications for soe classes of divergence easures such as Uegaki and Tsallis relative entropies are also given. In what follows we recall soe facts we need concerning the trace of operators and quantu f-divergence for trace class operators in infinite diensional coplex Hilbert spaces.. Soe inequalities for convex functions of self-adjoint operators Suppose that I is an interval of real nubers with interior I and f : I R is a convex function on I. Then f is continuous on I and has finite left and right derivatives at each point of I. Moreover, if x,y I and x < y, then f (x) f +(x) f (y) f +(y), which shows that both f and f + are nondecreasing function on I. It is also known that a convex function ust be differentiable except for at ost countably any points. Let A be a self-adjoint operator on the coplex Hilbert space (H,, ) with Sp(A) [,M]forsoerealnubers < M and let {E λ } λ be its spectralfaily. Then for any continuous function f : [,M] R, it is known that we have the following spectral representation in ters of the Rieann-Stieltjes integral (see for instance [14, p. 57]): f(a)x,y = f(λ)d( E λ x,y ), and for any x,y H. f(a)x = f(λ) d E λ x, Theore.1. Let A be a self-adjoint operator on the coplex Hilbert space (H,, ) and Sp((A) [,M] I where I is an interval. If the function f : I R

5 Quantu f-divergence Inequalities for Self-adjoint Operators 19 is convex on I, then f(a)x,x f(m) f() Ax,x [ ] M f(t)dt Mf(M) f() x. Proof. Without loosing the generality, we can assue that f is differentiable on I. By the convexity of f on I, we have f(t) f(s) f (s)(t s), (.1) for all t,s [,M] I. If {E t } t R is the spectral faily of A then the apping [,M] t E t x,x is onotonic real apping for any x H. If we integrate (.1) over t [,M] with the integraler E t x,x then we get f(t)d E t x,x f(s) d E t x,x [ f (s) td E t x,x s d E t x,x ] for any s [,M] and x H. Using the spectral representation theore for self-adjoint operators we get f(a)x,x f(s) x f (s) [ Ax,x s x ] (.) for any s [,M] and x H. Now if we integrate (.) over s on [,M] and divide by, we get f(a)x,x x 1 f(s)ds ( ) 1 M Ax, x sf (s)ds =: K (.3) However, we know that and 1 M f (s)ds = f(m) f() [ 1 M ] sf 1 M (s)ds = Mf(M) f() f(s)ds.

6 130 A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir Therefore K = Ax,x f(m) f() = Ax,x f(m) f() + x By (.3) we then get f(s)ds f(a)x,x f(m) f() Ax,x [ x Mf(M) f() ] M f(s)ds [ x Mf(M) f() ] M f(s)ds x f(s)ds x [Mf(M) f()] [ ] = x M f(s)ds Mf(M) f(). Reark.. Since f(t) = t 1 is convex function for t > 0, by above theore we have MA 1 x,x + Ax,x Mln(M ) for x = 1. Theore.3. Let A and B be two self-adjoint operators on the Hilbert space (H,, ) with Sp(A),Sp(B) [,M] I and f : I R a continuously differentiable convex function on I. Then f(a)x,x y f(b)y,y x ( ) Ax,x f (B) x f (B)B y,y for any x,y H. Proof. Since f : I R is convex on I we have f(t) f(s) f (s)(t s), for any t,s I. (.4) Let {E t } t R be the spectral faily of A and {F t } t R the spectral faily of B. The appings [,M] t E t x,x and [,M] s F s y,y are onotonic real apping for any x,y H.

7 Quantu f-divergence Inequalities for Self-adjoint Operators 131 Integrating (.4) over t on [,M] with integraler E t x,x we get f(t)d E t x,x f(s) d E t x,x ( f (s) td E t x,x s which is equivalent to f(a)x,x x f(s) f (s) Ax,x sf (s) x, d E t x,x for any s [,M] and x H. Integrating above inequality over s on [,M] with integraler F s x,x we then get f(a)x,x Ax,x which is equivalent to d F s y,y x f (s)d F s y,y x f(a)x,x y f(b)y,y x f(s)d F s y,y sf (s)d F s y,y Ax,x f (B)y,y x f (B)By,y. We know that ( ) Ax,x f (B) x f (B)B y,y = ( Ax,x I x B ) f (B) y,y and f(a)x,x y f(b)y,y x = ( f(a)x,x I x,x f(b))y,y On the other hand, we have ( Ax,x I x B ) f (B) f(a)x,x I x f(b) ) and f(a)x,x I x f(b) f (B)( Ax,x I x B) for all x H. We obtain the desired result. Reark.4. Let B = n α jx j in above theore and α j 0, j {1,...,n} such that n α j = 1, then we have

8 13 A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir n f(a)x,x I x f α j X j n n f α j X j Ax,x I x α j X j. Multiply above inequality by α k and suing on k {1,...,n}, we get n n α k f(x k )x,x I x f α j X j i=1 ( n n f α j X j α k X k )x,x n I x α j X j. k=1 Equivalently, n n α k f(x k )x,x I f α j X j i=1 ( n n f α j X j α k X k )x,x I k=1 n α j X j for x = 1 which is a Jensen type inequality. 3. Soe quantu f-divergence inequalities for convex functions Let (H,, ) be a coplex Hilbert space and {e i } an orthonoral basis of H. We say that A B(H) is a Hilbert-Schidt operator if Ae i <. (3.1) It is well know that, if {e i } and {f j } j J are orthonoral bases for H and A B(H) then Ae i = Af j = A f j (3.) j I j I showing that the definition (3.1) is independent of the orthonoral basis and A is a Hilbert-Schidt operator iff A is a Hilbert-Schidt operator. Let B (H) the set of Hilbert-Schidt operators in B(H). For A B (H) we define ( ) 1/ A := Ae i (3.3)

9 Quantu f-divergence Inequalities for Self-adjoint Operators 133 for {e i } an orthonoral basis of H. This definition does not depend on the choice of the orthonoral basis. Using the triangle inequality in l (I), one checks that B (H) is a vector space and that is a nor on B (H), which is usually called in the literature as the Hilbert-Schidt nor. Denote the odulus of an operator A B(H) by A := (A A) 1/. Because A x = Ax for all x H, A is Hilbert-Schidt iff A is Hilbert- Schidt and A = A. Fro (3.) we have that if A B (H), then A B (H) and A = A. We define the trace of a trace class operator A B 1 (H) to be Tr(A) := Ae i,e i, (3.4) where {e i } an orthonoral basis of H. Note that this coincides with the usual definition of the trace if H is finite-diensional. We observe that the series (3.4) converges absolutely and it is independent fro the choice of basis. Utilising the trace notation we obviously have that A,B = Tr(B A) = Tr(AB ) and A = Tr(A A) = Tr ( A ) for any A, B B (H). If A 0 and P B 1 (H) with P 0, then 0 Tr(PA) A Tr(P). (3.5) Indeed, since A 0, then Ax,x 0 for any x H. If {e i } an orthonoral basis of H, then 0 AP 1/ e i,p 1/ e i A P 1/ e i = A Pei,e i for any i I. Suing over i I we get 0 AP 1/ e i,p 1/ e i A Pe i,e i = A Tr(P) and since AP 1/ e i,p 1/ e i = P 1/ AP 1/ e i,e i ( = Tr P 1/ AP 1/) = Tr(PA) we obtain the desired result (3.5). This obviously iply the fact that, if A and B are self-adjoint operators with A B and P B 1 (H) with P 0, then Tr(PA) Tr(PB). (3.6)

10 134 A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir Now, if A is a self-adjoint operator, then we know that Ax,x A x,x for any x H. This inequality follows by Jensen s inequality for the convex function f (t) = t defined on a closed interval containing the spectru of A. If {e i } is an orthonoral basis of H, then Tr(PA) = AP 1/ e i,p 1/ e i AP 1/ e i,p 1/ e i (3.7) A P 1/ e i,p 1/ e i = Tr(P A ), for any A a self-adjoint operator and P B + 1 (H) := {P B 1(H) with P 0}. For the theory of trace functionals and their applications the reader is referred to []. For soe classical trace inequalities see [4], [5], [18] and [3], which are continuations of the work of Bellan [1]. On coplex Hilbert space (B (H),, ), where the Hilbert-Schidt inner product is defined by U,V := Tr(V U), U, V B (H), for A, B B + (H) consider the operators L A : B (H) B (H) and R B : B (H) B (H) defined by L A T := AT and R B T := TB. We observe that they are well defined and since L A T,T = AT,T = Tr(T AT) = Tr ( ) T A 0 and ( ) R B T,T = TB,T = Tr(T TB) = Tr T B 0 for any T B (H), they are also positive in the operator order of B(B (H)), the Banach ( algebra of all bounded operators on B (H) with the nor where T = Tr T ), T B (H). ( Since Tr X ) ( = Tr X ) for any X B (H), then also ( ) Tr(T AT) = Tr T A 1/ A 1/ T A = Tr( 1/ T ) = Tr (( = Tr A ) A T) 1/ 1/ T ( T ) A 1/ ( ( A 1/ T) ) = Tr

11 Quantu f-divergence Inequalities for Self-adjoint Operators 135 for A 0 and T B (H). We observe that L A and R B are coutative, therefore the product L A R B is a self-adjoint positive operator in B(B (H)) for any positive operators A,B B(H). For A,B B + (H) with B invertible, we define the Araki transfor A A,B : B (H) B (H) by A A,B := L A R B 1. We observe that for T B (H) we have A A,B T = ATB 1 and A A,B T,T = ATB 1,T = Tr( T ATB 1). Observe also, by the properties of trace, that Tr ( T ATB 1) = Tr (B 1/ T A 1/ A 1/ TB 1/) = Tr ((A 1/ TB 1/) ( A 1/ TB 1/)) ( A ) = Tr 1/ TB 1/ giving that A A,B T,T = Tr( A 1/ TB 1/ ) 0 (3.8) for any T B (H). We observe that, by the definition of operator order and by (3.8) we have r1 B(H) A A,B R1 B(H) for soe R r 0 if and only if ( A rtr T ) Tr( ) 1/ TB 1/ ( RTr T ) (3.9) for any T B (H). We also notice that a sufficient condition for (3.9) to hold is that the following inequality in the operator order of B(H) is satisfied r T A 1/ TB 1/ R T (3.10) for any T B (H). Let U be a self-adjoint linear operator on a coplex Hilbert space (K;, ). The Gelfand ap establishes a -isoetrically isoorphis Φ between the set C(Sp(U)) of all continuous functions defined on the spectru of U, denoted Sp(U), and the C -algebra C (U) generated by U and the identity operator 1 K on K as follows: For any f,g C(Sp(U)) and any α,β C we have (i) Φ(αf +βg) = αφ(f)+βφ(g); (ii) Φ(fg) = Φ(f)Φ(g) and Φ ( f) = Φ(f) ; (iii) Φ(f) = f := sup t Sp(U) f (t) ; (iv) Φ(f 0 ) = 1 K andφ(f 1 ) = U,wheref 0 (t) = 1andf 1 (t) = t,fort Sp(U). With this notation we define f (U) := Φ(f) for all f C(Sp(U))

12 136 A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir and we call it the continuous functional calculus for a self-adjoint operator U. If U is a self-adjoint operator and f is a real valued continuous function on Sp(U), then f (t) 0 for any t Sp(U) iplies that f (U) 0, i.e. f (U) is a positive operator on K. Moreover, if both f and g are real valued functions on Sp(U) then the following iportant property holds: f (t) g(t) for any t Sp(U) iplies that f (U) g(u) (P) in the operator order of B(K). Letf : [0, ) Rbe acontinuousfunction. Utilisingthe continuousfunctional calculus for the Araki self-adjoint operator A Q,P B(B (H)) we can define the quantu f-divergence for Q,P D 1 (H) := {P B 1 (H), P 0 with Tr(P) = 1 } and P invertible, by D f (Q,P) := f (A Q,P )P 1/,P 1/ ( = Tr P 1/ f (A Q,P )P 1/). (3.11) If we consider the continuous convex function f : [0, ) R, with f (0) := 0 and f (t) = tlnt for t > 0 then for Q,P D 1 (H) and Q,P invertible we have D f (Q,P) = Tr[Q(lnQ lnp)] =: U (Q,P), which is the Uegaki relative entropy. If we take the continuous convex function f : [0, ) R, f (t) = t 1 for t 0 then for Q,P D 1 (H) with P invertible we have D f (Q,P) = Tr( Q P ) =: V (Q,P), where V (Q,P) is the variational distance. If we take f : [0, ) R, f (t) = t 1 for t 0 then for Q,P D 1 (H) with P invertible we have D f (Q,P) = Tr ( Q P 1) 1 =: χ (Q,P), which is called the χ -distance Let q (0,1) and define the convex function f q : [0, ) R by f q (t) = 1 tq 1 q. Then D fq (Q,P) = 1 Tr( Q q P 1 q), 1 q which is Tsallis relative entropy. ( ), If we consider the convex function f : [0, ) R by f (t) = 1 t 1 then ( D f (Q,P) = 1 Tr Q 1/ P 1/) =: h (Q,P), which is known as Hellinger discriination.

13 Quantu f-divergence Inequalities for Self-adjoint Operators 137 If we take f : (0, ) R, f (t) = lnt then for Q,P D 1 (H) and Q,P invertible we have D f (Q,P) = Tr[P (lnp lnq)] = U (P,Q). The reader can obtain other particular quantu f-divergence easures by utilizing the noralized convex functions fro Introduction, naely the convex functions defining the dichotoy class, Matsushita s divergences, Puri-Vincze divergences or divergences of Arioto-type. We oit the details. In the iportant case of finite diensional space H and the generalized inverse P 1, nuerous properties of the quantu f-divergence, ostly in the case when f is operator convex, have been obtained in the recent papers [15], [16], [0], [1] and the references therein. In the following theore we apply the sae proof of Theore.3. Theore 3.1. Let A Q,P be Araki self-adjoint operator on B(B (H)) and Sp(A Q,P ) [,M] I where I is an interval. If the function f : I R is convex on I, then f(a Q,P )T,T f(m) f() A Q,PT,T [ ] M f(t)dt Mf(M) f() T. for T B (H). Reark 3.. Let T = P 1 in above theore we get f(a Q,P )P 1 1,P f(m) f() A Q,PP 1 1,P [ f(t)dt Mf(M) f() ]. for P 1 = 1. On the other hand, by (3.11) we have [ D f (Q,P) f(m) f() + ] f(t)dt Mf(M) f() (3.1) since Q D 1 (H). If we take the continuous convex function f : [0, ) R, f(t) = t 1 for t 0 then for P,Q D 1 (H) with P invertible we have the following for 1 D f (Q,P) = Tr( Q P ) = V(Q,P) (M + 1). In the following theore we apply the sae proof of Theore.1.

14 138 A. Taghavi, V. Darvish, H. M. Nazari and S. S. Dragoir Theore 3.3. LetA Q,P andb V,U be two Arakiself-adjoint operators on B(B (H)) with Sp(A Q,P ),Sp(B V,U ) [,M] I and f : I R a continuously differentiable convex function on I. Then f(a Q,P )T,T S f(b V,U )S,S T ) ( A Q,P T,T f (B V,U ) T f (B V,U )B V,U S,S for any x,y H. Reark 3.4. Let T = P 1 and S = U 1 in above theore we get f(a Q,P )P 1,P 1 f(b V,U )U 1,U 1 ( A Q,P P 1,P 1 f (B V,U ) f (B V,U )B V,U ) U 1 1,U for P 1 = U 1 = 1. Fro above inequality we have D f (Q,P) D f (V,U) Tr ( ) f (B V,U )U f (B V,U )V (3.13) since Q D 1 (H). Put f(t) = t 1 for t 0 in above inequality (3.13) we get χ (Q,P) χ (V,U) Tr(B V,U U B V,U V) where P,Q,U,V D 1 (H) and P,U are invertible. Equivalently, we have χ (Q,P) χ (V,U) Tr ( V V U 1). References 1. R. Bellan, Soe inequalities for positive definite atrices, in: E.F. Beckenbach (Ed.), General Inequalities, Proceedings of the nd International Conference on General Inequalities, Birkhäuser, Basel, 89 90, (1980).. P. Cerone and S. S. Dragoir, Approxiation of the integral ean divergence and f- divergence via ean results, Math. Coput. Modelling 4, 07 19, (005). 3. P. Cerone, S. S. Dragoir and F. Österreicher, Bounds on extended f-divergences for a variety of classes, Preprint, RGMIA Res. Rep. Coll. 6, Article 5, (003). [ONLINE: 4. D. Chang, A atrix trace inequality for products of Heritian atrices, J. Math. Anal. Appl. 37, 71 75, (1999). 5. I. D. Coop, On atrix trace inequalities and related topics for products of Heritian atrix, J. Math. Anal. Appl. 188, , (1994). 6. I. Csiszár, Eine inforationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffschen Ketten, (Geran) Magyar Tud. Akad. Mat. Kutató Int. Közl. 8, , (1963).

15 Quantu f-divergence Inequalities for Self-adjoint Operators S. S. Dragoir, An upper bound for the Csiszár f-divergence in ters of the variational distance and applications, Panaer. Math. J. 1, 43 54, (00). 8. S. S. Dragoir, Upper and lower bounds for Csiszár f-divergence in ters of Hellinger discriination and applications, Nonlinear Anal. Foru 7, 1 13, (00). 9. S. S. Dragoir, New inequalities for Csiszár divergence and applications, Acta Math. Vietna. 8, , (003). 10. S. S. Dragoir, A generalized f-divergence for probability vectors and applications, Panaer. Math. J. 13, 61 69, (003). 11. S. S. Dragoir, A converse inequality for the Csiszár Φ-divergence, Tasui Oxf. J. Math. Sci. 0, 35 53, (004). 1. S. S. Dragoir, Soe general divergence easures for probability distributions, Acta Math. Hungar. 109, , (005). 13. S. S. Dragoir, A refineent of Jensen s inequality with applications for f-divergence easures, Taiwanese J. Math. 14, , (010). 14. G. Helberg, Introduction to Spectral Theory in Hilbert Space, John Wiley, New York, (1969). 15. F. Hiai, Fuio and D. Petz, Fro quasi-entropy to various quantu inforation quantities, Publ. Res. Inst. Math. Sci. 48, 55 54, (01). 16. F. Hiai, M. Mosonyi, D. Petz and C. Bény, Quantu f-divergences and error correction, Rev. Math. Phys. 3, , (011). 17. F. Liese and I. Vajda, Convex Statistical Distances, Teubuer Texte zur Matheatik, Band 95, Leipzig, (1987). 18. H. Neudecker, A atrix trace inequality, J. Math. Anal. Appl. 166, , (199). 19. F. Österreicher and I. Vajda, A new class of etric divergences on probability spaces and its applicability in statistics, Ann. Inst. Statist. Math. 55, , (003). 0. D. Petz, Fro quasi-entropy, Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 55, 81 9, (01). 1. D. Petz, Fro f-divergence to quantu quasi-entropies and their use, Entropy 1, , (010).. B. Sion, Trace Ideals and Their Applications, Cabridge University Press, Cabridge, (1979). 3. Y. Yang, A atrix trace inequality, J. Math. Anal. Appl. 133, , (1988). A. Taghavi, V. Darvish, H. M. Nazari, Departent of Matheatic,s Faculty of Matheatical Sciences, University of Mazandaran, P. O. Box , Babolsar, Iran. E-ail address: taghavi@uz.ac.ir vahid.darvish@ail.co,.nazari@stu.uz.ac.ir and S. S. Dragoir, Matheatics, School of Engineering and Science, Victoria University P. O. Box 1448, Melbourne City, MC 8001, Australia. E-ail address: sever.dragoir@vu.edu.au