Some means inequalities for positive operators in Hilbert spaces

In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities. Moreover, we establish a matrix version of Heinz inequality for the Hilbert-Schmidt norm. Finally, we introduce a weighted multivariate geometric mean and show that the weighted multivariate operator geometric mean possess several attractive properties and means inequalities.


Introduction
Since Heinz proved a series of useful norm inequalities, which are closely related to the Cordes inequality and the Furuta inequality, in , many researchers have devoted themselves to sharping the Heinz inequalities and extending the Heinz norm inequalities to more general cases with the help of a Bernstein type inequality for nonselfadjoint operators, the convexity of norm functions, the Jensen functional and its properties, the Hermite-Hadamard inequality, and so on. With this kind of research, the study of various means inequalities, such as the geometric mean, the arithmetic mean, the Heinz mean, arithmetic-geometric means, and Arithmetic-Geometric-Harmonic (A-G-H) weighted means, has received much attention and development too. For recent interesting work in this area, we refer the reader to [-] and references therein.
Based on [-], in this paper, we are concerned with the further refinements of the geometric mean and the Heinz mean for operators in Hilbert spaces. Our purpose is to derive some new generalizations of Heinz operator inequalities by refining the ordering relations among Heinz means with different parameters, and of the geometric mean by investigating geometric means of several operator variables in a weighted setting. Moreover, we will obtain a matrix version of the Heinz inequality for the Hilbert-Schmidt norm.
as the geometric mean of T and S. When ν = / we write T∇S and T S in short, respectively. We refer the reader to Kubo and Ando [] for more information on the means of positive linear operators. Recall that, for any a, b ≥ , the number is called the Heinz mean H ν (a, b) of a and b. It is clear that For any T, S ∈ B ++ (H) and ν ∈ [, ], the operator is called the Heinz operator mean of T and S. Clearly, that is, the Heinz operator mean interpolates between the geometric mean and the arithmetic mean.

Improved Heinz means inequalities
In a very recent work [], we establish the following inequalities: and In this section, we improve the result and give two theorems as follows.
Theorem . Suppose T, S ∈ B ++ (H), and let s, t ∈ [, ] satisfy In view of the Taylor series of cosh x, we deduce that Let a = e x , b = e y . Then Taking a = x and b =  in the inequality (.), we get With the positive operator The proof is completed.
For the functions F ν : we have the following result.
Theorem . Suppose T, S ∈ B ++ (H) and let s, t ∈ [, ] satisfy Then Proof Writing α =  -s, β =  -t, we have It follows from the Taylor series of sinh x that .
Put a = e x , b = e y . Then we get Letting a = x and b =  in inequality (.), we see that Therefore,

Heinz inequality for the Hilbert-Schmidt norm
In this section, we let M n be the Hilbert space of n × n complex matrices and let · stand for any unitarily invariant norm on M n , i.e. UTV = T for all T ∈ M n and for all unitary matrices U, V ∈ M n . We suppose that T, S, X ∈ M n with T and S being positive semidefinite. For T = [a ij ] ∈ M n , the Hilbert-Schmidt norm of T is defined by It is well known that the Hilbert-Schmidt norm is unitarily invariant. Next, we prove the following matrix version of Heinz inequality for the Hilbert-Schmidt norm.
Proof Noting that T and S are positive semidefinite, we know by the spectral theorem that there exist unitary matrices U, V ∈ M n such that Then we have Hence, By a similar argument to the above, we deduce that By virtue of the inequalities (.) and (.), we obtain n i,j= Thus, the proof is completed.

The inductive weighted geometric means and means inequalities
are positive and invertible. Thus, the concavity of P F implies that For λ → , by continuity, we get P F (S  , . . . , S k+ ) ≥ P F (T  , . . . , T k+ ).
Letting ν → , we get the conclusion.
Theorem . Suppose that F : D k + → B(H) sa is a regular, concave, and positively homogeneous. Then the perspective function P F satisfies the property of congruence invariance: for any invertible operator W on H.
Proof It follows from Theorem . of [] that the perspective function P F is concave. Moreover, since F is positively homogeneous, it is easy to prove that P F is also positively homogeneous. Hence, by Proposition . in [], we get the conclusion.
for positive T  , . . . , T k , where β is the weight associated to T  , . . . , T k . (iii) Define the geometric mean G α k+ : D k+ where Particularly, the geometric means of two variables coincide with the weighted geometric means of two variables T  α  T  in the sense of Kubo and Ando [], where α = (α  , α  ) satisfy α  + α  = . In the above procedure, α i is determined by β i and α k+ in the following sense: Conversely for fixed α = (α  , . . . , α k+ ), we can set and hence trace back to the case of k = . Therefore for fixed weight we can define the corresponding weighted geometric mean. Proof By the definition of G α k , we know that G α k for each k = , , . . . is the perspective of a regular positively homogeneous map. Therefore, G α k are regular and positively homogeneous. Moreover, since G α k+ is the perspective of (G β k ) -α k+ , we see that (.) holds. Next, we prove that G α k is concave. Clearly, G α  is concave. Assume that G β k is concave for some k and the corresponding weight β. For α k+ ∈ [, ], the map x → x -α k+ is operator monotone (increasing) and operator concave. Then we have where T = (T  , . . . , T k ), S = (S  , . . . , S k ). So the auxiliary mapping is concave. Then by Theorem . in [] we see that its perspective G α k+ is also concave. By induction, we know that G α k is concave for all k = , , . . . .

Remark . A similar analysis to Theorem . in []
shows that the above conditions uniquely determine the Geometric means G α k for k = , , . . . by setting G α  (T) = T.
The means G α k constructed as above have the following properties: (P) (consistency with scalars) G α Proof If T  and T  commute, then Hence, (P) holds for k = , . Now assume that (P) holds for some k > . Since we see that (P) also holds for k + . By induction, we know that (P) holds for k = , , . . . . It is easy to verify (P) holds for k =  and k = . Assume that (P) holds for some k > . Then we have By induction, we get Hence (P) is true.
(P) and (P) follow from Theorems . and .. Clearly, (P) is true for k =  and k = . Assume that (P) is true for some k > . Then we have By induction, we get which verifies (P). The A-G-H weighted mean inequality, i.e. the arithmetic-geometric-harmonic weighted mean inequality reads for arbitrary (T  , . . . , T k ) ∈ D k + . Firstly, we show the second inequality. It is easy to see the second inequality holds for k = . Assume the inequality holds for some k. Then, by virtue of X p ≤  + p(X -) for X ∈ D  (the set of positive operators) and p ∈ [, ], we obtain Hence the property (P) holds. For T ∈ D  and p ∈ R, we have det T p = (det T) p due to det T = exp(Tr log T). For k =  and k = , (P) is obviously correct. Assume that (P) holds for some k > . Then, using