Reverses of Ando’s and Hölder–McCarty’s inequalities

In this paper, we give some reverse-types of Ando’s and Hölder–McCarthy’s inequalities for positive linear maps, and positive invertible operators. For this purpose, we use a recently improved Young inequality and its reverse.

Let A, B ∈ B(H) be positive. The operator t-weighted arithmetic, geometric, and harmonic means of operators A, B are defined by In particular, for t = 1 2 we get the usual operator arithmetic mean ∇, the geometric mean and the harmonic mean !.

Results and discussion
For positive real numbers a i and b i (i = 1, 2, . . . , n) the Hölder inequality states that for p, q > 1 such that 1 p + 1 q = 1. If p = q = 2 in (4), then we get the Cauchy-Schwarz inequality. The Hölder inequality for positive operators A i and B i (i = 1, 2, . . . , n) is where 0 ≤ t ≤ 1. In the case t = 1 2 , we get the operator Cauchy-Schwarz inequality. For further information as regards the Hölder and Cauchy-Schwarz inequalities we refer the reader to [3-5, 11, 12, 20] and the references therein. Ando [1] proved that if is a positive linear map, then for positive operators A, B ∈ B(H) and t ∈ [0, 1], we have Recently, some authors presented several reverse-types of Ando's inequality (see [13,17]).
The Hölder-McCarthy's inequality says that for any positive operator A and any unit vector x ∈ H, we have Furuta [8] showed that this inequality is equivalent to Young's inequality.

Conclusions
In this paper, we establish a reverse of Ando's inequality for positive (non-unital) linear maps and positive definite matrices by using an inequality due to Sababheh. We obtain some reverses of the matrix Hölder and Cauchy-Schwarz inequalities and a reverse of inequality (5) for t ∈ (0, 1 2 ] as follows:

Methods
We use the properties of inner product and the inequalities obtained in [16] and [19].

Main results
To prove our first result, we need the following lemmas.
For N = 2, we have the following lemma, which is shown in [19] for positive invertible operators. (i) If 0 < t ≤ 1 2 , then
For N = 2, we have the following lemma, which is shown in [19] for positive invertible operators.

Lemma 5 Let A, B ∈ B(H) be positive invertible operators and t
where r = min{ν, 1 -ν} and r 0 = min{2r, 1 -2r}. Now, we obtain a reverse of Ando's inequality for positive invertible operators as follows.

Theorem 6 Let A, B ∈ B(H) be positive invertible operators, be a positive linear map
where Proof The proof of inequality (13) is similar to the proof of inequality (12). Thus, we only prove inequality (12). Let 0 ≤ t ≤ 1 2 . Applying inequalities (10) and (9), we have Now, using the positive linear map on (14), we get Moreover, if we replace A and B by (A) and (B) in inequality (14), respectively, then From the first inequality of (16) and the second inequality of (15), we have Therefore, applying inequality (1), we get Similarly for N = 2 by applying Lemma 2 and Lemma 5, we can obtain a reverse of Ando's inequality for positive invertible operators.
We want to establish some inequalities for positive invertible operators.
Proof The proof of part (ii) is similar to the proof of part (i). Thus we only prove (i). Applying inequality (2) for any positive real numbers k, s, we have Fix s in (20). Then applying functional calculus to the operator A, we have If we apply the positive linear map and inner product for x ∈ H with x = 1 in inequality (21), we have Now, using the functional calculus to the operator B, we have Taking the positive linear map and the inner product for y ∈ H with y = 1, we get Now, if we put x = y, then we get the desired result.
Proof The proof of part (ii) is similar to the proof of part (i). Thus we just prove (i). For any positive real number k and any unit vector x ∈ H, we have Applying inequality (22) and the functional calculus for the operator A, we have Now, using the unital positive operator and the inner product for y ∈ H with y = 1 in inequality (23), we get Now, putting y = x, we get the desired result. Then for any unit vector x ∈ H (i) for 0 < t ≤ 1 2 , where r = min{t, 1 -t}, R = max{t, 1 -t}, r 0 = min{2r, 1 -2r}.
Proof Letting = and B = A in Theorem 9, we get the desired inequalities.
In the next result, we obtain a refinement of inequality (5) for t ∈ (0, 1 2 ].
Proof If we replace (A) = A, A ∈ B(H) and t with 1t in Corollary 10, then we get the desired result.

Some new results
In this section, we prove some difference reverse-types of the Hölder and Cauchy-Schwarz inequalities.
Since the function f (x) = x t (t ∈ [0, 1]) is an operator concave function, n i=1 w i T t i ≤ ( n i=1 w i T i ) t for positive operators T i and positive real numbers w i such that n i=1 w i = 1. Now, Theorem 12 yields a reverse of this inequality as follows. Theorem 12, where w i 's are positive real numbers such that n i=1 w i = 1, we obtain the following inequalities:

Example 13 If for positive operators
In [18], the Tsallis relative operator entropy T t (A|B) for positive invertible operators A, B and 0 < t ≤ 1 is defined as follows: For further information as regards the Tsallis relative operator entropy see [6] and the references therein. In [7,Proposition 2.3], it is shown that for any unital positive linear map the following inequality holds: In (25), by similar techniques of Theorem 12, for positive operators A i , B i (i = 1, 2, . . . , n), we have In the next theorem, we show a reverse of inequality (26).
(ii) If 1 2 < t < 1, then Proof Applying Theorem 12 for 0 < t ≤ 1 2 , we have whence we get the first inequality. The proof of the second inequality is similar.
Remark 15 We can present our results for non-invertible operators; see [6]. It is a direct consequence of the definition of the mean in the sense of Kubo-Ando [11] that A t (B + ε) is a monotone increasing net. Let B be a non-invertible operator and > 0. It follows from the set {A t (B + ) : > 0} being bounded above for 0 < ε < 1 that the limit exists in the strong operator topology. So by (27), A t B exists.