Received 8 February 2015; accepted 22 February 2015; published 26 February 2015
We denote by capital letter A, B et al. the bounded linear operators on a complex Hilbert space H. An operator T on H is said to be positive, denoted by if for all.
M. Ito and T. Yamazaki  obtained relations between two inequalities
and Yamazaki and Yanagida  obtained relation between two inequalities
for (not necessarily invertible) positive operators A and B and for fixed and. These results led M. Ito  to obtain relation between two operator inequalities
for (not necessarily invertible) positive operators A and B, where f and g are non-negative continuous functions on satisfying.
Remarks (1.1): The two inequalities in (1.1) are closely related to Furuta inequalities  .
The inequalities in (1.1) and (1.2) are equivalent, respectively, if A and B are invertibles; but they are not always equivalent. Their equivalence for invertible case was shown in  .
Motivated by the result (1.3) of M. Ito  , we obtain the results taking representing functions f and g as non-negative continuous invertible functions on satisfying.
2. Main Results
We denote by the kernel of an operator T.
Theorem 1: Let A and B be positive invertible operators, and let f and g be non-negative invertible continuous functions on satisfying. Then the following hold:
Here and denote orthoprojections to and respectively.
The following Lemma is helpful in proving our results:
Lemma 2: If is a continuous function on and T is an invertible operator with, then
Proof of Lemma: Since is a continuous function on, it can be uniformly approximated by a
sequence of polynomials on. We may assume that itself is a polynomial. Then
Hence the result.
Proof of Theorem 1: For, let and;
1) We suppose that. Then
Let and then
Further since increases as decreases and
2) We suppose that; i.e., then
With and, we have by Lemma 2
Now as and since
thus completing the proof of 2.
Corollary 3. Let A and B be positive invertible operators, and let f and g be non-negative continuous invertible functions on satisfying.
1) If or, then ensures.
2) If, then ensures.
Proof 1) This result follows from 1) of Theorem 1 because each of the conditions and
implies, so that
2) This result follows from 2) of Theorem (1) because, so that
Hence the proof is complete.
Remark (3.1) 1) If, then automatically since, so 1) of corollary 3 holds without any conditions.
2) The invertibility of positive operators A and B is necessary condition.
3) We have considered instead of because the requirement of the limit.
when is not fulfilled, rather it is fulfilled when because.
We have the following results as a consequence of corollary 3.
Theorem 4: Let A and B be positive invertible operators. Then for each and, the following hold
1) If then.
2) If and then.
In Theorem 4 we consider that for or when and we define for a positive invertible operator T.
Theorem 5: Let A and B be positive invertible operators. Then for each and, the following hold:
1) If, then.
2) If and, then .
Proof of Theorem 4: 1) First we consider the case when and.Replacing A with Ap and B with
and putting and in 1) of Corollary 3 so that, we have
if then. (5.1)
If and (5.1) means that
i.e., if then
i.e., if then
i.e., if then
or in other words, ensures.
But, since implies, it follows an equivalent assertion ensures, i.e., which is further equivalent to the trivial assertion ensures.
2) Again first we consider the case and. Replacing A with and B with Ap and putting
and in 2) of Corollary 3.
Since, we have
If p = 0 and r > 0, (5.2) means that ensures i.e.,
which implies that.
Hence (5.3) means that ensures, i.e. ensures.
Hence the result.
Proof of Theorem 5: We can prove by the similar way to Theorem 4 for and, replacing A with Ap and B with and putting and for 1) in 1) of Corollary 3 and replacing A with and B with Ap and putting and for 2) in 2) of Corollary 3.
Corollary 4: Let A and B be positive invertible operators, and let f and g be non-negative continuous invertible functions on satisfying. If, then
Proof: The proof follows directly by applying the condition, in 1) of Corollary 3 and for the proof we have only to interchange the roles of A and B and those of f and g in 2) of Corollary 3, Since if.
 Ito, M. and Yamazaki, T. (2002) Relations between Two Inequalities and and Their Applications. Integral Equations and Operator Theory, 44, 442-450.
 Ito, M. (2005) Relations between Two Operator Inequalities Motivated by the Theory of Operator Means. Integral Equations and Operator Theory, 53, 527-534.