^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

We shall show relation between two operator inequalities and for positive, invertible operators A and B, where f and g are non-negative continuous invertible functions on satisfying *f(t)g(t)=t*^{-1} .

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

M. Ito and T. Yamazaki [

and Yamazaki and Yanagida [

for (not necessarily invertible) positive operators A and B and for fixed

for (not necessarily invertible) positive operators A and B, where f and g are non-negative continuous functions on

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 denote by

Theorem 1: Let A and B be positive invertible operators, and let f and g be non-negative invertible continuous functions on

1)

2)

Here

The following Lemma is helpful in proving our results:

Lemma 2: If

Proof of Lemma: Since

sequence of polynomials on

Hence the result.

Proof of Theorem 1: For

1) We suppose that

Let

We have

Further since

we have

Then

i.e.

2) We suppose that

With

Now as

we have

Then

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

1) If

2) If

Proof 1) This result follows from 1) of Theorem 1 because each of the conditions

2) This result follows from 2) of Theorem (1) because

Hence the proof is complete.

Remark (3.1) 1) If

2) The invertibility of positive operators A and B is necessary condition.

3) We have considered

We have the following results as a consequence of corollary 3.

Theorem 4: Let A and B be positive invertible operators. Then for each

1) If

2) If

In Theorem 4 we consider that

Theorem 5: Let A and B be positive invertible operators. Then for each

1) If

2) If

Proof of Theorem 4: 1) First we consider the case when ^{p} and B with

if

If

if

i.e., if

i.e., if

i.e., if

or in other words,

But, since

2) Again first we consider the case ^{p} and putting

Since

If p = 0 and r > 0, (5.2) means that

ensures

which implies that

Hence (5.3) means that

Hence the result.

Proof of Theorem 5: We can prove by the similar way to Theorem 4 for ^{p} and B with ^{p} and putting

Corollary 4: Let A and B be positive invertible operators, and let f and g be non-negative continuous invertible functions on

Proof: The proof

MohammadIlyas,ReyazAhmad,ShadabIlyas, (2015) Relation between Two Operator Inequalities . Advances in Pure Mathematics,05,93-99. doi: 10.4236/apm.2015.52012