To reach our goal, we need the following lemmas.
Lemma 2.1
([10])
If
\(A, B\in B(\mathcal{H})\)
are accretive, then
$$\begin{aligned} \Re (A)\sharp \Re (B)\le \Re (A\sharp B). \end{aligned}$$
Lemma 2.2
([10])
If
\(A, B\in B(\mathcal{H})\)
are accretive, then
$$\begin{aligned} \Re \biggl( \biggl(\frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr)\ge \biggl( \frac{{\Re (A)}^{-1}+{\Re (B)}^{-1}}{2} \biggr)^{-1}. \end{aligned}$$
Lemma 2.3
([9])
If
\(A\in \mathbb{M}_{n}\)
has a positive definite real part, then
$$\begin{aligned} \Re \bigl(A^{-1} \bigr)\le \Re (A)^{-1}. \end{aligned}$$
Lemma 2.4
([4])
If
\(A\in \mathbb{M}_{n}\)
with
\(W(A)\subset S _{\alpha }\), then
$$\begin{aligned} \sec ^{2}(\alpha )\Re \bigl(A^{-1} \bigr)\ge \Re (A)^{-1}. \end{aligned}$$
It is easy to verify that \(\Re ( (\frac{A^{-1}+B^{-1}}{2} ) ^{-1} )\le \Re (A\sharp B)\le \Re (\frac{A+B}{2} )\) does not persist for two accretive operators A and B. However, Lin presented the following extension of the arithmetic–geometric mean inequality.
Lemma 2.5
([9])
Let
\(A ,B\in \mathbb{M}_{n}\)
be such that
\(W(A), W(B)\subset S_{\alpha }\). Then
$$\begin{aligned} \Re (A\sharp B)\le \sec ^{2}(\alpha )\Re \biggl( \frac{A+B}{2} \biggr). \end{aligned}$$
(6)
Lemma 2.6
([2])
Let
\(A, B\in B(\mathcal{H})\)
be positive. Then
$$\begin{aligned} \Vert AB \Vert \le \frac{1}{4} \Vert A+B \Vert ^{2}. \end{aligned}$$
Lemma 2.7
([1])
Let
\(A\in B(\mathcal{H})\)
be positive. Then, for every positive unital linear map Φ,
$$\begin{aligned} \varPhi ^{-1}(A)\le \varPhi \bigl(A^{-1} \bigr). \end{aligned}$$
Lemma 2.8
([1])
Let
\(A, B\in B(\mathcal{H})\)
be positive. Then, for
\(1\le r<+\infty \),
$$\begin{aligned} \bigl\Vert A^{r}+B^{r} \bigr\Vert \le \bigl\Vert (A+B)^{r} \bigr\Vert . \end{aligned}$$
An operator Kantorovich inequality obtained by Marshall and Olkin [12] reads as follows.
Let \(0< {mI}\le A\le {MI}\), then, for every positive unital linear map Φ,
$$\begin{aligned} \varPhi \bigl(A^{-1} \bigr)\le K(h)\varPhi (A)^{-1}, \end{aligned}$$
(7)
where \(K(h)=\frac{(h+1)^{2}}{4h}\) and \(h=\frac{M}{m}\).
Lin [7] showed that (7) can be squared as follows:
$$\begin{aligned} \varPhi ^{2} \bigl(A^{-1} \bigr)\le \bigl(K(h) \bigr)^{2}\varPhi (A)^{-2}, \end{aligned}$$
(8)
where \(K(h)=\frac{(h+1)^{2}}{4h}\) and \(h=\frac{M}{m}\).
Let \(A\in \mathbb{M}_{n}\) have a positive definite real part, \(0< {mI}_{n}\le \Re (A)\le {MI}_{n}\) and Φ be a unital positive linear map. By (7) and Lemma 2.3, we can obtain the following inequality:
$$\begin{aligned} \varPhi \bigl(\Re \bigl(A^{-1} \bigr) \bigr)\le K(h) \varPhi \bigl(\Re (A) \bigr)^{-1}, \end{aligned}$$
(9)
where \(K(h)=\frac{(h+1)^{2}}{4h}\) and \(h=\frac{M}{m}\).
As an analog of inequality (8), we show that inequality (9) can be squared nicely as follows.
Theorem 2.9
If
\(A\in \mathbb{M}_{n}\)
has a positive definite real part and
\(0< {mI}_{n}\le \Re (A)\le {MI}_{n}\), then, for every positive unital linear map
Φ,
$$\begin{aligned} \varPhi ^{2} \bigl(\Re \bigl(A^{-1} \bigr) \bigr) \le \bigl(K(h) \bigr)^{2}\varPhi \bigl(\Re (A) \bigr)^{-2}, \end{aligned}$$
(10)
where
\(K(h)=\frac{(h+1)^{2}}{4h}\)
and
\(h=\frac{M}{m}\).
Proof
Since
$$\begin{aligned} {mI}_{n}\le \Re (A)\le {MI}_{n}, \end{aligned}$$
we have
$$\begin{aligned} \bigl( {MI}_{n}-\Re (A) \bigr) \bigl( {mI}_{n}- \Re (A) \bigr){\Re (A)}^{-1}\le 0, \end{aligned}$$
which is equivalent to
$$\begin{aligned} \Re (A)+Mm\Re (A)^{-1}\le (M+m)I_{n}. \end{aligned}$$
(11)
By Lemma 2.3 and (11), we get
$$\begin{aligned} &\Re (A)+Mm\Re \bigl(A^{-1} \bigr) \\ &\quad \le \Re (A)+Mm\Re (A)^{-1} \\ &\quad \le (M+m)I_{n}. \end{aligned}$$
(12)
Inequality (10) is equivalent to
$$\begin{aligned} \bigl\Vert \varPhi \bigl(\Re \bigl(A^{-1} \bigr) \bigr)\varPhi \bigl( \Re (A) \bigr) \bigr\Vert \le K(h). \end{aligned}$$
By computation, we have
$$\begin{aligned} & \bigl\Vert Mm\varPhi \bigl(\Re \bigl(A^{-1} \bigr) \bigr)\varPhi \bigl(\Re (A) \bigr) \bigr\Vert \\ &\quad \le \frac{1}{4} \bigl\Vert Mm\varPhi \bigl(\Re \bigl(A^{-1} \bigr) \bigr)+\varPhi \bigl(\Re (A) \bigr) \bigr\Vert ^{2}\quad \text{(by Lemma 2.6)} \\ &\quad \le \frac{1}{4}(M+m)^{2}\quad \bigl(\text{by (12)}\bigr). \end{aligned}$$
That is,
$$\begin{aligned} \bigl\Vert \varPhi \bigl(\Re \bigl(A^{-1} \bigr) \bigr)\varPhi \bigl( \Re (A) \bigr) \bigr\Vert \le K(h). \end{aligned}$$
This completes the proof. □
Let \(A, B\in B(\mathcal{H})\) be accretive, \(0< {mI}\le \Re (A), \Re (B) \le {MI}\) and Φ be a unital positive linear map. By inequality (2) and Lemma 2.1, we can obtain the following inequality:
$$\begin{aligned} \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)\le K(h)\varPhi \bigl( \Re (A\sharp B) \bigr), \end{aligned}$$
(13)
where \(K(h)=\frac{(h+1)^{2}}{4h}\) and \(h=\frac{M}{m}\).
Following an idea of Lin [6], we give a squaring version of inequality (13) below.
Theorem 2.10
If
\(A, B\in \mathbb{M}_{n}\)
with
\(W(A), W(B) \subset S_{\alpha }\)
and
\(0< {mI}_{n}\le \Re (A), \Re (B)\le {MI}_{n}\), then, for every positive unital linear map
Φ,
$$\begin{aligned} \varPhi ^{2} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)\le \bigl(\sec ^{4}(\alpha )K(h) \bigr)^{2} \varPhi ^{2} \bigl(\Re (A\sharp B) \bigr), \end{aligned}$$
(14)
where
\(K(h)=\frac{(h+1)^{2}}{4h}\)
and
\(h=\frac{M}{m}\).
Proof
From Theorem 2.9 we have
$$\begin{aligned} \frac{1}{2}\Re (A)+\frac{1}{2}Mm\Re (A)^{-1}\le \frac{1}{2}(M+m)I_{n} \end{aligned}$$
(15)
and
$$\begin{aligned} \frac{1}{2}\Re (B)+\frac{1}{2}Mm\Re (B)^{-1}\le \frac{1}{2}(M+m)I_{n}. \end{aligned}$$
(16)
Summing up inequalities (15) and (16), we get
$$\begin{aligned} \Re \biggl( \frac{A+B}{2} \biggr)+Mm \biggl( \frac{\Re (A)^{-1}+ \Re (B)^{-1}}{2} \biggr)\le (M+m)I_{n}. \end{aligned}$$
(17)
Inequality (14) is equivalent to
$$\begin{aligned} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \varPhi ^{-1} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \le \sec ^{4}(\alpha )K(h). \end{aligned}$$
By computation, we have
$$\begin{aligned} & \biggl\Vert \sec ^{4}(\alpha )Mm\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)\varPhi ^{-1} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \\ &\quad \le \frac{1}{4} \biggl\Vert \sec ^{4}(\alpha )\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+Mm\varPhi ^{-1} \bigl(\Re (A \sharp B) \bigr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.6)} \\ &\quad \le \frac{1}{4} \biggl\Vert \sec ^{4}(\alpha )\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+Mm\varPhi \bigl( \bigl(\Re (A\sharp B) \bigr)^{-1} \bigr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.7)} \\ &\quad \le \frac{1}{4} \biggl\Vert \sec ^{4}(\alpha )\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+\sec ^{2}(\alpha )Mm\varPhi \bigl(\Re \bigl(A^{-1} \sharp B^{-1} \bigr) \bigr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.4)} \\ &\quad \le \frac{1}{4} \biggl\Vert \sec ^{4}(\alpha )\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+\sec ^{4}(\alpha )Mm\varPhi \biggl(\Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) \biggr) \biggr\Vert ^{2}\quad \bigl(\text{by (6)}\bigr) \\ &\quad =\frac{1}{4} \biggl\Vert \sec ^{4}(\alpha )\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr)+Mm\Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) \biggr) \biggr\Vert ^{2} \\ &\quad \le \frac{1}{4} \biggl\Vert \sec ^{4}(\alpha )\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr)+Mm \biggl( \frac{\Re (A)^{-1}+\Re (B)^{-1}}{2} \biggr) \biggr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.3)} \\ &\quad \le \frac{1}{4}\sec ^{8}(\alpha ) (M+m)^{2} \quad \bigl(\text{by (17)}\bigr). \end{aligned}$$
That is,
$$\begin{aligned} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \varPhi ^{-1} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \le \sec ^{4}(\alpha )K(h). \end{aligned}$$
This completes the proof. □
Next we give a pth (\(p\ge 2\)) powering of inequality (14).
Theorem 2.11
If
\(A, B\in \mathbb{M}_{n}\)
with
\(W(A), W(B) \subset S_{\alpha }\), \(0< {mI}_{n}\le \Re (A), \Re (B)\le {MI}_{n}\), \(1<\beta \le 2\)
and
\(p\ge 2\beta \), then, for every positive unital linear map
Φ,
$$\begin{aligned} \varPhi ^{p} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)\le \frac{( \sec ^{2\beta }(\alpha )K(h)^{\frac{\beta }{2}}(M^{\beta }+m^{\beta }))^{\frac{2p}{ \beta }}}{16M^{p}m^{p}}\varPhi ^{p} \bigl(\Re (A\sharp B) \bigr), \end{aligned}$$
(18)
where
\(K(h)=\frac{(h+1)^{2}}{4h}\)
and
\(h=\frac{M}{m}\).
Proof
Since
$$\begin{aligned} {mI}_{n}\le \varPhi \biggl(\Re \biggl(\frac{A+B}{2} \biggr) \biggr)\le {MI} _{n}, \end{aligned}$$
we have
$$\begin{aligned} M^{\beta }m^{\beta }\varPhi ^{-{\beta }} \biggl( \Re \biggl(\frac{A+B}{2} \biggr) \biggr)+ \varPhi ^{\beta } \biggl(\Re \biggl(\frac{A+B}{2} \biggr) \biggr)\le M^{ \beta }+m^{\beta }. \end{aligned}$$
(19)
By (14) and the L-H inequality [1], we obtain
$$\begin{aligned} \varPhi ^{-{\beta }} \bigl(\Re (A\sharp B) \bigr)\le \bigl( \sec ^{4}(\alpha )K(h) \bigr)^{\beta } \varPhi ^{-{\beta }} \biggl( \Re \biggl( \frac{A+B}{2} \biggr) \biggr). \end{aligned}$$
(20)
Inequality (18) is equivalent to
$$\begin{aligned} \biggl\Vert \varPhi ^{\frac{p}{2}} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \varPhi ^{-{\frac{p}{2}}} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \le \frac{( \sec ^{2\beta }(\alpha )K(h)^{\frac{\beta }{2}}(M^{\beta }+m^{\beta }))^{\frac{p}{ \beta }}}{4M^{\frac{p}{2}}m^{\frac{p}{2}}}. \end{aligned}$$
By computation, we have
$$\begin{aligned} & \biggl\Vert M^{\frac{p}{2}}m^{\frac{p}{2}}\varPhi ^{\frac{p}{2}} \biggl( \Re \biggl( \frac{A+B}{2} \biggr) \biggr)\varPhi ^{-{\frac{p}{2}}} \bigl(\Re (A \sharp B) \bigr) \biggr\Vert \\ &\quad \le \frac{1}{4} \biggl\Vert \bigl(\sec ^{4}(\alpha )K(h) \bigr)^{\frac{p}{4}} \varPhi ^{\frac{p}{2}} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+ \biggl(\frac{M^{2}m^{2}}{\sec ^{4}(\alpha )K(h)} \biggr)^{\frac{p}{4}} \varPhi ^{-{\frac{p}{2}}} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert ^{2} \\ &\quad \le \frac{1}{4} \biggl\Vert \bigl(\sec ^{4}(\alpha )K(h) \bigr)^{ \frac{\beta }{2}}\varPhi ^{\beta } \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+ \biggl(\frac{M^{2}m^{2}}{\sec ^{4}(\alpha )K(h)} \biggr)^{\frac{\beta }{2}} \varPhi ^{-{\beta }} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert ^{\frac{p}{\beta }} \\ &\quad \le \frac{1}{4} \biggl\Vert \bigl(\sec ^{4}(\alpha )K(h) \bigr)^{ \frac{\beta }{2}}\varPhi ^{\beta } \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \\ &\quad\quad{} + \bigl( \sec ^{4}(\alpha )K(h) \bigr)^{\frac{\beta }{2}}M^{\beta }m^{\beta } \varPhi ^{-{\beta }} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \biggr\Vert ^{\frac{p}{\beta }} \\ &\quad =\frac{1}{4} \biggl\Vert \bigl(\sec ^{4}(\alpha )K(h) \bigr)^{\frac{\beta }{2}} \biggl(\varPhi ^{\beta } \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+M ^{\beta }m^{\beta }\varPhi ^{-{\beta }} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \biggr) \biggr\Vert ^{\frac{p}{\beta }} \\ &\quad \le \frac{1}{4} \bigl(\sec ^{2\beta }(\alpha )K(h)^{\frac{\beta }{2}} \bigl(M ^{\beta }+m^{\beta } \bigr) \bigr)^{\frac{p}{\beta }}, \end{aligned}$$
where the first inequality is by Lemma 2.6, the second one is by Lemma 2.8, the third one is by (20) and the last one is by (19).
That is,
$$\begin{aligned} \biggl\Vert \varPhi ^{\frac{p}{2}} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \varPhi ^{-{\frac{p}{2}}} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \le \frac{( \sec ^{2\beta }(\alpha )K(h)^{\frac{\beta }{2}}(M^{\beta }+m^{\beta }))^{\frac{p}{ \beta }}}{4M^{\frac{p}{2}}m^{\frac{p}{2}}}. \end{aligned}$$
This completes the proof. □
We are not satisfied with the factor \((\sec ^{4}(\alpha )K(h))^{2}\) in Theorem 2.10, the ideal factor should be \((K(h))^{2}\). We shall prove it in the following theorem.
Theorem 2.12
If
\(A, B\in B(\mathcal{H})\)
are accretive and
\(0< {mI}\le \Re (A), \Re (B)\le {MI}\), then, for every positive unital linear map
Φ,
$$\begin{aligned} \varPhi ^{2} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)\le \bigl(K(h) \bigr)^{2} \varPhi ^{2} \bigl( \Re (A\sharp B) \bigr), \end{aligned}$$
(21)
where
\(K(h)=\frac{(h+1)^{2}}{4h}\)
and
\(h=\frac{M}{m}\).
Proof
From Theorem 2.10 one can get
$$\begin{aligned} \Re \biggl( \frac{A+B}{2} \biggr)+Mm \biggl( \frac{\Re (A)^{-1}+ \Re (B)^{-1}}{2} \biggr)\le (M+m)I. \end{aligned}$$
(22)
Inequality (21) is equivalent to
$$\begin{aligned} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \varPhi ^{-1} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \le K(h). \end{aligned}$$
By computation, we have
$$\begin{aligned} & \biggl\Vert Mm\varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \varPhi ^{-1} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \\ &\quad \le \frac{1}{4} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+Mm \varPhi ^{-1} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.6)} \\ &\quad \le \frac{1}{4} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+Mm \varPhi \bigl( \bigl(\Re (A\sharp B) \bigr)^{-1} \bigr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.7)} \\ &\quad \le \frac{1}{4} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+Mm \varPhi \bigl( \bigl(\Re (A)\sharp \Re (B) \bigr)^{-1} \bigr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.1)} \\ &\quad =\frac{1}{4} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+Mm \varPhi \bigl(\Re (A)^{-1}\sharp \Re (B)^{-1} \bigr) \biggr\Vert ^{2} \\ &\quad \le \frac{1}{4} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+Mm \varPhi \biggl( \frac{\Re (A)^{-1}+\Re (B)^{-1}}{2} \biggr) \biggr\Vert ^{2}\quad \text{(by AM-GM inequality)} \\ &\quad \le \frac{1}{4}(M+m)^{2}\quad\bigl(\text{by (22)}\bigr). \end{aligned}$$
That is,
$$\begin{aligned} \biggl\Vert \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr) \varPhi ^{-1} \bigl(\Re (A\sharp B) \bigr) \biggr\Vert \le K(h). \end{aligned}$$
This completes the proof. □
Remark 2.13
Letting \(A, B\ge 0\) in Theorem 2.12, inequality (21) coincides with inequality (3).
Next we give a pth (\(p\ge 2\)) powering of inequality (21) along the same line as in Theorem 2.11.
Theorem 2.14
If
\(A, B\in B(\mathcal{H})\)
are accretive and
\(0< {mI}\le \Re (A), \Re (B)\le {MI}\)
\(1<\beta \le 2\)
and
\(p\ge 2\beta \), then, for every positive unital linear map
Φ,
$$\begin{aligned} \varPhi ^{p} \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)\le \frac{(K(h)^{\frac{ \beta }{2}}(M^{\beta }+m^{\beta }))^{\frac{2p}{\beta }}}{16M^{p}m^{p}} \varPhi ^{p} \bigl(\Re (A\sharp B) \bigr), \end{aligned}$$
(23)
where
\(K(h)=\frac{(h+1)^{2}}{4h}\)
and
\(h=\frac{M}{m}\).
Remark 2.15
Letting \(A, B\ge 0\) and \(\beta =2\) in Theorem 2.14, inequality (23) coincides with inequality (4).
The following theorem corrects Theorem 1.2 of Liu et al. [11].
Theorem 2.16
Let
\(A, B\in \mathbb{M}_{n}\)
be such that
\(W(A),W(B)\subset S_{\alpha }\), then
$$\begin{aligned} \Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \le \sec ^{4}(\alpha )\Re (A\sharp B). \end{aligned}$$
(24)
Proof
We can get
$$\begin{aligned} \biggl( \Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) \biggr)^{-1} \le \sec ^{2}(\alpha ) \bigl( \Re \bigl(A^{-1} \bigr)\sharp \Re \bigl(B^{-1} \bigr) \bigr) ^{-1} \end{aligned}$$
(25)
along the same line as Liu et al. did in [11] by Lemma 2.1 and Lemma 2.5.
Thus we have
$$\begin{aligned} \Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) &\le \biggl( \Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) \biggr)^{-1} \quad \text{(by Lemma 2.3)} \\ &\le \sec ^{2}(\alpha ) \bigl( \Re \bigl(A^{-1} \bigr)\sharp \Re \bigl(B^{-1} \bigr) \bigr) ^{-1} \quad \bigl(\text{by (25)}\bigr) \\ &=\sec ^{2}(\alpha ) \bigl(\Re \bigl(A^{-1} \bigr) \bigr)^{-1}\sharp \bigl( \Re \bigl(B^{-1} \bigr) \bigr)^{-1} \\ &\le \sec ^{4}(\alpha ) \bigl(\Re (A)\sharp \Re (B) \bigr)\quad \text{(by Lemma 2.4)} \\ &\le \sec ^{4}(\alpha )\Re (A\sharp B)\quad \text{(by Lemma 2.1).} \end{aligned}$$
This completes the proof. □
Remark 2.17
Maybe it is just a clerical error in Theorem 1.2 of their work [11]. However, the authors present the following inequalities in their proof:
$$\begin{aligned} \Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) &\le \sec ^{2}(\alpha ) \bigl( \Re \bigl({A^{-1}} \bigr) \bigr)^{-1}\sharp \bigl( \Re \bigl({B^{-1}} \bigr) \bigr)^{-1} \\ &\le \sec ^{2}(\alpha ) \bigl(\Re (A)\sharp \Re (B) \bigr). \end{aligned}$$
Obviously, such a deduction in their proof collapses given the property of geometric mean for positive definite matrices. Thus we give Theorem 2.16 and the proof.
Let \(A, B\in \mathbb{M}_{n}\) with \(W(A), W(B)\subset S_{\alpha }\), \(0< {mI}_{n}\le \Re (A^{-1}),\Re (B^{-1})\le {MI}_{n}\) and Φ be a unital positive linear map. As a complement of inequalities (13) and (24), we have the following reverse harmonic–geometric mean inequality:
$$\begin{aligned} \varPhi \bigl(\Re (A\sharp B) \bigr)\le \sec ^{2}( \alpha )K(h)\varPhi \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr), \end{aligned}$$
(26)
where \(K(h)=\frac{(h+1)^{2}}{4h}\) and \(h=\frac{M}{m}\).
Proof
Compute
$$\begin{aligned} \Re (A\sharp B) &=\Re \bigl( \bigl(A^{-1}\sharp B^{-1} \bigr)^{-1} \bigr) \\ &\le \Re \bigl(A^{-1}\sharp B^{-1} \bigr)^{-1} \\ &\le K(h)\Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \\ &\le \sec ^{2}(\alpha )K(h)\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr), \end{aligned}$$
in which the first inequality is by Lemma 2.3, the second one is by inequality (13) and the last one is by Lemma 2.4.
Imposing Φ on both sides of the inequalities above, we thus obtain inequality (26). □
As an analog of Theorem 2.12, we shall present a squaring version of inequality (26).
Theorem 2.18
If
\(A, B\in \mathbb{M}_{n}\)
with
\(W(A), W(B) \subset S_{\alpha }\)
and
\(0< {mI}_{n}\le \Re (A^{-1}), \Re (B^{-1}) \le {MI}_{n}\), then, for every positive unital linear map
Φ,
$$\begin{aligned} \varPhi ^{2} \bigl(\Re (A\sharp B) \bigr)\le \bigl(\sec ^{4}(\alpha )K(h) \bigr)^{2}\varPhi ^{2} \biggl( \Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr), \end{aligned}$$
(27)
where
\(K(h)=\frac{(h+1)^{2}}{4h}\)
and
\(h=\frac{M}{m}\).
Proof
From Theorem 2.10 we have
$$\begin{aligned} \frac{1}{2}\Re \bigl(A^{-1} \bigr)+ \frac{1}{2}Mm\Re \bigl(A^{-1} \bigr)^{-1}\le \frac{1}{2}(M+m)I _{n} \end{aligned}$$
(28)
and
$$\begin{aligned} \frac{1}{2}\Re \bigl(B^{-1} \bigr)+ \frac{1}{2}Mm\Re \bigl(B^{-1} \bigr)^{-1}\le \frac{1}{2}(M+m)I _{n}. \end{aligned}$$
(29)
Summing up inequalities (28) and (29), we get
$$\begin{aligned} &\Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)+Mm\Re \biggl(\frac{A+B}{2} \biggr) \\ &\quad \le \Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)+Mm \biggl( \frac{ \Re (A^{-1})^{-1}+\Re (B^{-1})^{-1}}{2} \biggr) \\ &\quad \le (M+m)I_{n}. \end{aligned}$$
Inequality (27) is equivalent to
$$\begin{aligned} \biggl\Vert \varPhi \bigl(\Re (A\sharp B) \bigr)\varPhi ^{-1} \biggl( \Re \biggl( \biggl( \frac{A ^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr) \biggr\Vert \le \sec ^{4}(\alpha )K(h). \end{aligned}$$
By computation, we have
$$\begin{aligned} & \biggl\Vert Mm\varPhi \bigl(\Re (A\sharp B) \bigr)\varPhi ^{-1} \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr) \biggr\Vert \\ &\quad \le \frac{1}{4} \biggl\Vert Mm\varPhi \bigl(\Re (A\sharp B) \bigr)+\varPhi ^{-1} \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.6)} \\ &\quad \le \frac{1}{4} \biggl\Vert Mm\varPhi \bigl(\Re (A\sharp B) \bigr)+\varPhi \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr)^{-1} \biggr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.7)} \\ &\quad \le \frac{1}{4} \biggl\Vert Mm\varPhi \bigl(\Re (A\sharp B) \bigr)+\sec ^{2}(\alpha )\varPhi \biggl(\Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) \biggr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.4)} \\ &\quad \le \frac{1}{4} \biggl\Vert \sec ^{2}(\alpha )Mm \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+\sec ^{2}( \alpha )\varPhi \biggl(\Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) \biggr) \biggr\Vert ^{2}\quad \bigl(\text{by (6)}\bigr) \\ &\quad =\frac{1}{4} \biggl\Vert \sec ^{2}(\alpha )\varPhi \biggl(Mm\Re \biggl( \frac{A+B}{2} \biggr)+\Re \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) \biggr) \biggr\Vert ^{2} \\ &\quad \le \frac{1}{4}\sec ^{4}(\alpha ) (M+m)^{2}. \end{aligned}$$
That is,
$$\begin{aligned} \biggl\Vert \varPhi \bigl(\Re (A\sharp B) \bigr)\varPhi ^{-1} \biggl( \Re \biggl( \biggl( \frac{A ^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr) \biggr\Vert \le \sec ^{4}(\alpha )K(h). \end{aligned}$$
This completes the proof. □
Obviously, the optimal factor in Theorem 2.18 should be \((\sec ^{2}( \alpha )K(h))^{2}\). We note that it is affirmative under the condition \({mI}_{n}\le \Re (A^{-1})\le \Re (A)^{-1}\le {MI}_{n}\) and \({mI}_{n}\le \Re (B^{-1})\le \Re (B)^{-1}\le {MI}_{n}\) by presenting the following theorem.
Theorem 2.19
If
\(A, B\in \mathbb{M}_{n}\)
with
\(W(A), W(B) \subset S_{\alpha }\)
and
\(0< {mI}_{n}\le \Re (A)^{-1}, \Re (B)^{-1} \le {MI}_{n}\), then, for every positive unital linear map
Φ,
$$\begin{aligned} \varPhi ^{2} \bigl(\Re (A\sharp B) \bigr)\le \bigl(\sec ^{2}(\alpha )K(h) \bigr)^{2}\varPhi ^{2} \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr), \end{aligned}$$
(30)
where
\(K(h)=\frac{(h+1)^{2}}{4h}\)
and
\(h=\frac{M}{m}\).
Proof
From Theorem 2.10 we obtain
$$\begin{aligned} \frac{{\Re (A)}^{-1}+{\Re (B)}^{-1}}{2}+Mm\Re \biggl( \frac{A+B}{2} \biggr) \le (M+m)I_{n}. \end{aligned}$$
(31)
Inequality (30) is equivalent to
$$\begin{aligned} \biggl\Vert \varPhi \bigl(\Re (A\sharp B) \bigr)\varPhi ^{-1} \biggl( \Re \biggl( \biggl( \frac{A ^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr) \biggr\Vert \le \sec ^{2}(\alpha )K(h). \end{aligned}$$
By computation, we have
$$\begin{aligned} & \biggl\Vert \sec ^{2}(\alpha )Mm\varPhi \bigl(\Re (A\sharp B) \bigr) \varPhi ^{-1} \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr) \biggr\Vert \\ &\quad \le \frac{1}{4} \biggl\Vert Mm\varPhi \bigl(\Re (A\sharp B) \bigr)+\sec ^{2}(\alpha )\varPhi ^{-1} \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr) ^{-1} \biggr) \biggr) \biggr\Vert ^{2} \quad \text{(by Lemma 2.6)} \\ &\quad \le \frac{1}{4} \biggl\Vert Mm\varPhi \bigl(\Re (A\sharp B) \bigr)+\sec ^{2}(\alpha )\varPhi \biggl(\Re \biggl( \biggl( \frac{A^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) ^{-1} \biggr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.7)} \\ &\quad \le \frac{1}{4} \biggl\Vert Mm\varPhi \bigl(\Re (A\sharp B) \bigr)+\sec ^{2}(\alpha )\varPhi \biggl(\frac{{\Re (A)}^{-1}+{\Re (B)}^{-1}}{2} \biggr) \biggr\Vert ^{2}\quad \text{(by Lemma 2.2)} \\ &\quad \le \frac{1}{4} \biggl\Vert \sec ^{2}(\alpha )Mm \varPhi \biggl(\Re \biggl( \frac{A+B}{2} \biggr) \biggr)+\sec ^{2}( \alpha )\varPhi \biggl(\frac{ {\Re (A)}^{-1}+{\Re (B)}^{-1}}{2} \biggr) \biggr\Vert ^{2} \quad \bigl(\text{by (6)}\bigr) \\ &\quad =\frac{1}{4} \biggl\Vert \sec ^{2}(\alpha )\varPhi \biggl(Mm\Re \biggl( \frac{A+B}{2} \biggr)+\frac{{\Re (A)}^{-1}+{\Re (B)}^{-1}}{2} \biggr) \biggr\Vert ^{2} \\ &\quad \le \frac{1}{4}\sec ^{4}(\alpha ) (M+m)^{2} \quad \bigl(\text{by (31)}\bigr). \end{aligned}$$
That is,
$$\begin{aligned} \biggl\Vert \varPhi \bigl(\Re (A\sharp B) \bigr)\varPhi ^{-1} \biggl( \Re \biggl( \biggl( \frac{A ^{-1}+B^{-1}}{2} \biggr)^{-1} \biggr) \biggr) \biggr\Vert \le \sec ^{2}(\alpha )K(h). \end{aligned}$$
This completes the proof. □