Abstract
In this paper, we prove the operator inequalities as follows: Let \(A,B\) be positive operators on a Hilbert space with \(0 < m \le A,B \le M\) and \(\sqrt{\frac{M}{m}} \le2.314\). Then for every positive unital linear map Φ,
and
Moreover, we prove Lin’s conjecture when \(\sqrt{\frac{M}{m}} \le 2.314\).
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1 Introduction
Let \(\mathcal{B(H)}\) be the \(C^{*}\)-algebra of all bounded linear operators on a Hilbert space \(\mathcal{H}\). Throughout this paper, a capital letter denotes an operator in \(\mathcal{B(H)}\), we identify a scalar with the identity operator I multiplied by this scalar. We write \(A\ge0\) to mean that the operator A is positive. A is said to be strictly positive (denoted by \(A>0\) ) if it is a positive invertible operator. A linear map \(\Phi:\mathcal{B(H)} \to\mathcal {B(K)}\) is called positive if \(A\ge0\) implies \(\Phi(A)\ge0\). It is said to be unital if \(\Phi(I)=I\). For \(A,B>0\), the geometric mean \(A\mathrel{\sharp} B\) is defined by
Let \(0 < m \le A,B \le M\). Tominaga [1] showed that the following operator reverse AM-GM inequality holds:
where \(S ( h ) = \frac{{h^{\frac{1}{{h - 1}}} }}{{e\log h^{\frac{1}{{h - 1}}} }}\) is called Specht’s ratio and \(h = \frac {M}{m}\). Indeed,
was observed by Lin [2, (3.3)].
Let Φ be a positive linear map and \(A,B > 0\). Ando [3] gave the following inequality:
By (1.1), (1.2) and (1.3), it is easy to obtain the following inequalities:
and
Lin [2] proved that (1.4) and (1.5) can be squared:
and
Meanwhile, Lin [2] conjectured that the following inequalities hold:
and
For more information on operator inequalities, the reader is referred to [4–7].
In this paper, we will present some operator reverse AM-GM inequalities which are refinements of (1.1), (1.6) and (1.7). Furthermore, we will prove (1.8) and (1.9) if the condition number \(\sqrt{\frac{M}{m}} \) is not too big.
2 Main results
We begin this section with the following lemmas.
Lemma 1
([8])
Let \(A,B>0\). Then the following norm inequality holds:
Lemma 2
([9])
Let \(A>0\). Then for every positive unital linear map Φ,
Theorem 1
If \(0 < m \le A,B \le M\) for some scalars \(m\le M \), then
Proof
Put \(C = A^{ - \frac{1}{2}} BA^{-\frac{1}{2}} \). Since \(\frac{m}{M} \le C \le\frac{M}{m}\), it follows that
and hence
This implies
Thus
This completes the proof. □
Remark 1
By (1.2), it is easy to know that (2.3) is tighter than (1.1).
Theorem 2
If \(0 < m \le A,B \le M\) and \(\sqrt{\frac {M}{m}} \le2.314\) for some scalars \(m\le M \), then
Proof
Inequality (2.4) is equivalent to
If \(0 < m \le A,B \le\frac{{M + m}}{2}\), we have
and
Compute
That is,
Since \(1 \le\sqrt{\frac{M}{m}} \le2.314 \), it follows that
It is easy to know that \(\frac{{ ( {\frac{{M + m}}{2} + m} )^{2} }}{{4 \frac{{M + m}}{2} m}} \le\frac{{M + m}}{{2\sqrt{Mm} }}\) is equivalent to (2.8).
Thus,
If \(\frac{{M + m}}{2} \le A,B \le M\), we have
and
Similarly, we get
If \(m \le A \le\frac{{M + m}}{2} \le B \le M \), we have
That is,
If \(m \le B \le\frac{{M + m}}{2} \le A \le M \), similarly, by (2.1), (2.7) and (2.9), we have
This completes the proof. □
Theorem 3
Let Φ be a positive unital linear map. If \(0 < m \le A,B \le M\) and \(\sqrt{\frac{M}{m}} \le2.314\) for some scalars \(m\le M \), then
and
Proof
Inequality (2.11) is equivalent to
If \(0 < m \le A,B \le\frac{{M + m}}{2}\), compute
By \(1 \le\sqrt{\frac{M}{m}} \le2.314 \) and (2.8), we have
If \(0 < \frac{{M + m}}{2} \le A,B \le M\), similarly, by (2.1), (2.2), (2.8), (2.9), (2.10) and \(\frac {{ ( {\frac{{M + m}}{2} + M} )^{2} }}{M} \le\frac{{ ({\frac{{M + m}}{2} + m} )^{2} }}{m}\), we have
If \(m \le A \le\frac{{M + m}}{2} \le B \le M \), we have
That is,
If \(m \le B \le\frac{{M + m}}{2} \le A \le M \), similarly, by (2.1), (2.2), (2.7), (2.9), we have
Thus (2.11) holds.
A and B are replaced by \(\Phi ( A )\) and \(\Phi (B )\) in (2.4), respectively, we get (2.12).
This completes the proof. □
Remark 2
Since \(0 < m \le M\), then \(\frac{{ ( {M + m} )^{2} }}{{4Mm}} \le [ {\frac{{ ( {M + m} )^{2} }}{{4Mm}}} ]^{2} \). Thus (2.11) and (2.12) are refinements of (1.6) and (1.7), respectively, when \(\sqrt{\frac{M}{m}} \le2.314\).
By (1.2) and Theorem 3, we know that Lin’s conjecture (1.8) and (1.9) hold when \(\sqrt{\frac{M}{m}} \le2.314\).
Corollary 1
Let Φ be a positive unital linear map. If \(0 < m \le A,B \le M\) and \(\sqrt{\frac{M}{m}} \le2.314\) for some scalars \(m\le M \), then
and
where \(S ( h ) = \frac{{h^{\frac{1}{{h - 1}}} }}{{e\log h^{\frac{1}{{h - 1}}} }}\), \(h = \frac{M}{m}\).
References
Tominaga, M: Specht’s ratio in the Young inequality. Sci. Math. Jpn. 55, 583-588 (2002)
Lin, M: Squaring a reverse AM-GM inequality. Stud. Math. 215, 187-194 (2013)
Ando, T: Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26, 203-241 (1979)
Xue, J, Hu, X: A note on some inequalities for unitarily invariant norms. J. Math. Inequal. 9, 841-846 (2015)
Xue, J, Hu, X: Some generalizations of operator inequalities for positive linear maps. J. Inequal. Appl. 2016, 27 (2016)
Zou, L: An arithmetic-geometric mean inequality for singular values and its applications. Linear Algebra Appl. 528, 25-32 (2017)
Hu, X: Some inequalities for unitarily invariant norms. J. Math. Inequal. 6, 615-623 (2012)
Bhatia, R, Kittaneh, F: Notes on matrix arithmetic-geometric mean inequalities. Linear Algebra Appl. 308, 203-211 (2000)
Bhatia, R: Positive Definite Matrices. Princeton University Press, Princeton (2007)
Acknowledgements
This research was supported by the Scientific Research Fund of Yunnan Provincial Education Department (No. 2014Y645).
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Xue, J. Some refinements of operator reverse AM-GM mean inequalities. J Inequal Appl 2017, 283 (2017). https://doi.org/10.1186/s13660-017-1557-y
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DOI: https://doi.org/10.1186/s13660-017-1557-y