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J. Kiefer,
1959
Earlier work this paper cites.
H. Epstein,
1973
Earlier work this paper cites.
E. H. Lieb,
1973
Earlier work this paper cites.
E. H. Lieb and M. B. Ruskai,
1974
Earlier work this paper cites.

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T. Ando,
1979
Cited alongside, same era.
T. N. Bekjan,
2004
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E. A. Carlen and E. H. Lieb,
Original
2008
Cited alongside, same era.
E. A. Carlen,
2010
Cited alongside, same era.

Then

V. Jaksic, Y. Ogata, Y. Pautrat and C.-A. Pillet,
2012
Later among the works it cites.
R. L. Frank and E. H. Lieb,
Original
2013
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F. Hiai,
Original
2013
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K. M. R. Audenaert and N. Datta,
Original
2015
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2014

Some Operator and Trace Function Convexity Theorems

Carlen, Eric A., Frank, Rupert L., Lieb, Elliott H.

Understand

We consider convex trace functions $\Phi_{p,q,s} = Trace[ (A^{q/2}B^p A^{q/2})^s]$ where $A$ and $B$ are positive $n\times n$ matrices and ask when these functions are convex or concave.

We also consider operator convexity/concavity of $A^{q/2}B^p A^{q/2}$ and convexity/concavity of the closely related trace functional $Trace[ A^{q/2}B^p A^{q/2} C^r]$.
For concavity, these questions are completely settled, thereby settling cases left open by Hiai, while the convexity questions are settled in many cases.
As a consequence, the Audenaert-Datta R\'enyi entropy conjectures are proved for some cases.