**Not an answer, but a random thought** This looks very close to the Lieb/Wigner/Yamase inequality, as in [these very nice notes of Eric Carlen's.][1] 


  [1]: https://www.evernote.com/shard/s24/sh/75e8c889-96da-49af-834a-816bfb6debb3/5a45aa4413e1852987d59b4c47f8db44