Letters in Mathematical Physics

, Volume 19, Issue 2, pp 167–170

On an inequality of Lieb and Thirring


  • Huzihiro Araki
    • Research Institute for Mathematical SciencesKyoto University

DOI: 10.1007/BF01045887

Cite this article as:
Araki, H. Lett Math Phys (1990) 19: 167. doi:10.1007/BF01045887


The following generalization of an inequality of Lieb and Thirring is proved:
$$Tr\{ b^{1 2} ab^{1 2} )^{qk} \} \leqslant Tr\{ (b^(q, 2) a^(q) b^(q 2)^k \} $$
for all positive selfadjoint operatorsa andb and for positive numbersq>1 andk>0. More generally,
$$Tr\varphi ((b^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ab^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} )q) \leqslant Tr\varphi (b^{qk} a^q b^{{q \mathord{\left/ {\vphantom {q 2}} \right. \kern-\nulldelimiterspace} 2}} q)$$
for any monotone increasing continuous function ϕ on (0, ∞) such that ϕ(0)=0 and ξ→ϕ(eξ) is convex.

AMS subject classification (1980)


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© Kluwer Academic Publishers 1990