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On an inequality of Lieb and Thirring

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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.

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Araki, H. On an inequality of Lieb and Thirring. Lett Math Phys 19, 167–170 (1990).

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