On the Araki-Lieb-Thirring inequality

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We prove an inequality that complements the famous Araki-Lieb-Thirring (ALT) inequality for positive matrices A and B, by giving a lower bound on the quantity [Aʳ Bʳ Aʳ]q in terms of [ABA]rq for 0≤ r≤ 1 and q≥0, whereas the ALT inequality gives an upper bound. The bound contains certain norms of A and B as additional ingredients and is therefore of a different nature than the Kantorovich type inequality obtained by Bourin (Math. Inequal. Appl. 8(2005) pp. 373--378) and others. Secondly, we also prove a generalisation of the ALT inequality to general matrices.