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Oct 26, 2014 at 23:58 comment added Suvrit Actually your answer does answer the original question, by creating weak majorization inequalities for the entire range, rather than just the $k=1,..,n$ case.
Oct 26, 2014 at 16:25 comment added Yanqi QIU I realize I didn't really answer the original question. Sorry.
Oct 26, 2014 at 16:21 history edited Yanqi QIU CC BY-SA 3.0
added 17 characters in body
Oct 26, 2014 at 16:14 history answered Yanqi QIU CC BY-SA 3.0