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Supported by the National Natural Science Foundation of China under grant 10871051, Shanghai Municipal Science & Technology Committee under grant 09DZ2272900, Shanghai

Research supported by National Natural Science Foundation of China (under Grant 10571047) and the Doctorate Foundation of the Ministry of Education of China (under Grant

The work of this author was supported by the NSFC Tianyuan Mathematics Youth Fund (10926086), and in part by Independent Innovation Research Plan of CUP (09CX04004A). ‡ School

This work is financially supported by self-determined research funds of CCNU (CCNU09Y01005, CCNU09Y01018) from the colleges’ basic research and operation of MOE. Li’s

This work is Project 10671026 supported by National Natural Science Foundation of China, Postdoctoral Scientific Research Foundation of Heilongjiang Province (no. HB200801165) and

This work was supported in part by the National Natural Science Foundation of China 10971176, the Guangdong Provincial Natural Science Foundation of China 10152104101000008 and the

This work was partially supported by the Ministerio de Ciencia e Innovaci´ on of Spain through grant MTM-2009-09281.. † Departamento de Matem´ aticas, Universidad Carlos III de

A key tool to obtain these bounds is using some Hermitian matrices which are, in the sense of Loewner matrix ordering, below the Hermitian part of A or, more generally, below