A difference counterpart to a matrix Hölder inequality
- Fujii, Masatoshi
- Lee, Eun-Young
- Seo, Yuki
DOIAbstract
AbstractWe point out a sharp reverse Cauchy-Schwarz/Hölder matrix inequality. The Cauchy-Schwarz version involves the usual matrix geometric mean: Let Ai and Bi be positive definite matrices such that 0<mAi⩽Bi⩽MAi for some scalars 0<m⩽M and i=1,2,⋯,n. Then∑i=1nAi♯∑i=1nBi-∑i=1nAi♯Bi⩽(M-m)24(M+m)∑i=1nAi,where the matrix geometric mean of positive definite A and B is defined byA♯B=A12A-12BA-1212A12
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