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| In mathematics, the '''Heinz mean''' (named after [[Erhard Heinz|E. Heinz]]<ref>E. Heinz (1951), "Beiträge zur Störungstheorie der Spektralzerlegung", ''Math. Ann.'', '''123''', pp. 415–438.</ref>) of two non-negative [[real number]]s ''A'' and ''B'', was defined by Bhatia<ref>{{citation|first=R.|last=Bhatia|title=Interpolating the [[arithmetic-geometric mean inequality]] and its operator version|journal=Linear Algebra and its Applications|volume=413|issue=2–3|pages=355–363|year=2006|doi=10.1016/j.laa.2005.03.005}}.</ref> as:
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| :<math>H_x(A, B) = \frac{A^x B^{1-x} + A^{1-x} B^x}{2}.</math>
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| with 0 ≤ ''x'' ≤ 1/2.
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| For different values of ''x'', this Heinz mean interpolates between the [[arithmetic mean|arithmetic]] (''x'' = 0) and [[geometric mean|geometric]] (''x'' = 1/2) means such that for 0 < ''x'' < 1/2:
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| :<math> \sqrt{A B} = H_{1/2}(A, B) < H_x(A, B) < H_0(A, B) = \frac{A + B}{2}. </math> | |
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| The Heinz mean may also be defined in the same way for [[positive semidefinite matrix|positive semidefinite matrices]], and satisfies a similar interpolation formula.<ref>{{citation|first1=R.|last1=Bhatia|first2=C.|last2=Davis|authorlink2=Chandler Davis|title=More matrix forms of the [[arithmetic-geometric mean inequality]]|journal=SIAM Journal on Matrix Analysis and Applications|volume=14|issue=1|pages=132–136|year=1993|doi=10.1137/0614012}}.</ref><ref>{{citation|first=Koenraad M.R.|last=Audenaert|title=A singular value inequality for Heinz means|arxiv=math/0609130 |journal=Linear Algebra and its Applications|volume=422|issue=1|pages=279–283|year=2007|doi=10.1016/j.laa.2006.10.006}}.</ref>
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| ==See also==
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| *[[Mean]]
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| *[[Muirhead's inequality]]
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| *[[Inequality of arithmetic and geometric means]]
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| ==References==
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| {{reflist}}
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| [[Category:Means]]
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| {{Mathapplied-stub}}
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Latest revision as of 15:51, 23 November 2014
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