Heinz mean

In mathematics, the Heinz mean (named after E. Heinz[1]) of two non-negative real numbers A and B, was defined by Bhatia[2] as:

${\displaystyle \operatorname {H} _{x}(A,B)={\frac {A^{x}B^{1-x}+A^{1-x}B^{x}}{2}}.}$

with 0 ≤ x ≤ 1/2.

For different values of x, this Heinz mean interpolates between the arithmetic (x = 0) and geometric (x = 1/2) means such that for 0 < x < 1/2:

${\displaystyle {\sqrt {AB}}=\operatorname {H} _{\frac {1}{2}}(A,B)<\operatorname {H} _{x}(A,B)<\operatorname {H} _{0}(A,B)={\frac {A+B}{2}}.}$

The Heinz means appear naturally when symmetrizing ${\textstyle \alpha }$-divergences.[3]

The Heinz mean may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.[4][5]

See also

• Mean
• Muirhead's inequality
• Inequality of arithmetic and geometric means

References

1. E. Heinz (1951), "Beiträge zur Störungstheorie der Spektralzerlegung", Math. Ann., 123, pp. 415–438.
2. Bhatia, R. (2006), "Interpolating the arithmetic-geometric mean inequality and its operator version", Linear Algebra and Its Applications, 413 (2–3): 355–363, doi:10.1016/j.laa.2005.03.005.
3. Nielsen, Frank; Nock, Richard; Amari, Shun-ichi (2014), "On Clustering Histograms with k-Means by Using Mixed α-Divergences", Entropy, 16 (6): 3273–3301, Bibcode:2014Entrp..16.3273N, doi:10.3390/e16063273.
4. Bhatia, R.; Davis, C. (1993), "More matrix forms of the arithmetic-geometric mean inequality", SIAM Journal on Matrix Analysis and Applications, 14 (1): 132–136, doi:10.1137/0614012.
5. Audenaert, Koenraad M.R. (2007), "A singular value inequality for Heinz means", Linear Algebra and Its Applications, 422 (1): 279–283, arXiv:math/0609130, doi:10.1016/j.laa.2006.10.006.