# 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:

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:

The Heinz mean may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.[3][4]

## References

- E. Heinz (1951), "Beiträge zur Störungstheorie der Spektralzerlegung",
*Math. Ann.*,**123**, pp. 415–438. - 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. - 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. - 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.

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