Heinz mean explained

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:

\operatorname{H} _{x(A,} B) = (A ^{x} B ^{1-x} + A ^{1-x} B ^{x})/(2) ,

with 0 ≤ x ≤ .

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 < :

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

The Heinz means appear naturally when symmetrizing $\backslash alpha$-divergences.^[3]

It 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

Notes and References

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