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]

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