Heinz mean

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

${\mathrm{H}}_x(A,B)=\frac{A^x{B}^{1-x}+A^{1-x}{B}^x}{2},$

with 0 ≤ x ≤ Template:Sfrac.

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 < Template:Sfrac:

$\sqrt{AB}={\mathrm{H}}_{\frac{1}{2}}(A,B)<{\mathrm{H}}_x(A,B)<{\mathrm{H}}_0(A,B)=\frac{A+B}{2}.$

The Heinz means appear naturally when symmetrizing $\alpha$-divergences.

It may also be defined in the same way for positive semidefinite matrices, and satisfies a similar interpolation formula.

See also

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

References

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