Given two   $n\times n$ positive definite matrices $A$ and $B$, the metric geometric mean   introduced by Pusz and Woronowicz in 1975 is

$$ A\sharp B :=   A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$

and the spectral   geometric mean introduced by Fiedler and Pt\'ak in 1997 is

$$ A \natural B   := (A^{-1}\sharp B)^{1/2}A(A^{-1}\sharp B)^{1/2}.$$

For $0\le t\le   1$, the $t$-metric geometric mean ($t$-geometric mean, for short) and   $t$-spectral geometric mean ($t$-spectral mean, for short) of $A$ and $B$ are   defined by

$$ A \sharp_t B   & := &A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2},

A \natural_t B   & := &(A^{-1}\sharp B)^{t}A(A^{-1}\sharp B)^{t}.$$

Some log majorization result involving $t$-metric geometric mean and $t$-spectral   geometric mean are given. Some questions are asked.