The calculation of effective mass for silicon

From what it was written, the calculation of effective mass in semiconductor had two different types, one is effective mass for density of states, through geometrical mean, the other one was the effective mass for conductivity, through harmonic mean.

Question 1:
Why it was that for density of states, the calculation of effective mass was thorough geometrical mean, while, for the conductivity, it was done through harmonic mean?

Question 2:
In the calculation of geometrical mean, $m^*_{density}=(g^2 m_x m_y m_z)^{1/3}$. Why there was a power of $2$ for the multiplicity factor, i.e. $g^2$, but not a simply summation, i.e. $g$?

Question 3:
I saw an expression for the effective mass for valance band $n=3,4$ of silicon, written as
$m_h^*=(m_{lh}^{3/2} +m_{hh}^{3/2})^{2/3}$. Could you explain what kind of effective mass it was? and how was it derived, please?