A note on diastatic entropy and balanced metrics

We give un upper bound Ent(\Omega, g)<\lambda\ of the diastatic entropy Ent(\Omega, g) of a complex bounded domain (\Omega, g) in terms of the balanced condition (in Donaldson terminology) of the Kaehler metric \lambda g. When (\Omega, g) is a homogeneous bounded domain we show that the converse holds true, namely if Ent(\Omega, g)<1 then g is balanced. Moreover, we explcit compute Ent(\Omega, g) in terms of Piatetski-Shapiro constants.