A note on diastatic entropy and balanced metrics

We give an upper bound Entd(Ω,g)<λ of the diastatic entropy Entd(Ω,g) (defined by the author in Mossa (2012) of a complex bounded domain (Ω,g) in terms of the balanced condition (in Donaldson terminology) of the Kähler metric λg. When (Ω,g) is a homogeneous bounded domain we show that the converse holds true, namely if Entd(Ω,g)<1 then g is balanced. Moreover, we explicitly compute Entd(Ω,g) in terms of Piatetski-Shapiro constants.