Ando-Hiai and Golden-Thomspon inequalities

The original Ando-Hiai and Golden-Thompson inequalities present comparisons for the operator geometric mean ${\mathrm{♯}}_v$ when $0\le v\le 1.$ Our main target in this article is to study these celebrated inequalities for means other than the geometric mean and for the geometric mean when $v\in ̸[0,1]\mathrm{.}$