  The main goal of this article is to present new inequalities for the spectral
geometric mean $A\natural_t B$ of two positive definite operators $A, B$ on a
Hilbert space. The obtained results complement many known inequalities for the
geometric mean $A\sharp_t B$. In particular, explicit comparisons between
  $A\natural_t B$ and $A\sharp_t B$ are given, with applications towards
Ando-type inequalities and Ando-Hiai inequalities for $A\natural_t B$ and some
other consequences.
