Let $G$ be a nonempty bounded domain in a finite-dimensional Euclidean space. The main results are general estimates from below at points from $G$ for an arbitrary subharmonic function $u\not\equiv -\infty$ on the closure of the domain $G$ through the maximum of the function $u$ on the boundary of the domain $G$. These results are new for planar domains $G$, and for intervals of $G$ on the numerical line have also not been previously noted. They show that the Harnack distance plays a key role in these estimates. Further applications to subharmonic, convex, holomorphic functions, as well as to meromorphic functions and differences of subharmonic functions in domains of a particular type are supposed to be outlined in the continuation of this article.