Monotonicity formulas and (S${}_{+}$)-property: old and new

The connection between monotonicity formulas and the (S${}_{+}$)-property is that, for some popular differential operators, the former is used to prove the latter. The purpose of this paper is to explore this connection, remark how in the past both the monotonicity formulas and the (S${}_{+}$)-property were focused on power-law growth, and prove the same type of results for a more general class of operators.