  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
exclusively on power-law growth and prove the same type of results for a more
general class of operators.
