L\u27hôpital\u27s Monotone rule, Gromov\u27s theorem, and operations that preserve the monotonicity of quotients

We study several operators T that when applied to both the numerator and denominator of an increasing, or decreasing, function u/v produce another increasing, or decreasing, function T(u)/T(v). We also give new proofs of the monotone form of L\u27Hôpital\u27s rule and of Gromov\u27s theorem