monotonicity alike functions

Question

assuming we have two smooth function ${f_1},{f_2}:{R^N} \to R$,

1. under what condition, we have

${f_1}\left( {{{\bf{x}}_1}} \right) \ge {f_1}\left( {{{\bf{x}}_2}} \right) \leftrightarrow {f_2}\left( {{{\bf{x}}_1}} \right) \ge {f_2}\left( {{{\bf{x}}_2}} \right), \forall \bf{x}_1\neq \bf{x}_2\in{R^N}$

2. what property is called in mathematical terminology about $f_1,f_2$?

we can assume both of the function are convex if needed. I know that monotone transformation preserve the above property. but I am not sure if it is necessary to be able to define a monotone transformation between these two functions or not?

Answer 1

Let Property A be the property described above.

Let Property B be the property that $f_2=g \circ f_1$ for some strictly monotonically increasing function $g:R\rightarrow R$.

Then Property A and Property B are equivalent.

To see that Property B implies Property A, suppose that $f_1(x_1)\geq f_1(x_2)$. Since $g$ is monotonic, if we apply $g$ we get $f_2(x_1)\geq f_2(x_2)$. Since $g$ strictly monotonic implies that $g^{-1}$ is also strictly monotonic, we get the other direction.

To see that Property A implies Property B, assume that Property A holds. Then $f_1(x_1)=f_1(x_2)$ iff $f_2(x_1)=f_2(x_2)$. Therefore, we can define $g$ on $Im(f_1)$ by setting $g(f_1(x))=f_2(x)$; by the preceding observation, $g$ is well-defined, and $f_2=g\circ f_1$.