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

- under what condition, we have

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

- 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?

comonotone$\endgroup$