Relative and mutual monotonicity

Abstract

In this work we first consider a certain monotonicity relative to some given one-to-one operator and prove the counterparts, adjusted to this new context, of most results obtained before in the joint work with G. Kassay \cite{Kassay-Pintea}.
For two operators with the same status relative to injectivity,
such as two local injective operators, we define what we call mutual $h$-monotonicity and prove that every two  mutual $h$-monotone local diffeomorphisms can be obtained from each other via a composition with a $h$-monotone diffeomorphism.