Most of the inequalities I know, concerning inner products induced by positive a definite matrix $P\in\mathbb{R}^{n\times n}$ (the identity as a simple example) on $\mathbb{R}^n$, involve majorizing with the norms of the entries.
My question is, given a function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ (nonlinear), is there some relevant inequality of the form $$ \langle x,f(Wx+y)-f(y) \rangle_P\leq L\langle x,Wx\rangle_P$$ where $W\in\mathbb{R}^{n\times n}$, $x,y\in\mathbb{R}^n$, and $L\in\mathbb{R}$ possibly depends on $P$, $x$ and anything needed?
My guess is that there can be something involving the Logarithmic norm of either $P$ or $W$. Of course I do not need necessarily a result holding for any $f$, it could be really helpful even some particular case or some property which, when satisfied by $f$, gives out this kind of inequality.
An interesting case might even be when $f$ is made by the same function acting componentwise. Indeed, if there is a $g:\mathbb{R}\rightarrow\mathbb{R}$ so that $f(x) = (g(x_1),...,g(x_n))$.