A What type of function satisfy a type of growth condition?

Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established:
\begin{equation}
||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),
\end{equation}
with ##\mathcal{X}:= \{x:f(x)=0\}## (zero set) and some function ##g## (like a homogeneous function).

I am interested to know the class of functions. Any help will help a lot. Thanks in advance
 

mathman

Science Advisor
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For ##n=1,## any of the form ##x^p## where ##p\ge 1 ## will do. ##g(u)=u## works. This can easily be generalized for ##n\gt 1##.
 
Yes, that is true. But I am looking for more general class of function and a well-established theory on the conditions and properties of such functions. This is a bit similar to holder conditions but on the inverse of ##f##.
 

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