A function space satisfying $ |f(x) -f(y)| \le ||x|^\theta -|y|^\theta| $

Question

Consider a class of function on the real line such that $$ |f(x) -f(y)| \le ||x|^\theta -|y|^\theta | $$ for a $\theta >0$.

Does this class of function space have a name When $ |f(x) -f(y)| \le |x-y|^\theta $, it is the $\theta-Holder $ function space. My condition is a little bit stronger. I think it should have be considered somewhere. Can anyone provide any information about this class of function space? or some function spaces which satisfy this condition.