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Under the additional condition that $$|f(x)-f(y)|\le|g(x)-g(y)| \tag{1}$$ for all $x$ and $y$, your desired inequality ($\ast$) holds with $C=1$. This follows from Lemma 5, page 7 (note also the remark "Both definitions agree for function classes which are closed under negation" in line -10 on the same page). The proof of that lemma is short and very clever. I am afraid that without (1) your inequality (*) may fail to hold, even though it may be hard to construct a counterexample, given the presence of the factor $C$.