Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be a monotonic function that is smooth. I am considering the quotient $\frac{g(f(x_1))}{g(f(x_2))}$ for some $x_1,x_2 \in \mathbb{R}$. Is is it possible to express or bound this quotient using knowledge on $\frac{f(x_1)}{f(x_2)}$? My inspiration comes from Lipschitz functions in which we have the definition $|g(f(x_1)) - g(f(x_2))| \le K|f(x_1)-f(x_2)|$ for some constant $K$. Are similar expressions available for quotients?
2026-03-25 09:27:32.1774430852
quotient of monotonic transformation of a function
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