Exchanging max, log and absolute value

Question

Let $N \in \mathbb{N}$, $f,g: [N] \to (0,1]$. It is a consequence of the triangle inequality that :

$$\left|\max_{n \in [N]} f(n) - \max_{n \in [N]} g(n)\right| \leq \max_{n \in [N]} \left| f(n) - g(n) \right|$$

But is the following also true ?

$$\left|\ln\left( \frac{\max_{n \in [N]} f(n)} {\max_{n \in [N]} g(n)} \right) \right| \leq \max_{n \in [N]} \left| \ln \left( \frac{f(n)}{g(n)} \right) \right|$$

Answer 1

Logarithm is an increasing function, hence

$$\ln (\max_{n \in [N]} f(n))=\max_{n \in [N]} \ln (f(n))$$

Your problem is equivalent to

$$\left|\max_{n \in [N]} \ln (f(n)) - \max_{n \in [N]} \ln (g(n)) \right| \le \max_{n \in [N]} \left|\ln (f(n)) - \ln (g(n)) \right|$$

Hence, if the proof of the first inequality doesn't depend on the range of $f$ or $g$ and if it is true for all real number, then the second inequalities hold.