How to prove that $|\ln(1+x^4)) - \ln(1+y^4)| <= 3^\frac34|x-y| \space \forall \space x,y \in \mathbb R$

Question

Question states

Prove that for all $x$ and $y$ $\in R$, the following inequality is true:

$\lvert \ln(1+x^4) - \ln(1+y^4)\rvert \le 3^\frac34\lvert x-y\rvert$

can you please solve this using mean value theorem? (it should be done this way)

Answer 1

By Lagrange's theorem, it is enough to show that the derivative of $f(z)=\log(1+z^4)$ is bounded by the constant $3^{3/4}$. Indeed $|f'(z)|=\frac{4|z|^3}{1+z^4}\leq 3^{3/4}$ is a consequence of the AM-GM inequality:

$$|z|^3 = \frac{1}{3^{1/4}}\text{GM}(3,z^4,z^4,z^4)\leq \frac{1}{3^{1/4}}\text{AM}(3,z^4,z^4,z^4)=\frac{3^{3/4}}{4}(1+z^4). $$