In a lecture on Applied Functional Analysis, the professor showed us some properties of the Takagi function from this paper. He wrote at the end the following property and said it could be easily done using induction. However, I can not still figure out how.
Let $$f:\mathbb{R}\to\mathbb{R}, \; f(x):= \sum_{n=0}^\infty 2^{-n}d(2^nx)$$ be the Takagi function, where $d(x) = \min_{k\in \mathbb{N}} \{ |x-k| \}$. Show that for all $x,y \in \mathbb{R}$ such that $|x-y|<1$ holds $$|f(x)-f(y)| \leq |x-y| \left( 1-\frac{\ln|x-y|}{\ln2} \right).$$
"The hints": The inequality holds for $f_m(x):= \sum_{n=0}^m 2^{-n}d(2^nx)$. And consider the operator $[Tf](x) = \frac{1}{2}f(2x)+d(x)$ with the property $f_{m+1} = Tf_m$.
I know that the $f_m$ approach the Takagi function pointwise from below. Then I can take limit pointwise to get the result since this holds for all $m\in\mathbb{N}$. This other paper at 8.1 shows somehow an approach to this hölder continuity but using $\log_{2}$. However, this is still for me unclear how to come up with this $\ln$ term by induction.
Thanks.