Let $f:[0,T]\to\Bbb R$ be an $\alpha$-Holder continuous function with $1/2<\alpha\le1$, such that $f(0)=0$.
I was asking myself a couple of things:
1) what can we say about the regularity of $$ t\mapsto\max_{0\le s\le t}|f(s)|=:h(t)\;? $$ I proved that it is $\alpha$-Holder too, but I don't know if it is exactly $\alpha$-Holder, or if there exists $\alpha<\alpha'\le1$ such that it is $\alpha'$-Holder too.
2)fixed $s=0$, in MO someone said me that (observe that $h(0)=0$) $$ \lim_{t\to 0^+}\frac{|h(t)|}{t^{\alpha}} $$ doesn't exist in general. It is easy to prove that the $\limsup_{t\to 0^+}$ is finite but what I'm really interested in, is to show that $$ \liminf_{t\to 0^+}\frac{|h(t)|}{t^{\alpha}}>0. $$
Ant hint is welcome!
Consider two scenarios:
$f$ is positive and increasing. Then $h=f$, so there is no improvement in Hölder exponent.
$f$ is positive and decreasing. Then $h\equiv f(0)$, which shows $$\liminf_{t\to s^+}\frac{|h(t)-h(s)|}{(t-s)^{\alpha}}$$
may well be zero.
So, the continuity may stay the same or improve a lot.