Behaviour of $\|f\|_{\lambda,[0,T]}$ as $T\to0$.

Question

Let $f:[0,T]\to\Bbb R$ be $\alpha$-Holder continuous function, for some $0<\alpha\le1$. Then $$ \|f\|_{\lambda,[0,T]}:=\sup_{0\le s<t\le T}\frac{|f(t)-f(s)|}{|t-s|^{\alpha}}<+\infty. $$

Now it's clear that $\|f\|_{\lambda,[0,T]}\to0$ as $T\to0$.

Is it known how it goes to zero? For example, does exists some function of $\alpha$, say $\eta(\alpha)$ such that $$ \lim_{T\to0}\frac{\|f\|_{\lambda,[0,T]}}{T^{\eta(\alpha)}}=1? $$

EDIT: above I wrongly wrote "now it's clear that...". This is not true as pointed out by Davide Giraudo and Fred.

However I am interested in the same question supposing that: that is, IF $\|f\|_{\lambda,[0,T]}\to0$ as $T\to0$, is it known how it goes to zero?

Answer 1

Let $f(x)=x$ and $ \alpha=1$. Then $\|f\|_{\lambda,[0,T]}=1$ for all $T>0$ !