Consider a function $f(x)$ with a point-wise Holder exponent $\beta \leq 1$.
Definition of point-wise Holder exponent: $$ \beta_x: = \sup \left\lbrace \beta: \limsup_{h \rightarrow 0^+} \left|\frac{f(x+h) -f(x)}{h^{\beta} } \right| =0 \right\rbrace $$
That is $|f(x+h) -f(x) | \leq C h^{\beta_x}$ uniformly in $[x, \delta]$, $\delta>x$.
Is the reverse Holder inequality true: that is $$ K h^{\beta_x} \leq |f(x+h) -f(x) | $$ for some $K>0$?
Remark:
Constant functions $f(x)=C $ are of any order $\beta> 1$ so they are not admissible.
To avoid confusion, let me answer the question now that you've edited it: Let $f=0$. Clearly $f$ is $\beta$-Holder. However for any $h$, $|f(x+h)-f(x)|=0$ so there can be no such positive $K$.