I am looking for a example of a function in $C_0(\mathbb{R})$ such that $f',f'' \,\text{and}\, f''' \in C_0(\mathbb{R})$ with 
$$ \inf f < \inf (f-a*f''')$$ for some $a>0$, but I couldn't find one yet. I've tried functions like $$-\frac{1}{1+x^2}, -\frac{x^3}{1+x^4}, -e^{-x^2}$$, but none of these worked ... thank you for your answers in advance!