Please I have this two conditions: $$\limsup_{|t|\rightarrow\infty}\frac{f(t)}{|t|^s}=0~~\text{and}~~ \lim_{t\rightarrow0} \frac{f(t)}{|t|^{p-1}}=0$$ where $2\leq p<N$ $f:\mathbb{R}\rightarrow \mathbb{R}$ is a $C^1$ function and $s\in (0,\frac{N(p-1)+p}{N-p})$ and the function $\frac{f(t)}{t^{p-1}}$ is increasing in $(0,+\infty)$
How to deduce that: $$\forall\varepsilon>0, \exists C_{\varepsilon}>0~~\text{such that}~~|f(t)|\leq\varepsilon|t|+C_{\varepsilon} t^s, \forall t\in (0,+\infty)$$
Thank you.