Let $p>1,s>0$ and $F$ be a function such that $$F(t)\le \frac{|t|^p}{p} +|t|^{p+p^{\prime}}e^{|t|^{p^{\prime}}}\quad\mbox{ for all } t\in\mathbb{R},$$ where $p^{\prime}$ denotes the conjugate exponent of $p$. I would like to prove that the quantity $$\frac{|t|^p}{s+1} e^{|t|^{p^{\prime}}} -2N F(t)$$ is bounded from below.
I tried using the above inequality in that way $$\frac{|t|^p}{s+1} e^{|t|^{p^{\prime}}} -2N F(t)\ge \frac{|t|^p}{s+1} e^{|t|^{p^{\prime}}}-2N\frac{|t|^p}{p} -2N|t|^{p+p^{\prime}}e^{|t|^{p^{\prime}}} $$ but I don't know how to go from here on out.
Could someone please help?
Thank you in advance!
$\bf EDIT:$ thanks to the answer of @uniquesolution, I see that the quantity is not bounded from below. I am wondering if assuming that $$F(t)\le \frac{|t|^{p-p^{\prime}}}{p^{\prime}}e^{|t|^{p^{\prime}}}\quad\mbox{ for all } t\in\mathbb{R},$$ the quantity $$\frac{|t|^p}{s+1} e^{|t|^{p^{\prime}}} -2N F(t)$$ is bounded from below. I guess the answer would be yes in this case because I have $$\frac{|t|^p}{s+1} e^{|t|^{p^{\prime}}} -2N F(t)\ge |t|^p e^{|t|^{p^{\prime}}}(1/(s+1)-2(N-1))|t|^{-p^{\prime}}$$ which to me seems to be bounded from below.
Could someone please help me to understand if am I wrong or not? Thank you.
Assuming $N$ to be independent of $t$, for $p=p'=2$ and $F(t)=t^2/2+t^4e^{t^2}$ the quantity in question becomes $$e^{t^2}t^2/(s+1)-2N(t^2/2+t^4e^{t^2})$$ which is unbounded from below.