Can we find a function $f:\mathbb{R}\to(0,\infty)$ which satisfies
$$\limsup_{|x|\to + \infty}\frac{f(x+c)}{f(x)}<+\infty, \ \ \forall c\in \mathbb{R},(\text{limit in }+\infty\text{ and }-\infty),\tag{1}\label{1}$$
$$\sup_{t\in \mathbb{R}}\int_t^{t+1}f(x)dx<+\infty,\tag{2}\label{2}$$ and $$\limsup_{r\to + \infty}\frac{r}{\int_{-r}^r f(x)dx}<+\infty,\tag{3}\label{3}$$ but not $$\limsup\limits _{r\to\infty}\frac{r^{\frac{1}{p}}\left(\int_{-r}^{r}f^{q}\left(x\right)dx\right)^{\frac{1}{q}}}{\int_{-r}^{r}f\left(x\right)dx}<+\infty,\tag{4}\label{4}$$ where $q>1.$
Note: I see that $f$ must not be bounded because in this case $\eqref{3}$ will automatically imply $\eqref{4}$. $f$ also should not touch $0$ because of $\eqref{1}$.
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