Let $\alpha \in (0, \infty)$ and $\beta \in (0, 1]$. We assume $f : [0, 1] \to [0, \infty)$ is a measurable and bounded function such that
$$
f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s,
\quad \forall t \in [0, 1].
$$

There [exists][1] a constant $C$ (depending on $\alpha, \beta$) such that $\sup_{t \in [0, 1]} f(t) \le C$.
>Can we pick such constant $C$ such that if $\alpha=0$ then $f=0$?

Thank you so much for your elaboration!


  [1]: https://mathoverflow.net/questions/467786/gr%C3%B6nwall-type-inequality-for-ft-le-alpha-int-0t-t-s-frac12-f