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 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!