Say $X\sim\mathrm{Beta}(\alpha,\beta)$. Are there any "nice" closed form upper bounds for the tail probability $P(X\geq\epsilon)$, that are reasonably tight when $\beta$ is large? By "nice" I mean involving only elementary functions, and not for instance the incomplete beta function.

In my specific setting I have $\alpha=1/2$ and $\beta=d/2$ for some (typically large) integer $d$, and so one way of bounding the tail is to write $X=\frac{Y}{Y+Z}$ where $Y\sim\chi^2_1$ and $Z\sim\chi^2_d$, and using an upper bound on $Y$ and lower bound on $Z$ (e.g. from here: What are the sharpest known tail bounds for $\chi_k^2$ distributed variables?). This gives an answer, but it's somewhat messy (requiring the small $d$ case to be handled separately, etc). Is there a cleaner way?

Thanks.