Bound on tail of beta distribution

Question

Let $X$ be a random variable with a beta distribution $\beta(j,k)$. Is there a convenient upper bound for the left tail when $j$ and $k$ are large: $$ \mathbb{P}(X \leq \varepsilon) \leq ?? $$

Answer 1

\begin{align} P(X\leq \epsilon) &= \frac{1}{B(j,k)}\int_0^\epsilon t^{j-1}(1-t)^{k-1}dt\\ &< \frac{1}{B(j,k)}\int_0^\epsilon t^{j-1} dt\\ &= \frac{\epsilon^j}{jB(j,k)} \end{align}