Consider a Beta random variable $X$ with shape parameters $k/2$ and $(d-k)/2$, where the parameters $k, d$ are integers that satisfy $0 < k < d$. What is the best possible upper bound for the following probability? $$ \mathbb{P}\{ X < \mathbb{E}(X) \} $$ So far, after several days of toil, the best result I have been able to prove is a wretched 0.922. This is based on a nontrivial normal approximation proposed by Alfers & Dinges (1984) and refined by Artstein (2002).
Numerical examples suggest that the extremal case occurs when $k = 1$ for every $d \geq 1$. Furthermore, for this case, the limiting value as $d \to \infty$ appears to be equal to $\Phi(1) - \Phi(-1)$, where $\Phi$ is the standard normal cdf.
I can obtain considerably better bounds for this special case, but I have no idea how to prove that these parameters are extremal. Any thoughts?