Tight bounds for the median of a gamma distribution

The median of a standard gamma distribution, as a function of its shape parameter $k$, has no known representation in terms of elementary functions. In this work we prove the tightest upper and lower bounds of the form $2^{-1/k} (A + k)$: an upper bound with $A = e^{-\gamma}$ that is tight for low $k$ and a lower bound with $A = \log(2) - \frac{1}{3}$ that is tight for high $k$. These bounds are valid over the entire domain of $k > 0$, staying between 48 and 55 percentile. We derive and prove several other new tight bounds in support of the proofs.