Gaunt–Merkle conjectured bounds for the median of the variance-gamma distribution

Let $X\sim\mathrm{VG}(r,\theta,\sigma,\mu)$ with $r,\theta,\sigma>0$, and let $\mathrm{Med}(X)$ denote its median. The parameters $r$, $\theta$, $\sigma$, and $\mu$ are the shape, skewness, scale, and location parameters, respectively.

Gaunt–Merkle median bounds conjecture. For $r>0$,

$$\mu+(r-1)\theta<\mathrm{Med}(X)<\mu+r\theta \mathrm{e}^{-2/3r}<\mu+\left(r-\frac{2}{3}+\frac{2}{9r}\right)\theta,$$

and, for $r\geq 2$,

$$\mathrm{Med}(X)\leq\mu+(r+2\log 2-2)\theta.$$

Exact median formulas are available for some parameter values, but no general closed-form formula is known. The conjectured bounds are supported by numerical results, while accurate general bounds for the median remain unresolved.