Lower bound on the radii of black-hole photonspheres

The existence of closed null circular geodesics around black holes is one of the most intriguing predictions of general relativity. It has recently been conjectured that the radii of black-hole photonspheres are bounded from below by the simple relation $r_{\text{ph}}\geq {3\over2}r_{\text{H}}$, where $r_{\text{H}}$ is the radius of the outer black-hole horizon. We here prove the validity of this conjecture for spherically symmetric hairy black-hole configurations whose radial pressure function $P\equiv |r^3p|$ decreases monotonically.