  In this article we extend a recent theorem proven by Hod (Phys. Lett. B, {\bf
727}, 345--348, 2013) to $n$-dimensional Einstein and Einstein-Gauss-Bonnet
theories, which gives an upper bound for the photon sphere radii of spherically
symmetric black holes. As applications of these results we give a universal
upper bound for the real part of quasinormal modes in the WKB limit and a
universal lower bound for the position of the first relativistic image in the
strong lensing regime produced by these type of black holes. For the
axially-symmetric case, we also make some general comments (independent of the
underlying gravitational theory) on the relation between circular null
geodesics and the fastest way to circle a black hole.
