 Solution

For an axisymmetric gravitational lens, the reduced deflection is $\alpha (\theta )=\bar{\kappa }(\theta )\theta$. The specified mean lensing convergence therefore givesThe derivative is smallest at the centre, where ${\beta }^{\prime }(0)=1-{\theta }_E/{\theta }_c$. If ${\theta }_c\ge {\theta }_E$, the mapping is monotone and an offset source has only one image. If $0<{\theta }_c<{\theta }_E$, the derivative is negative near zero and positive at large radius, producing two stationary points and a range of source offsets with three images. Thus the condition for a nonempty multiple-image region isThis condition permits multiple images; a particular source must additionally lie inside the multiple-image caustic of a softened isothermal lensing potential. At equality the critical radii collapse to the origin and there is no finite three-image region.