= Solution
For an axisymmetric <gravitational lens>, the reduced deflection is $\alpha(\theta)=\bar\kappa(\theta)\theta$. The specified mean <lensing convergence> therefore gives
$$
\beta(\theta)=\theta\left(1-\frac{\theta_E}{\sqrt{\theta^2+\theta_c^2}}\right),\qquad
\beta'(\theta)=1-\frac{\theta_E\theta_c^2}{(\theta^2+\theta_c^2)^{3/2}}.
$$
The <derivative> is smallest at the centre, where $\beta'(0)=1-\theta_E/\theta_c$. If $\theta_c\geq\theta_E$, 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 \b[the condition for a nonempty multiple-image region is]
$$
\boxed{0<\theta_c<\theta_E,\quad\text{equivalently }\kappa(0)>1.}
$$
This 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.
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