  For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the
logarithmic capacity $c_{\beta}$, and the analytic capacity $c_{B}$ satisfy the
inequality chain $\pi K \geq c^2_{\beta} \geq c^2_B$; moreover, equality holds
at a single point between any two of the three quantities if and only if $X$ is
biholomorphic to a disk possibly less a relatively closed polar set. We extend
the inequality chain by showing that $c_{B}^2 \geq \pi v^{-1}(X)$ on planar
domains, where $v(\cdot)$ is the Euclidean volume, and characterize the
extremal cases when equality holds at one point. Similar rigidity theorems
concerning the Szeg\"{o} kernel, the higher-order Bergman kernels, and the
sublevel sets of the Green's function are also developed. Additionally, we
explore rigidity phenomena related to the multi-dimensional Suita conjecture.
