Let \(D\) be a domain in the Riemann sphere \(\widehat{C}\) possessing at least two boundary points. For a point \(z\in D\), let \(\lambda_D(z)\) be the hyperbolic density in \(D\) and let \[ \mu_D(z):=\inf\{\lambda_\Delta(z):z\in \Delta\subset D,\;D\text{ is a disk in } \widehat{C}\} \] be the Möbius metric-density in \(D\). The authors present numerous uniform and pointwise estimates involving these densities. For example: 1. If \(D\) is simply connected, \(\lambda_D(z)\geq \mu_D(z)/2\) with equality if and only if \(D\) is a Möbius image of a slit plane. 2. If \(\inf_{z\in D}\lambda_D(z)/\mu_D(z)\geq \sqrt{3}/2\), then \(D\) is simply connected. (The constant \(\sqrt{3}/2\) is not sharp; the authors conjecture that \(\pi/4\) is the sharp constant). 3. \(\sup_{z\in D}\lambda_D(z)/\mu_D(z)\geq M\), where \(M\) is a certain constant (specified in the paper), with equality when \(D\) is the twice punctured plane. 4. For a planar domain \(D\), \(\sup_{z\in D}\text{ dist}(z,\partial D)\lambda_D(z)\geq H\), where \(H\) is a certain constant (specified in the paper), with equality when \(D\) is the twice punctured plane. This result verifies an unpublished conjecture of J. R. Hilditch.