Let \(q\gt 2\), \(0\lt \lambda \lt 1/2\), and let \(X\) be a centered measurable real random variable on a probability space, supported almost surely on a finite set whose atoms each have mass at least \(\lambda \). Then \(H_{2,q,\rho _*(q,\lambda )}(X)\) holds. Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\). Write \(H_{p,q,\rho }(X)\) for \(1\le p\le q\le \infty \), \(0\le \rho \lt 1\), \(X\in L^q(\mu )\), and \(\| a+\rho bX\| _q\le \| a+bX\| _p\) for all \(a,b\in \mathbb {R}\).