Let \(q\gt 2\) and \(\lambda \gt 0\). Let \(X\) be a measurable real random variable on a probability space, supported almost surely on a finite set whose listed atoms all have probability at least \(\lambda \). Then \(\| X\| _q\le (1/\lambda )^{1/2-1/q}\| X\| _2\).

Let \(q\gt 2\), \(0\lt \lambda \le 1\), and let \(X\) be measurable and centered on a probability space, supported almost surely on a finite set whose atoms each have mass at least \(\lambda \). Setting \(\rho =\lambda ^{1/2-1/q}/(2\sqrt{q-1})\), both \(H_{2,q,\rho }(X)\) and \(H_{q/(q-1),2,\rho }(X)\) hold. 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 \(a,b\in \mathbb {R}\).

Let \(q\gt 2\), \(0\lt \lambda \le 1\), and let measurable \(X\) have a symmetric law on a probability space, supported almost surely on a finite set whose atoms each have mass at least \(\lambda \). Setting \(\rho =\lambda ^{1/2-1/q}/\sqrt{q-1}\), both \(H_{2,q,\rho }(X)\) and \(H_{q/(q-1),2,\rho }(X)\) hold. 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 \(a,b\in \mathbb {R}\).

Let \(q\gt 2\) and \(0\lt \lambda \lt 1/2\), and let \(X\) be measurable and centered on a probability space, supported almost surely on a finite set whose atoms each have probability at least \(\lambda \). Both \(H_{2,q,\rho _*}(X)\) and \(H_{q/(q-1),2,\rho _*}(X)\) hold. 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 \(a,b\in \mathbb {R}\).

Let \(q\gt 2\), \(0\lt \lambda \lt 1/2\), and \(0\le \rho \lt 1\). Let measurable \(X\) on a probability space satisfy \(\Pr [X=1-\lambda ]=\lambda \) and \(\Pr [X=-\lambda ]=1-\lambda \). Then \(H_{2,q,\rho }(X)\) holds if and only if \(\rho \le \rho _*(q,\lambda )\), and the same equivalence holds for \(H_{q/(q-1),2,\rho }(X)\). 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 \(a,b\in \mathbb {R}\).