Let \(q\gt 2\), \(0\lt \lambda \lt 1/2\), and let \(X\) be measurable on a probability space with \(\Pr [X=1-\lambda ]=\lambda \) and \(\Pr [X=-\lambda ]=1-\lambda \). Then \(H_{q/(q-1),2,\rho _*(q,\lambda )}(X)\) 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 \(a,b\in \mathbb {R}\).

Let \(q\gt 2\), \(0\lt \lambda \lt 1/2\), and let \(X\) be measurable on a probability space with \(\Pr [X=1-\lambda ]=\lambda \) and \(\Pr [X=-\lambda ]=1-\lambda \). If \(H_{q/(q-1),2,\rho }(X)\) holds, then \(\rho \le \rho _*(q,\lambda )\). 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 let \(X\) be measurable on a probability space with \(\Pr [X=1-\lambda ]=\lambda \) and \(\Pr [X=-\lambda ]=1-\lambda \). 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 \(a,b\in \mathbb {R}\).