Let \(X\in L^4\) have a symmetric law on a probability space, \(\| X\| _2=1\), and \(\| X\| _4=C\gt 0\). For \(0\le \rho \lt 1\), \(H_{2,4,\rho }(X)\) holds if and only if \(\rho \le \min (1/\sqrt3,1/C)\). 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 let \(X\in L^q\) have a symmetric law on a probability space, with \(\| X\| _2=1\) and \(\| X\| _q=C\gt 0\). Then \(H_{2,q,\rho }(X)\) holds for \(\rho =1/(C\sqrt{q-1})\). 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 \(a,b\in \mathbb {R}\).