Let \(X,Y\) be independent real random variables on a probability space. If both satisfy \(H_{p,q,\rho }\), then \(X+Y\) also satisfies \(H_{p,q,\rho }\). 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 \(X\in L^q\) be centered on a probability space, with \(\| X\| _2=1\) and \(\| X\| _q=C\gt 0\). Then \(H_{2,q,\rho }(X)\) holds for \(\rho =1/(2C\sqrt{q-1})\). 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 \(a,b\in \mathbb {R}\).