Let \(q\gt 2\) and \(z\in \mathbb {R}\), and put \(g=z|z|^{q-2}\). Then \(zg=|z|^q\) and \(|g|^{q/(q-1)}=|z|^q\).

Let \(I\) be finite, let real weights satisfy \(\sum _iw_i=1\), and let \(\rho \in \mathbb {R}\). For real vectors \(f,g\), write \(m_h=\sum _iw_ih_i\) and \((T_\rho h)_i=\rho h_i+(1-\rho )m_h\). Then \(\sum _iw_i(T_\rho g)_i^2=\rho ^2\sum _iw_ig_i^2+(1-\rho ^2)m_g^2\) and \(\sum _iw_i(T_\rho f)_ig_i=\sum _iw_if_i(T_\rho g)_i\).

Let \(I\) be finite, \(w_i\ge 0\), \(\sum _iw_i=1\), \(q\gt 2\), and \(\rho \in \mathbb {R}\). Put \(p=q/(q-1)\) and \((T_\rho h)_i=\rho h_i+(1-\rho )\sum _jw_jh_j\). If \(\rho ^2\sum _iw_ig_i^2+(1-\rho ^2)(\sum _iw_ig_i)^2\le (\sum _iw_i|g_i|^p)^{2/p}\) for every \(g\), then \((\sum _iw_i|(T_\rho f)_i|^q)^{2/q}\le \sum _iw_if_i^2\) for every \(f\).