Let \(I\) be finite, let real weights satisfy \(\sum _iw_i=1\), and let \(f:I\to \mathbb {R}\) take values in \(\{ a,b\} \), where \(a,b\in \mathbb {R}\). For any \(\varphi :\mathbb {R}\to \mathbb {R}\), setting \(\gamma =\sum _{i:f_i=a}w_i\) gives \(\sum _iw_i\varphi (f_i)=\gamma \varphi (a)+(1-\gamma )\varphi (b)\).

Let \(I\) be finite, \(w_i\gt 0\), \(\sum _iw_i=1\), and \(0\lt \lambda \le w_i\) for every \(i\). If \(f:I\to \mathbb {R}\) takes values in \(\{ a,b\} \), then either \(f\) is identically \(a\), or identically \(b\), or \(\lambda \le \sum _{i:f_i=a}w_i\le 1-\lambda \).

Let \(I\) be finite, \(w_i\gt 0\), \(p\gt 0\), and \(\rho \in \mathbb {R}\). Define \(G(f)=\sum _iw_i|f_i|^p\) and \(F(f)=\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\). If \(G(f)=1\) implies \(F(f)\le 1\) for every real vector \(f\), then \(F(f)\le G(f)^{2/p}\) for every \(f\).

Let \(I\) be finite, let \(\sum _iw_i=1\), let \(q\gt 2\), \(0\lt \lambda \lt 1/2\), and assume \(w_i\ge \lambda \) for all \(i\). Put \(p=q/(q-1)\) and \(\rho =\rho _*(q,\lambda )\). Every real vector \(f\) satisfies \(\rho ^2\sum _iw_if_i^2+(1-\rho ^2)(\sum _iw_if_i)^2\le (\sum _iw_i|f_i|^p)^{2/p}\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).