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

Let \(I\) be a finite set with nonnegative weights \(w_i\), and let \(p,q\gt 1\) satisfy \(1/p+1/q=1\). For \(f,g:I\to \mathbb R\), \(\sum _iw_if_ig_i\le (\sum _iw_i|f_i|^p)^{1/p}(\sum _iw_i|g_i|^q)^{1/q}\).

Let \(I\) be a finite set with nonnegative weights \(w_i\), \(q\gt 1\), \(z:I\to \mathbb R\), and \(N\ge 0\). If \(\sum _iw_iz_ig_i\le N(\sum _iw_i|g_i|^{q/(q-1)})^{(q-1)/q}\) for every \(g:I\to \mathbb R\), then \((\sum _iw_i|z_i|^q)^{1/q}\le N\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) and define \(\| f\| _{r,w}=(\sum _iw_i|f_i|^r)^{1/r}\). If an operator \(T\) satisfies \(\sum _iw_i(Tf)_ig_i=\sum _iw_if_i(Tg)_i\) for all \(f,g\) and \(\| Tg\| _{q/(q-1),w}\le \| g\| _{q/(q-1),w}\) for every \(g\), where \(q\gt 1\), then \(\| Tf\| _{q,w}\le \| f\| _{q,w}\) for every \(f\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) and let \(g:I\to \mathbb R\) satisfy \(\sum _iw_ig_i=0\). Then \((\sum _iw_i|g_i|)^2\le 4(\sum _iw_ig_i^2)(\sum _{i:g_i\gt 0}w_i)\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) with \(\sum _iw_i\gt 0\), let \(d_i\in \mathbb R\), and let \(\rho \gt 0\). Then \((\sum _iw_i)\rho ^{(\sum _iw_id_i)/(\sum _iw_i)}\le \sum _iw_i\rho ^{d_i}\).

Let \(I\) be finite and let \(a_i\ge 0\) have positive sum. If \(A\ge 0\) and \(A\le \sum _i a_i^{3/2}\), then some \(i\in I\) satisfies \((A/\sum _ja_j)^2\le a_i\).

Let \(\alpha \) be finite, let \(E(S)\ge 0\) for every \(S\subseteq \alpha \), let \(J\subseteq \alpha \), \(k\in \mathbb N\), and \(0\le \rho \le 1\). Then \(\rho ^k\sum _{|S|\le k,\ S\nsubseteq J}E(S)\le \sum _{i\notin J}\sum _{S\ni i}\rho ^{|S|}E(S)\).