Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(p\ge 1\), \(q\gt 1\), and \(0\le \rho \le 1\). If every coordinate probability law satisfies \(\| \rho g+(1-\rho )\mathbb E_i g\| _{q,i}\le \| g\| _{p,i}\) for every real \(g\) on that coordinate, then \(\| T_\rho f\| _{q,P}\le \| f\| _{p,P}\) for every \(f:\Omega \to \mathbb R\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\| g\| _{p,i}=(\sum _{a\in \Omega _i}w_i(a)|g(a)|^p)^{1/p}\) and \(\mathbb E_i g=\sum _{a\in \Omega _i}w_i(a)g(a)\), where \(w_i\) gives the probability masses in coordinate \(i\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\), \(1\le p\le q\), and \(0\le \rho \le 1\). If every coordinate satisfies \(\| \rho g+(1-\rho )\mathbb E_i g\| _{q,i}\le \| g\| _{p,i}\) for every real coordinate function \(g\), then \(\| T_\rho f\| _{q,P}\le \| f\| _{p,P}\) for every \(f:\Omega \to \mathbb R\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\); \(\| g\| _{p,i}=(\sum _{a\in \Omega _i}w_i(a)|g(a)|^p)^{1/p}\) and \(\mathbb E_i g=\sum _{a\in \Omega _i}w_i(a)g(a)\), where \(w_i\) gives the probability masses in coordinate \(i\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

For every \(A,B\ge 0\) and \(p,q\in \mathbb R\), the inequality \(A^{2/q}\le B^{2/p}\) implies \(A^{1/q}\le B^{1/p}\).

Let \(I\) be a finite set with nonnegative weights \(w_i\) satisfying \(\sum _iw_i=1\), and let \(g:I\to \mathbb R\) and \(0\le \rho \le \sigma \). Then \(\rho ^2\sum _iw_ig_i^2+(1-\rho ^2)(\sum _iw_ig_i)^2\le \sigma ^2\sum _iw_ig_i^2+(1-\sigma ^2)(\sum _iw_ig_i)^2\).

Let \(I\) be finite with weights \(w_i\ge \lambda \), \(\sum _iw_i=1\), \(0\lt \lambda \lt 1/2\), and \(q\gt 2\). Put \(r=\sqrt{\sinh (\log ((1-\lambda )/\lambda )/q)/\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}\). For \(0\le \rho \le r\), the operator \(Ng=\rho g+(1-\rho )\sum _iw_ig_i\) satisfies \(\| Ng\| _{q,w}\le \| g\| _{2,w}\) and \(\| Ng\| _{2,w}\le \| g\| _{q/(q-1),w}\). Here \(\| h\| _{p,w}=(\sum _{i\in I}w_i|h_i|^p)^{1/p}\) for \(p\gt 0\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(\lambda \), where \(0\lt \lambda \lt 1/2\), and let \(q\gt 2\). Put \(r=\sqrt{\sinh (\log ((1-\lambda )/\lambda )/q)/\sinh (\log ((1-\lambda )/\lambda )/(q/(q-1)))}\). For \(0\le \rho \le r\) and every \(f:\Omega \to \mathbb R\), \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(a,b\) be real numbers with \(0\lt a\le b\). Then \((a/b)e^{a-b}\le \sinh (a)/\sinh (b)\).

Let \(q\gt 2\) and \(0\lt \lambda \lt 1\). For \(r_q(t)=\sqrt{\sinh (\log ((1-t)/t)/q)/\sinh (\log ((1-t)/t)/(q/(q-1)))}\), one has \(\lambda ^{1/2-1/q}/\sqrt{q-1}\le r_q(\lambda /(1+\lambda ))\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(1\), and let \(q\gt 2\) and \(\rho \in \mathbb R\). For every \(f:\Omega \to \mathbb R\), \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a product of \(n\) finite probability spaces with strictly positive atom masses and outcome space \(\Omega =\prod _i\Omega _i\) with every coordinate atom at least \(\lambda \), where \(0\lt \lambda \le 1\), and let \(q\gt 2\). For \(0\le \rho \le \lambda ^{1/2-1/q}/\sqrt{q-1}\) and every \(f:\Omega \to \mathbb R\), \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\). For a real function \(h\) on \(\Omega \), use \(E_Uh(x)=\mathbb E_P[h(X)\mid X_U=x_U]\) for \(X\sim P\) and \(U\subseteq [n]\); \(h_S=\sum _{U\subseteq S}(-1)^{|S|-|U|}E_Uh\) for \(S\subseteq [n]\) (the Hoeffding–Efron–Stein components); \(T_a h=\sum _{S\subseteq [n]}a^{|S|}h_S\) for every real rate \(a\); \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\). The powers indexed by subset cardinalities are natural powers, including \(0^0=1\).

Let \(P\) be a probability law on a finite outcome space \(\Omega \), with every atom of mass at least \(\lambda \gt 0\), where \(0\lt \lambda \le 1\), and let \(q\gt 2\). For \(0\le \rho \le \lambda ^{1/2-1/q}/\sqrt{q-1}\), the operator \(T_\rho f=\rho f+(1-\rho )\mathbb E_Pf\) satisfies \(\| T_\rho f\| _{q,P}\le \| f\| _{2,P}\) and \(\| T_\rho f\| _{2,P}\le \| f\| _{q/(q-1),P}\) for every real \(f\). For a real function \(h\) on \(\Omega \), use \(\| h\| _{p,P}=(\mathbb E_P|h|^p)^{1/p}\) for \(p\gt 0\).