306 Boolean Analysis — Two Point Contraction

306.1 Overview

The biased two-point inequalities hold for arbitrary real values and have explicit extremizers. They give both forward contraction and the optimal radius bound, uniformly over two-point laws with a minimum atom mass.

306.2 Declarations

Theorem 306.1 Biased two-point inequality for real values

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Let \(1\lt p\lt 2\), \(0\lt \alpha \lt \beta \), \(\alpha ^p+\beta ^p=1\), and \(u,v\in \mathbb {R}\). Put \(r=(\alpha ^p\beta ^{2-p}-\alpha ^{2-p}\beta ^p)/(\alpha ^2-\beta ^2)\). Then \((\alpha ^pu+\beta ^pv)^2+r\alpha ^p\beta ^p(u-v)^2\le (\alpha ^p|u|^p+\beta ^p|v|^p)^{2/p}\).

Theorem 306.2 Sharp weighted two-point contraction

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Let \(1\lt p\lt 2\), \(0\lt \lambda \lt 1/2\), and \(u,v\in \mathbb {R}\). Setting \(\rho =\rho _*(p/(p-1),\lambda )\), one has \((\lambda u+(1-\lambda )v)^2+\rho ^2\lambda (1-\lambda )(u-v)^2\le (\lambda |u|^p+(1-\lambda )|v|^p)^{2/p}\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).

Theorem 306.3 An extremizer for the biased inequality

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Let \(1\lt p\lt 2\), \(0\lt \alpha \lt \beta \), and \(\alpha ^p+\beta ^p=1\). Put \(r=(\alpha ^p\beta ^{2-p}-\alpha ^{2-p}\beta ^p)/(\alpha ^2-\beta ^2)\). For \(u=\beta /\alpha \) and \(v=\alpha /\beta \), one has \(u\ne v\), \(\alpha ^pu^p+\beta ^pv^p=1\), and \((\alpha ^pu+\beta ^pv)^2+r\alpha ^p\beta ^p(u-v)^2=1\).

Theorem 306.4 Necessary radius for weighted contraction

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Let \(1\lt p\lt 2\), \(0\lt \lambda \lt 1/2\), and \(\rho \ge 0\). If \((\lambda u+(1-\lambda )v)^2+\rho ^2\lambda (1-\lambda )(u-v)^2\le (\lambda |u|^p+(1-\lambda )|v|^p)^{2/p}\) for every \(u,v\in \mathbb {R}\), then \(\rho \le \rho _*(p/(p-1),\lambda )\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).

Theorem 306.5 Forward weighted two-point contraction

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Let \(q\gt 2\), \(0\lt \lambda \lt 1/2\), and \(u,v\in \mathbb {R}\). Put \(\rho =\rho _*(q,\lambda )\) and \(m=\lambda u+(1-\lambda )v\). Then \((\lambda |\rho u+(1-\rho )m|^q+(1-\lambda )|\rho v+(1-\rho )m|^q)^{2/q}\le \lambda u^2+(1-\lambda )v^2\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).

Theorem 306.6 Two-point contraction from an atom bound

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Let \(q\gt 2\), \(0\lt \lambda \lt 1/2\), \(\lambda \le \gamma \le 1-\lambda \), and \(u,v\in \mathbb {R}\). Put \(p=q/(q-1)\) and \(\rho =\rho _*(q,\lambda )\). Then \((\gamma u+(1-\gamma )v)^2+\rho ^2\gamma (1-\gamma )(u-v)^2\le (\gamma |u|^p+(1-\gamma )|v|^p)^{2/p}\). Here \(\rho _*(q,\lambda )=\sqrt{\sinh (\ell /q)/\sinh (\ell /(q/(q-1)))}\) with \(\ell =\log ((1-\lambda )/\lambda )\).