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Write $X_{-i}$ for all coordinates except $X_i$, let $X_i^{\prime }$ be an independent random variable with the same distribution as $X_i$, and let $Z_i^{\prime }=f(X_1,\dots ,X_i^{\prime },\dots ,X_n)$. Three equivalent forms of the Efron–Stein inequality arefor arbitrary square-integrable $Z_i$ measurable with respect to $X_{-i}$, andThe first is the sharp choice within the second because conditional expectation is the least-squares projection. The first and third right sides are equal because two conditionally independent copies have expected squared difference twice their conditional variance.

The bounded differences property with constants $c_i$ meanswhenever $x$ and $x^{\prime }$ differ only in coordinate $i$. Conditional on $X_{-i}$, the range of $Z$ is therefore at most $c_i$. The range bound on variance givesSubstitution into the Efron–Stein inequality yields

Let ${\ell }^{\ast }$ be a maximizing index for the original sample. Since $Z_i^{\prime }$ is at least the value of its ${\ell }^{\ast }$th linear form,and henceThe one-sided replacement form of the Efron–Stein inequality isFor independent uniform signs, $\mathbb{E}[(X_i-X_i^{\prime }{)}^2\mid X]=2$. It follows that

Fix every coordinate except $X_i$. As $X_i$ varies, the longest increasing subsequence length can change by at most one: deleting the changed term leaves a common subsequence of length at least the larger value minus one. The conditional range therefore has length at most one, so the Popoviciu inequality on variances givesThe coordinatewise conditional-variance form of the Efron–Stein inequality now yields

We prove the Convex Poincaré inequality. For a differentiable convex function $h:[0,1]\to \mathbb{R}$ and independent copies $U,U^{\prime }$ supported on $[0,1]$, convexity givesTaking expectations and using $\mathrm{Var}(h(U))=\frac{1}{2}\mathbb{E}(h(U)-h(U^{\prime }){)}^2$ givesApplying this conditional inequality coordinate by coordinate in the Efron–Stein inequality provesSince $-g$ is convex and has the same gradient norm as $g$, the same argument gives $\mathrm{Var}(g(X))\le 1$.