Manuscript Oai119-A@1 · Oai119-A.tex · lemma · fdff7292

Lemma: Scalar profile facts

Stage 1: the five proof interfaces › Gate 1: scalar rate and effective amplitude · lines 355–403 · proof included or linked

Oai119-A@1#lem:scalar, Oai119-A@1#eq:F

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The function $ψ$ is even, continuous on $[- 1, 1]$ , strictly increasing on $[0, 1]$ , and strictly convex. For $∣ r ∣ < 1$ ,

ψ^{'} (r) = atanh r, ψ^{''} (r) = \frac{1}{1 - r ^{2}} .

Furthermore,

0 \leq I (u) \leq h (E u) \leq ℓ,

and $I (u) = 0$ if and only if $u$ is constant.

Define $F : [0, ℓ) \to [0, \infty)$ by

F (ψ (r)) = r atanh r, 0 \leq r < 1.

Then $F (0) = 0$ , $F^{'} (0) = 2$ , $F$ is convex, and $F (s) \geq 2 s$ .

Proof. The derivatives follow by differentiation. The range of $ψ$ on $[0, 1]$ is $[0, ℓ]$ . Jensen's inequality proves nonnegativity of $I$ , with equality precisely for constant $u$ , by strict convexity. Its upper bound follows from $h \geq 0$ .

For $r > 0$ , put $V = atanh r$ . The chain rule gives

F^{'} (ψ (r)) = 1 + \frac{r}{( 1 - r ^{2} ) V},

and

F^{''} (ψ (r)) = \frac{( 1 + r ^{2} ) V - r}{( 1 - r ^{2} ) ^{2} V ^{3}} > 0,

because $V \geq r$ . The expansion $V = r + O (r^{3})$ gives $F^{'} (0) = 2$ . Convexity and the tangent at zero give $F (s) \geq 2 s$ .

Displayed formulas in this passage

Oai119-A@1#eq:F · line 372
$F (ψ (r)) = r atanh r, 0 \leq r < 1.$
unlabeled · line 360
$ψ^{'} (r) = atanh r, ψ^{''} (r) = \frac{1}{1 - r ^{2}} .$
unlabeled · line 366
$0 \leq I (u) \leq h (E u) \leq ℓ,$
unlabeled · line 389
$F^{'} (ψ (r)) = 1 + \frac{r}{( 1 - r ^{2} ) V},$
unlabeled · line 395
$F^{''} (ψ (r)) = \frac{( 1 + r ^{2} ) V - r}{( 1 - r ^{2} ) ^{2} V ^{3}} > 0,$

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section: Gate 1: scalar rate and effective amplitude