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

Lemma: Convex entropy conversion

The Courtade–Kumar implication as a corollary · lines 1040–1083 · proof included or linked

Oai119-A@1#lem:conversion, Oai119-A@1#eq:conversionidentity, Oai119-A@1#eq:conversionlower

Define

φ (t) = h_{2} (\frac{1 - 1 - t ^{2}}{2}), 0 \leq t \leq 1.

Then $φ$ is increasing and convex, and

h_{2} (\frac{1 + u}{2}) = φ (H (u)) (- 1 \leq u \leq 1) .

For $0 \leq a \leq b \leq 1$ ,

φ (b) - φ (a) \geq \frac{b ^{2} - a ^{2}}{2 ℓ} .

Proof. For $0 < t < 1$ , put $v = 1 - t^{2}$ . Differentiation gives

φ^{'} (t) = \frac{t}{2 v ℓ} ln \frac{1 + v}{1 - v} \geq 0,

φ^{''} (t) = \frac{ln (( 1 + v ) / ( 1 - v )) - 2 v}{2 v ^{3} ℓ} \geq 0.

Indeed,

ln \frac{1 + v}{1 - v} = 2 atanh v \geq 2 v .

Continuity extends monotonicity and convexity to the endpoints. The same inequality gives $φ^{'} (t) \geq t / ℓ$ ; integration proves (eq:conversionlower).

Finally, $1 - H (u)^{2} = ∣ u ∣$ , and binary entropy is symmetric about $1/2$ . This proves (eq:conversionidentity).

Displayed formulas in this passage

Oai119-A@1#eq:conversionidentity · line 1049
$h_{2} (\frac{1 + u}{2}) = φ (H (u)) (- 1 \leq u \leq 1) .$
Oai119-A@1#eq:conversionlower · line 1055
$φ (b) - φ (a) \geq \frac{b ^{2} - a ^{2}}{2 ℓ} .$
unlabeled · line 1043
$φ (t) = h_{2} (\frac{1 - 1 - t ^{2}}{2}), 0 \leq t \leq 1.$
unlabeled · line 1064
$φ^{'} (t) = \frac{t}{2 v ℓ} ln \frac{1 + v}{1 - v} \geq 0,$
unlabeled · line 1068
$φ^{''} (t) = \frac{ln (( 1 + v ) / ( 1 - v )) - 2 v}{2 v ^{3} ℓ} \geq 0.$
unlabeled · line 1073
$ln \frac{1 + v}{1 - v} = 2 atanh v \geq 2 v .$

Lemma: Convex entropy conversion

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