Manuscript Oai211-A@1 · Oai211-A.tex · lemma · 3f3e4f63

Lemma: Quadratic inverse criterion

Conditional transfer to weighted planar-map blocks · lines 1278–1344 · proof included or linked

Oai211-A@1#lem:inverse

Under Definition def:blockcontract and Assumption ass:endpoint, put $v = u τ^{2}$ . Then

Φ (v) = τ, 2 v Φ^{'} (v) = Φ (v), Φ^{''} (v) < \infty,

where derivatives at the endpoint denote nonnegative coefficient series.

Proof. Integrate (eq:lowerinput) to obtain, near $u$ ,

τ - T (z) = \int_{z / u}^{1} \frac{1}{x} n \geq 1 \sum n a_{n} u^{n} x^{n} d x \geq c_{1} u - z .

All exchanges are justified by nonnegativity.

The function $y (z) = z T (z)^{2}$ increases from zero to $v$ . The formal substitution identity extends to interior positive arguments and then to $Φ (v) = τ$ by monotone convergence. Its inverse is

f (y) = \frac{y}{Φ ( y ) ^{2}} .

Since $f$ is increasing,

f^{'} (y) = \frac{Φ ( y ) - 2 y Φ ^{'} ( y )}{Φ ( y ) ^{3}} \geq 0.

This bounds $Φ^{'} (y)$ near $v$ , so its increasing derivative series has a finite endpoint limit.

Moreover,

v - y (z) = (u - z) τ^{2} + z (τ^{2} - T (z)^{2}) \geq c_{2} (τ - T (z))

near $u$ . Hence

0 \leq u - f (y) \leq C (v - y)^{2} .

The finite endpoint limit of $f^{'}$ must be zero, proving $2 v Φ^{'} (v) = Φ (v)$ .

Finally,

f^{''} (y) = - \frac{4 Φ ^{'} ( y )}{Φ ( y ) ^{3}} - \frac{2 y Φ ^{''} ( y )}{Φ ( y ) ^{3}} + \frac{6 y Φ ^{'} ( y ) ^{2}}{Φ ( y ) ^{4}} .

If $Φ^{''} (v) = \infty$ , then $f^{''} (y) \to - \infty$ . For every large $K$ , sufficiently near $v$ , $f^{''} (y) \leq - K$ . Integrating twice, using $f^{'} (v -) = 0$ , gives

u - f (y) \geq \frac{K}{2} (v - y)^{2},

contradicting the preceding fixed quadratic upper bound.

Displayed formulas in this passage

unlabeled · line 1283
$Φ (v) = τ, 2 v Φ^{'} (v) = Φ (v), Φ^{''} (v) < \infty,$
unlabeled · line 1294
$τ - T (z) = \int_{z / u}^{1} \frac{1}{x} n \geq 1 \sum n a_{n} u^{n} x^{n} d x \geq c_{1} u - z .$
unlabeled · line 1306
$f (y) = \frac{y}{Φ ( y ) ^{2}} .$
unlabeled · line 1310
$f^{'} (y) = \frac{Φ ( y ) - 2 y Φ ^{'} ( y )}{Φ ( y ) ^{3}} \geq 0.$
unlabeled · line 1317
$v - y (z) = (u - z) τ^{2} + z (τ^{2} - T (z)^{2}) \geq c_{2} (τ - T (z))$
unlabeled · line 1323
$0 \leq u - f (y) \leq C (v - y)^{2} .$
unlabeled · line 1330
$f^{''} (y) = - \frac{4 Φ ^{'} ( y )}{Φ ( y ) ^{3}} - \frac{2 y Φ ^{''} ( y )}{Φ ( y ) ^{3}} + \frac{6 y Φ ^{'} ( y ) ^{2}}{Φ ( y ) ^{4}} .$
unlabeled · line 1340
$u - f (y) \geq \frac{K}{2} (v - y)^{2},$

Lemma: Quadratic inverse criterion

Displayed formulas in this passage

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