ScalingStacks

Lemma 5.11 . [0547]

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Lemma 5.11.

Let

(5.148) F^​(z)\displaystyle\widehat{F}(z) ≡−j​zn2+F⁡(t0​(z)),\displaystyle\equiv-\frac{jz^{n}}{2}+F(t_{0}(z)),
(5.149) G^​(z)\displaystyle\widehat{G}(z) ≡−j​zn2+G⁡(u0​(z)),\displaystyle\equiv-\frac{jz^{n}}{2}+G(u_{0}(z)),

then for all η≥0\eta\geq 0, when z≥η2nz\geq\eta^{\frac{2}{n}}, F^​(z)+η⋅zn2\widehat{F}(z)+\eta\cdot z^{\frac{n}{2}} is decreasing and G^​(z)−η⋅zn2\widehat{G}(z)-\eta\cdot z^{\frac{n}{2}} is increasing.

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