Proposition 5.5 . [053T] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Proposition 5.5 .
The following hold
(1)
For all ν ∈ ℝ \nu\in\mathbb{R} , there is a constant C ( ν ) > 1 C(\nu)>1 such that
(5.47) C − 1 ( ν ) ⋅ e − y y ≤ K ν ( y ) ≤ C ( ν ) ⋅ e − y y , y ≥ 1 ; \displaystyle C^{-1}(\nu)\cdot\frac{e^{-y}}{\sqrt{y}}\leq K_{\nu}(y)\leq C(\nu)\cdot\frac{e^{-y}}{\sqrt{y}},\qquad y\geq 1;
(5.48) I ν ( y ) ≤ { C ( ν ) ⋅ e y y , y ≥ 1 , C ( ν ) ⋅ y ν , 0 < y ≤ 1 . \displaystyle I_{\nu}(y)\leq\begin{cases}C(\nu)\cdot\frac{e^{y}}{\sqrt{y}},&y\geq 1,\\
C(\nu)\cdot y^{\nu},&0<y\leq 1.\end{cases}
(2)
For all ν > − 1 \nu>-1 , we have
(5.49) I ν ( y ) ≥ { C ( ν ) − 1 ⋅ e y y , y ≥ 1 , C ( ν ) − 1 ⋅ y ν , 0 < y ≤ 1 . \displaystyle I_{\nu}(y)\geq\begin{cases}C(\nu)^{-1}\cdot\frac{e^{y}}{\sqrt{y}},&y\geq 1,\\
C(\nu)^{-1}\cdot y^{\nu},&0<y\leq 1.\end{cases}