7.14 Verify the following properties for the function defined by (7.22):; (a) (u) is nonincreasing for

7.14 Verify the following properties for the function ψ defined by (7.22):;

(a) ψ(u) is nonincreasing for u ≥ −1 with ψ(0) = 1;

(b) uψ(u) is nondecreasing for u ≥ −1;

(c) ψ(u) ∼ (2 log u)/u as u→∞;

(d) 0 ≤ 1 − ψ(u) ≤ u/3 for 0 ≤ u ≤ 3;

(e) 0 ≤ ψ(u) − 1 ≤ |u| for −1 ≤ u ≤ 0;

(f) ψ



(0) = −1/3, ψ(−1) = 2 and ψ



(−1) = −∞;

(g) uψ(u) equals 0 and −2 respectively for u = 0 and −1 and has derivative 1 for u = 0;

(h) for |u| < 1, we have the Taylor expansion

ψ(u) = 1 − u 3

+ u2 6

− u3 10

+· · ·+ (−1)k2uk

(k + 1)(k + 2)

+· · · .