Factorial–Furuta

Factorial Function Inequalities (a) [Gautschi](i) If 0 ≤ y ≤ 1 then

(x+ 1)y ≥ x! (x+ y)!

≥ xy;

(ii)

x!

( 1

x

) ! ≥ 1.

(iii) There is a constant α,−1 < α < 0 such that if a > α then Gn(a!;w) ≥

( Gn(a;w)

) !,

with equality if and only if a is constant. The inequality is reversed if −1 < a < α. (b) [Watson] If x ≥ 0 then√

x+ 1

4 ≤ x!