ボーア・モレルップの定理
log  G ( x + c ) ≤ ( 1 − c ) log  G ( x ) + c log  G ( x + 1 ) ( x ≥ 1 , 0 ≤ c ≤ 1 ) G ( x + c ) ≤ G ( x ) 1 − c G ( x + 1 ) c = G ( x ) x c {\displaystyle {\begin{aligned}&{\log {G(x+c)}}\leq {(1-c)\log {G(x)}+c\log {G(x+1)}}\qquad (x{\geq }1,0{\leq }c{\leq }1)\\&{G(x+c)}\leq {G(x)^{1-c}G(x+1)^{c}=G(x)x^{c}}\\\end{aligned}}}
である。また、
log  G ( x + 1 ) ≤ c log  G ( x + c ) + ( 1 − c ) log  G ( x + 1 + c ) ( x ≥ 1 , 0 ≤ c ≤ 1 ) G ( x + 1 ) ≤ G ( x + c ) c G ( x + 1 + c ) 1 − c = G ( x + c ) ( x + c ) 1 − c G ( x + 1 ) ( x + c ) − ( 1 − c ) ≤ G ( x + c ) {\displaystyle {\begin{aligned}&{\log {G(x+1)}}\leq {c\log {G(x+c)}+(1-c)\log {G(x+1+c)}}\qquad (x{\geq }1,0{\leq }c{\leq }1)\\&{G(x+1)}\leq {G(x+c)^{c}G(x+1+c)^{1-c}=G(x+c)(x+c)^{1-c}}\\&{G(x+1)(x+c)^{-(1-c)}}\leq {G(x+c)}\\\end{aligned}}}
であるから、合わせて
G ( x + 1 ) ( x + c ) − ( 1 − c ) ≤ G ( x + c ) ≤ G ( x ) x c ( x ≥ 1 , 0 ≤ c ≤ 1 ) G ( x ) x c ( x x + c ) 1 − c ≤ G ( x + c ) ≤ G ( x ) x c {\displaystyle {\begin{aligned}&G(x+1)(x+c)^{-(1-c)}\leq {G(x+c)}\leq {G(x)x^{c}}\qquad (x{\geq }1,0{\leq }c{\leq }1)\\&G(x)x^{c}\left({\frac {x}{x+c}}\right)^{1-c}\leq {G(x+c)}\leq {G(x)x^{c}}\\\end{aligned}}}
となる。 x = n {\displaystyle x=n} を整数とし、 n → ∞ {\displaystyle n\to \infty } とすれば不等式の両端が一致して
( n − 1 ) ! n c ( n n + c ) 1 − c ≤ G ( c ) ∏ k = 0 n − 1 ( c + k ) ≤ ( n − 1 ) ! n c ( 0 ≤ c ≤ 1 ) G ( c ) = lim n → ∞ ( n − 1 ) !