Sharp Inequalities Involving $$(n!)^{1/n}$$

We prove that, for all integers (nge 1),

$$begin{aligned} Big (sqrt{2pi n}Big )^{frac{1}{n(n+1)}}left( 1-frac{1}{n+a}right)

and

$$begin{aligned} big (sqrt{2pi n}big )^{1/n}left( 1-frac{1}{2n+alpha }right)

with the best possible constants

$$begin{aligned}&a=frac{1}{2},quad b=frac{1}{2^{3/4}pi ^{1/4}-1}=0.807ldots ,quad alpha =frac{13}{6} &text {and}quad beta =frac{2sqrt{2}-sqrt{pi }}{sqrt{pi }-sqrt{2}}=2.947ldots . end{aligned}$$