A few answers here on math.SE have used as an intermediate step the following inequality that is due to Walter Gautschi:

$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > 0,\; 0 < s < 1$$

Unfortunately, the paper that the DLMF is pointing to is not easily accessible. How might this inequality be proven?

**Answer**

The strict log-convexity of $\Gamma$ (see the end of this answer) implies that for $0< s <1$,

$$

\Gamma(x+s)<\Gamma(x)^{1-s}\Gamma(x+1)^s=x^{s-1}\Gamma(x+1)\tag{1}

$$

which yields

$$

x^{1-s}<\frac{\Gamma(x+1)}{\Gamma(x+s)}\tag{2}

$$

Again by the strict log-convexity of $\Gamma$,

$$

\Gamma(x+1)<\Gamma(x+s)^s\Gamma(x+s+1)^{1-s}=(x+s)^{1-s}\Gamma(x+s)\tag{3}

$$

which yields

$$

\frac{\Gamma(x+1)}{\Gamma(x+s)}<(x+s)^{1-s}<(x+1)^{1-s}\tag{4}

$$

Combining $(2)$ and $(4)$ yields

$$

x^{1-s}<\frac{\Gamma(x+1)}{\Gamma(x+s)}<(x+1)^{1-s}\tag{5}

$$

**Attribution***Source : Link , Question Author : J. M. ain’t a mathematician , Answer Author : Community*