Here's my thinking. Please let me know if any of my assumptions are incorrect.
(1) From Gautschi's Inequality, if $x > 0$ is real and $0 < s < 1$:
$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s}$$
For this example, let's assume that $x$ is a positive integer.
(2) Inverting both sides:
$$x^{s-1} > \frac{\Gamma(x+s)}{\Gamma(x+1)} > (x+1)^{s-1}$$
(3) From the properties of the Gamma function:
$$\frac{\Gamma(x+1)}{\Gamma(x)}=x$$
(4) Multiplying step(2) and step(3):
$$x^s > \frac{\Gamma(x+s)}{\Gamma(x+1)}\frac{\Gamma(x+1)}{\Gamma(x)} = \frac{\Gamma(x+s)}{\Gamma(x)} > (x)(x+1)^{s-1}$$
Is my reasoning correct? Did I make any mistakes or make any bad assumptions about the Gamma function?