Prove inequality on quotient of $\Gamma$-functions

Question

$$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n + 1}{2}\right)} > \frac {\sqrt{2n + 1}}{n}, \space\space\forall{}n\in\Bbb{N}$$

Answer 1

With $s = \frac{n}{2}$, by taking reciprocals and using $\Gamma (s + 1) = s\Gamma (s)$, the inequality is equivalent to $$ \frac{1}{\sqrt {s + \frac{1}{4}} } > \frac{{\Gamma \left( {s + \frac{1}{2}} \right)}}{{\Gamma (s + 1)}}. $$ We shall show that this holds for all positive $s$. Indeed, by using the Beta integral, the right-hand side is \begin{align*} &\frac{{\Gamma \left( {s + \frac{1}{2}} \right)}}{{\Gamma (s + 1)}} = \frac{1}{{\sqrt \pi }}B\left( {s + \tfrac{1}{2},\tfrac{1}{2}} \right) \\ & = \frac{1}{{\sqrt \pi }}\int_0^1 {t^{s - 1/2} (1 - t)^{ - 1/2} dt} = \frac{1}{{\sqrt \pi }}\int_0^{ + \infty } {e^{ - (s - 1/2)u} (1 - e^{ - u} )^{ - 1/2} e^{ - u} du} \\ & = \frac{1}{{\sqrt \pi }}\int_0^{ + \infty } {e^{ - (s + 1/4)u} u^{ - 1/2} \left( {\frac{{e^{u/2} - e^{ - u/2} }}{u}} \right)^{ - 1/2} du} \\ & = \frac{1}{{\sqrt \pi }}\int_0^{ + \infty } {e^{ - (s + 1/4)u} u^{ - 1/2} \sqrt {\frac{u}{{\sinh u}}} du} \\ & < \frac{1}{{\sqrt \pi }}\int_0^{ + \infty } {e^{ - (s + 1/4)u} u^{ - 1/2} du} = \frac{1}{\sqrt {s + \frac{1}{4}} } . \end{align*}

Answer 2

By using $\Gamma(\frac{n}{2}+1) = \frac{n}{2}\Gamma(\frac{n}{2})$, the inequality is written as $$\frac{\Gamma(\frac{n}{2}+1)}{\Gamma(\frac{n+1}{2})} > \sqrt{\frac{n}{2} + \frac{1}{4}}.$$

Recall Kershaw's improvement of Gautschi's inequality [1]: for $x > 0$ and $s \in (0, 1)$, $$\left(x + \frac{s}{2}\right)^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < \left(x - \frac{1}{2} + \sqrt{s + \frac{1}{4}}\right)^{1-s}.$$ By letting $x = \frac{n}{2}$ and $s = \frac{1}{2}$ in Kershaw's improvement of Gautschi's inequality, we have $$\sqrt{\frac{n}{2}+\frac{1}{4}} < \frac{\Gamma(1 + \tfrac{n}{2})}{\Gamma(\tfrac{1}{2} + \tfrac{n}{2})} < \sqrt{\frac{n}{2}+\frac{\sqrt{3}-1}{2}}.$$ We are done.

[1] D. Kershaw, "Some extensions of W. Gautschi’s inequalities for the gamma function", Math. Comp. 41 (1983) 607–611. https://en.wikipedia.org/wiki/Gautschi%27s_inequality