Bound for Gamma Function Expression

Question

In Sobolev's The Efimov Effect. Discrete Spectrum Asymptotics, the following inequality is stated on page 112 $$\int_{1}^{+\infty}\frac{x^{-(n+\frac{1}{2})}}{(x^2-1)^{\frac{3}{4}}}dx=\frac{\Gamma(\frac{n}{2}+\frac{1}{2})\Gamma(\frac{1}{4})}{2\Gamma(\frac{n}{2}+\frac{3}{4})}\le\frac{8}{3^{\frac{3}{4}}(2n+1)^{\frac{1}{4}}},$$ where $n$ is a natural number. Using Gautschi's inequality I was able to prove the looser bound $$\le\frac{\Gamma(\frac{1}{4})\sqrt{10}}{2(2n+1)^{\frac{1}{4}}},$$ but I can't get that one. Can anybody help me to find that out?

Answer 1

We have $$ \frac{{\Gamma \left( {\frac{n}{2} + \frac{1}{2}} \right)\Gamma \left( {\frac{1}{4}} \right)}}{{\Gamma \left( {\frac{n}{2} + \frac{3}{4}} \right)}} = \int_0^1 {t^{\frac{n}{2} - \frac{1}{2}} (1 - t)^{ - \frac{3}{4}} dt} = \int_0^{ + \infty } {e^{ - s\left( {\frac{n}{2} + \frac{1}{8}} \right)} s^{ - \frac{3}{4}} \left( {\frac{{s/2}}{{\sinh (s/2)}}} \right)^{\frac{3}{4}} ds} $$ for all $n\geq 0$. Thus, $$ \frac{{\Gamma \left( {\frac{n}{2} + \frac{1}{2}} \right)\Gamma \left( {\frac{1}{4}} \right)}}{{\Gamma \left( {\frac{n}{2} + \frac{3}{4}} \right)}} \le \int_0^{ + \infty } {e^{ - s\left( {\frac{n}{2} + \frac{1}{8}} \right)} s^{ - \frac{3}{4}} ds} = \frac{{\sqrt 2 \Gamma \left( {\frac{1}{4}} \right)}}{{\left( {2n + \frac{1}{2}} \right)^{1/4} }} $$ for all $n\geq 0$. Finally, $$ \frac{{\Gamma \left( {\frac{n}{2} + \frac{1}{2}} \right)\Gamma \left( {\frac{1}{4}} \right)}}{{2\Gamma \left( {\frac{n}{2} + \frac{3}{4}} \right)}} \le \frac{8}{{3^{3/4} (2n + 1)^{1/4} }}\frac{{\sqrt 2 3^{3/4} \Gamma \left( {\frac{1}{4}} \right)(2n + 1)^{1/4} }}{{16\left( {2n + \frac{1}{2}} \right)^{1/4} }} < \frac{8}{{3^{3/4} (2n + 1)^{1/4} }} $$ for all $n\geq 0$.