Exercise 4, § 1

Show that the function $f(x) = \frac{1}{\sqrt{2}} \frac{\Gamma(x/2)}{\Gamma\left(\frac{x+1}{2}\right)}$ is the unique convex solution of the equation $u(x+1) = 1/x\,u(x)$ (remark that this equation implies $u(x+2) = \frac{x}{x+1}\,u(x)$ and apply exerc. 1 a)).