Let $f(m,n,w)$ be the probability density function of F variable with m numerator df and n denominator df, i.e.
$$f(m,n,w)=\frac{\Gamma\left(\frac{m+n}{2}\right)(m/n)^{m/2}}{\Gamma(m/2)\Gamma(n/2)}w^{(m/2)-1}\left(1+\frac{mw}{n}\right)^{-(m+n)/2}$$
I am interested in the infimum of $\int_1^\infty f(m,n,w) dw$ over all $m,n$. From my exploration, this seems to be $$1-erf(1/\sqrt{2})\approx 0.3173$$ where $erf(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} dt$.
Can anyone prove this or point me to a reference?
