Let $p>1$.
Consider $\phi(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^pdt$. Function $\phi(p)$ is analytic on its domain.
It's derivative, $\phi'(p)=\int_0^{\infty}\left|\frac{\sin t}{t}\right|^p\ln\left|\frac{\sin t}{t}\right|dt$.
Let $g(p)=\frac{1}{2\sqrt p}\phi(p)$ and $f(p)=\sqrt p \phi'(p)$. Using Rouche's theorem http://en.wikipedia.org/wiki/Rouch%C3%A9's_theorem , show that $f(p)+g(p)$ has one zero.
Thank you.