Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 24J / b

For conjugate exponents $1<p,q<\infty$, $1/p+1/q=1$, Hölder's inequality is $\boxed{\mathbb{E}|XY|\le \Vert X{\Vert }_p\Vert Y{\Vert }_q}$. If either finite norm is zero, the corresponding variable is zero almost surely. Otherwise normalize $u=|X|/\Vert X{\Vert }_p$, $v=|Y|/\Vert Y{\Vert }_q$. Young inequality $uv\le u^p/p+v^q/q$ follows by minimizing the right side minus $uv$ in $u\ge 0$, whose derivative is $u^{p-1}-v$. Integration gives $\mathbb{E}(uv)\le 1/p+1/q=1$, proving the result. Infinite norms give the usual extended-value inequality whenever the right side is unambiguous. At the endpoints, $\mathbb{E}|XY|\le \Vert X{\Vert }_1\Vert Y{\Vert }_{\infty }$ follows from the almost-sure bound for $Y$, with the symmetric version for $p=\infty$.