= Solution
For conjugate exponents $1<p,q<\infty$, $1/p+1/q=1$, <Hölder's inequality> is $\boxed{\mathbb E|XY|\leq\|X\|_p\|Y\|_q}$. If either finite <norm> is zero, the corresponding variable is zero <almost surely>. Otherwise normalize $u=|X|/\|X\|_p$, $v=|Y|/\|Y\|_q$. <Young inequality> $uv\leq u^p/p+v^q/q$ follows by minimizing the right side minus $uv$ in $u\geq0$, whose <derivative> is $u^{p-1}-v$. Integration gives $\mathbb E(uv)\leq1/p+1/q=1$, proving the result. Infinite <norms> give the usual extended-value inequality whenever the right side is unambiguous. At the endpoints, $\mathbb E|XY|\leq\|X\|_1\|Y\|_\infty$ follows from the almost-sure bound for $Y$, with the symmetric version for $p=\infty$.
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