Nemotron-IMO-Bench 2026 A20

Let $f, g:\mathbb{Z}\rightarrow [0,\infty )$ be two functions such that $f(n)=g(n)=0$ for all but finitely many $n$. Define $h:\mathbb{Z}\rightarrow [0,\infty )$ by $$h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}.$$ Let $p$ and $q$ be two positive reals such that $1/p+1/q=1$. Prove that $$\sum_{n\in\mathbb{Z}}h(n)\geq \Bigg(\sum_{n\in\mathbb{Z}}f(n)^p\Bigg)^{1/p}\Bigg(\sum_{n\in\mathbb{Z}}g(n)^q\Bigg)^{1/q}.$$