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Inequality involving some functions

Source: Romania TST 4 2012, Problem 2

May 16, 2012
inequalitiesfunctioninequalities proposed

Problem Statement

Let f,g:Z→[0,∞)f, g:\mathbb{Z}\rightarrow [0,\infty ) be two functions such that f(n)=g(n)=0f(n)=g(n)=0 with the exception of finitely many integers nn. Define h:Z→[0,∞)h:\mathbb{Z}\rightarrow [0,\infty ) by h(n)=max⁡{f(n−k)g(k):k∈Z}.h(n)=\max \{f(n-k)g(k): k\in\mathbb{Z}\}. Let pp and qq be two positive reals such that 1/p+1/q=11/p+1/q=1. Prove that ∑n∈Zh(n)≥(∑n∈Zf(n)p)1/p(∑n∈Zg(n)q)1/q. \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}.
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