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- Jan 26, 2012

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Prove that, for any two bounded functions $g_1, g_2: \mathbb{R} \to [1, \infty)$, there exist functions $h_1, h_2: \mathbb{R} \to \mathbb{R}$ such that, for every $x \in \mathbb{R},$

$$

\sup_{s \in \mathbb{R}} (g_1(s)^x g_2(s)) = \max_{t \in \mathbb{R}} (x h_1(t) + h_2(t)).

$$

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