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Young’s complement of $ x \mapsto x \, {\log^{+}}(x) $, $ N $-functions and Orlicz spaces

The function $ \Phi: \mathbb{R} \to \mathbb{R} $ is an $ N $-function if and only if it is continuous, even and convex with: $ \displaystyle \lim_{x \to 0} \frac{\Phi(x)}{x} = 0 $. $ \displaystyle \ …