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Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article

To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question. We consider the ...
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