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Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\\in\\mathcal{H}ol(B_d)$ such that the standard integral means $M_p(f, r)$ and $w(r)$ are equivalent for any $0<p\\le\\infty$. 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Given a log-convex strictly positive weight $w(r)$ on $[0,1)$, we construct a function $f\\in\\mathcal{H}ol(B_d)$ such that the standard integral means $M_p(f, r)$ and $w(r)$ are equivalent for any $0<p\\le\\infty$. 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