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We answer an open problem posed by Mossel--Oleszkiewicz--Sen regardingrelations between $p$-log-Sobolev inequalities for $p\in(0,1]$. We show thatfor any interval $I\subset(0,1]$, there exist $q,p\in I$, $q<p$, and a measure$\mu$ for which the $q$-log-Sobolev inequality holds, while the $p$-log-Sobolevinequality is violated. As a tool we develop certain necessary and closelyrelated sufficient conditions characterizing those inequalities in the case ofbirth-death processes on $\mathbb{N}$.

$P$-log-Sobolev inequalities on $\mathbb{N}$