Analytic proof of the invariant-density derivative inequalities

Prove analytically that the invariant density h of the Liverani–Saussol–Vaienti map satisfies \(\bigl(h'/h\bigr)' > 0\) and \(\bigl(h''/h\bigr)' < 0\) on \((0,1]\), rather than establishing these inequalities through rigorous numerical verification.

Background

The paper identifies the inequalities $\bigl(h'/h\bigr)' > 0$ and $\bigl(h''/h\bigr)' < 0$ as key properties needed for the authors’ optimal memory-loss results for Liverani–Saussol–Vaienti intermittent maps. The authors establish the inequalities on subintervals using asymptotic estimates near zero and Chebyshev approximations away from zero, while covering the remaining interval through rigorous numerical verification and recursive bounds. They explicitly state that an analytic proof of the full inequalities is not known to them, leaving the replacement of the numerical argument by a purely analytic proof unresolved.