Pre-threshold fractional susceptibility functions at Misiurewicz parameters

We show that the response, frozen and semifreddo fractional susceptibility functions of certain real-analytic unimodal families, at Misiurewicz parameters and for fractional differentiation index $0\le \eta <1\mathrm{/}2$, are holomorphic on a disk of radius greater than one. This is a step towards solving a conjecture of Baladi and Smania, in the case of the aforementioned susceptibility functions.