We study the absolutely continuous invariant probability (SRB) measure $m_t$ of $f_t$, as a function of $t$ on the set of Collet-Eckmann (CE) parameters:
Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set $D$ of CE parameters, and, for each $s$ in $D$, a subset $D_0$ of $D$ of polynomially recurrent parameters containing $s$ as a Lebesgue density point, and constants $C>1$, $G >4$, so that, for every 1/2-Holder function $A4 (of 1/2-Holder norm $|A|$) and all $t$ in $D_0$,
$|\int A dm_t -\int A dm_s| < C |A| |t-s|^{1/2} |\log|t-s||^G$
(If $f_t(x)=tx(1-x)$, the set $D$ contains almost all CE parameters.)
Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter s, we find a set of CE parameters $D'$ accumulating at $s$, a constant $C >1$, and an infinitely differentiable function $B$, so that for all $t\in D'$
$C |t-s|^{1/2} > |\int B dm_t -\int B dm_s| > |t-s|^{1/2}/C$