I am currently reading the work of Belaga on upper bounds on minimal cyclic iterates in the $3x+d$ problem.
In the paper, the author gives an upper bound on the perigee as $$ dk^{C_2} $$ where $k$ is the number of odd elements in the cycle, and $C_2$ is an effectively computable constant not exceeding $32$ (as per Corollary 2 in the cited paper of Laurent et al).
Later in the introduction, Belaga mentions an upper bound for the apogee (maximal element)
$$dk^C (3/2)^k $$
(the author and Mignotte derive this upper bound in another paper )
Question: Does 32 still apply as an upper bound for the effectively computable constant C (when bounding the apogee)?
The author writes that he corresponded with another author in the derivation of this constant (for the perigee).
One can apply the results of Rhin (as provided by Lemma 12 in the work of Simons and De Weger) to derive sharp constants.
Assume $k+l>k$. Lemma 12 in Simons/De Weger demonstrates the inequality
$$ (k+l)\log 2 - k\log 3 > e^{-13.3(0.46507)}k^{-13.3}.$$ This inequality provides means for deriving a lower bound on the denominator $2^{k+l}-3^{k}$ of a periodic orbit element; the argument in the abovementioned paper of Belaga/Mignotte demonstrates how this lower bound can be applied to derive an upper bound on the maximal iterate element.