The main purpose of this work is to establish some logarithmic estimates of optimal type in the Hardy-Sobolev space $H^{k,\infty }$; $k\in {\mathbb{N}}^{*}$ of an annular domain. These results are considered as a continuation of a previous study in the setting of the unit disk by L. Baratchart and M. Zerner, On the recovery of functions from pointwise boundary values in a Hardy-Sobolev class of the disk, J. Comput. Appl. Math. 46 (1993), 255–269 and by S. Chaabane and I. Feki, Optimal logarithmic estimates in Hardy-Sobolev spaces...