On – estimates of logarithmic-type in Hardy–Sobolev spaces of the disk and the annulus

The aim of this paper is to prove some optimal estimates of logarithmic-type in Hardy–Sobolev spaces H 1 , p , 1 ≤ p ≤ ∞ , of both the unit disk and the annulus. These estimates extend those previously established by Baratchart and Zerner in the Hardy–Sobolev space H 1 , 2 of the unit disk and by Chaabane and the first author of the paper in the case of the spaces H 1 , ∞ of the unit disk and of the annulus.