On separated Carleson sequences in the unit disc of $\mathbb{C}\mathrm{.}$

The interpolating sequences for $H^{\mathrm{\infty }}(\mathbb{D}),$ the bounded holomorphic function in the unit disc $\mathbb{D}$ of the complex plane $\mathbb{C},$ {\small where characterised by L. Carleson by metric conditions on the points. They are also characterised by “dual boundedness” conditions which imply an infinity of functions. A. Hartmann proved recently that just one function in $H^{\mathrm{\infty }}(\mathbb{D})$ was enough to characterize interpolating sequences for $H^{\mathrm{\infty }}(\mathbb{D})\mathrm{.}$ In this work we use the “hard” part of the proof of Carleson for the Corona theorem, to extend Hartman’s result and answer a question he asked in his paper.}\ \par