The critical point and the $p$-norm of $A_s$ and $C$-matrices

The $L$-matrix $A_s=[1/(n+s)]$ was introduced in \cite{MRtmp}. As a surprising property, we showed that its 2-norm is constant for $s \geq s_0$, where the critical point $s_0$ is unknown but relies in the interval $(1/4,1/2)$. In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding $s_0$. Moreover, we show that the same property persists for the $p$-norm of $A_s$ matrices. We also obtain the 2-norm of a family of $C$-matrices with lacunary sequences.