Throughout this paper let \(p= (p_k)\) be any sequence of strictly positive numbers, the Köthe-Toeplitz dual of the sequence space \[ \ell_\infty(p)= \Biggl\{(x_k)\in w:\sup_k | x_k|^{p_k} 1} \Biggl\{(x_k)\in w: \sum_k| x_k| N^{1/p_k} 0\) for every \(k\). Then \(A\in(M_\infty(p),\overline c\cap\ell_\infty)\) if and only if there exists an integer \(B>1\) such that \(D= \sup_{n,k} | a_{nk}| B^{-1/p_k}<\infty\) holds and \(a_k= \text{stat}-\lim_{n\to\infty} a_{nk}\) exists \(\forall k\in \mathbb{N}\). Theorem 2. \(A\in(\varphi, \overline c\cap\ell_\infty)\) if and only if there are numbers \(t\) and \(r\) \((0