  In [8] (Nakagawa, et.al., IEEE Trans. IT, 2021), we investigated the
convergence speed of the Arimoto-Blahut algorithm. In [8], the convergence of
the order $O(1/N)$ was analyzed by focusing on the second-order nonlinear
recurrence formula consisting of the first- and second-order terms of the
Taylor expansion of the defining function of the Arimoto-Blahut algorithm.
However, in [8], an infinite number of inequalities were assumed as a
"conjecture," and proofs were given based on the conjecture. In this paper, we
report a proof of the convergence of the order $O(1/N)$ for a class of channel
matrices without assuming the conjecture. The correctness of the proof will be
confirmed by several numerical examples.
