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As a characterization of chaotic order, we showed “If MI B mI > 0 , then log A log B is equivalent to Mh(p)A p Bp for all p > 0 , where h = m > 1 and Mh(p) = h p hp−1 e log h p hp−1 ” in [11]. In this paper, we shall show the following characterization of chaotic order as a parallel result to the result mentioned above: “If MI B mI > 0 , then log A log B is equivalent to Ap + L(mp, Mp) log Mh(p)I Bp for all p > 0 , where L(m,M) = M−m log M−log m .” And we shall discuss the relations among this result and some related results. Mathematics subject classification (1991): 47A63.

Further characterizations of chaotic order via Specht's ratio