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We shall give some applications of Tanahashi’s result which states the best possibility of Furuta inequality. Firstly, we shall discuss the best possibility of a well-known characterization of chaotic order: log A log B if and only if Ar (A r 2 BpA r 2 ) r p+r holds for all p 0 and r 0 . Secondly, we shall discuss the best possibility of p-hyponormality of generalized Aluthge transformation Ts,t = |T|sU|T|t for p-hyponormal or log-hyponormal operator T whose polar decomposition is T = U|T| . Mathematics subject classification (1991): 47B20, 47A63.

Some applications of Tanahashi's result on the best possibility of Furuta inequality