Abstract. Recently Gao--Yang showed the following result. Let $T$ be a $p$-hyponormal operator for $0< p \le1$. Then $$ (T^{n+1^*}T^{n+1})^{n+p\over n+1} \ge(T^{n^*}T^n)^{n+p\over n} \mbox{ and } (T^nT^{n^*})^{n+p\over n} \ge(T^{n+1}T^{n+1^*})^{n+p\over n+1} $$ hold for all positive integers $n$. Moreover, parallel results to invertible log-hyponormal operators have already been shown by Yamazaki, and also it was known that Yamazaki's result holds even for class $A$ operators. In this paper, as a parallel result to that of class $A$ operators, we shall show that the above inequalities hold under weaker conditions than $p$-hyponomality, that is, class $F(p,r,q)$ defined by Fujii--Nakamoto or class $wF(p,r,q)$ defined by Yang--Yuan under appropriate conditions of $p$, $r$ and $q$.

AMS Subject Classification (1991): 47B20, 47A63

Keyword(s): p, -hyponormal operators, log-hyponormal operators, A, classoperators, F(p, class, r, operators and class, q)wF(p, r, operators, q)

Received December 20, 2007. (Registered under 6076/2009.)