Abstract. In this paper we show the following extensions of the results by Yamazaki and Furuta--Yanagida: If $T$ is a $p$-hyponormal operator for $p\in(0,1]$, then $(T^{n+1^{\ast }}T^{n+1})^{n+p\over n+1}\geq(T^{n^{\ast }}T^n)^{n+p\over n}$ and $(T^nT^{n^{\ast }})^{n+p\over n}\geq(T^{n+1}T^{n+1^{\ast }})^{n+p\over n+1}$ hold for all positive integer $n$. And if $T$ is a $p$-hyponormal operator for $p>1$, then $T^{n+1^{\ast }}T^{n+1}\geq(T^{n^{\ast }}T^n)^{n+1\over n}$ and $(T^nT^{n^{\ast }})^{n+1\over n}\geq T^{n+1}T^{n+1^{\ast }}$ hold for all positive integer $n$. And we also discuss the best possibility of our results.

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

Received January 3, 2005, and in final form May 30, 2006. (Registered under 5943/2009.)