We introduce a new family of classes of operators which is derived  

from the Furuta inequality.  It is closely related to log-hyponormal  

operators defined by Tanahashi, the classes $A(p)$ and absolute  

$p$-paranormal operators defined by Furuta-Ito-Yamazaki.  We discuss  

some properties of operators in such classes and relations among  

them and extend results due to Furuta-Ito-Yamazaki.