Extensions of the results on powers of -hyponormal and -hyponormal operators

Firstly, we will show the following extension of the results on powers of-hyponormal and-hyponormal operators: letandbe positive integers, ifis-hyponormal for, then: (i) in case,andhold, (ii) in case,andhold. Secondly, we will show an estimation on powers of-hyponormal operators forwhich implies the best possibility of our results. Lastly, we will show a parallel estimation on powers of-hyponormal operators as follows: let, then the following hold for each positive integerand: (i) there exists a log-hyponormal operatorsuch that, (ii) there exists a-hyponormal operatorsuch that.