An operator \(T\) is said to be log-hyponormal

  if \(T\) is invertible and \(\log T^*T \ge \log TT^*\),

  and \(p\)-paranormal for \(p > 0\)

  if \(\||T|^p U |T|^p x\| \ge \||T|^p x\|^2\)

  for every unit vector \(x\),

  where the polar decomposition of \(T\) is \(T = U|T|\).

  We show that \(T\) is log-hyponormal if and only if

  \(T\) is invertible and \(p\)-paranormal for all \(p > 0\).