we introduce a new concept of general
$g$-$eta$-monotone operator generalizing the general
$(h,eta)$-monotone operator cite{arvar2, arvar1}, general
$h-$ monotone operator cite{xiahuang} in banach spaces, and also
generalizing $g$-$eta$-monotone operator cite{zhang}, $(a,
eta)$-monotone operator cite{verma2}, $a$-monotone operator
cite{verma0}, $(h, eta)$-monotone operator cite{fanghuang}...
