Relationships between monotonicity and complex rotundity properties with some consequences

It is proved that a point $f$ of the complexification $E^C$ of a real Köthe space $E$ is a complex extreme point if and only if $|f|$ is a point of upper monotonicity in $E$. As a corollary it follows that $E$ is strictly monotone if and only if $E^C$ is complex rotund. It is also shown that $E$ is uniformly monotone if and only if $E^C$ is uniformly complex rotund. Next, the fact that $|x|\in S(E^+)$ is a ULUM-point of $E$ whenever $x$ is a $C$-LUR-point of $S(E^C)$ is proved, whence the relation that $E$ is a ULUM-space whenever $E^C$ is $C$-LUR is concluded. In the second part of this paper these general results are applied to characterize complex rotundity of properties Calderón-Lozanovskiĭ spaces, generalized Calderón-Lozanovskiĭ spaces and Orlicz-Lorentz spaces.