  We propose the study of a conformally invariant functional for surfaces of
complex projective plane which is closely related to the classical Willmore
functional. We show that minimal surfaces of complex projective plane are
critical for this functional and construct some minima for it via the twistors
spaces of complex projective plane. Also, we find lower bounds for this
functional and for its restriction to the class of Lagrangian surfaces and
characterize the complex lines and the Lagrangian totally geodesic surfaces and
the Whitney spheres as the only attaining those bounds.
