Inspired by the geometric mean due to Atteia and Ra\"\i ssouli,
we discuss a general theory of convex functional means 
on a Hilbert space like the Kubo-Ando theory of operator means. 
Though our construction is based on the integral representation 
in Kubo-Ando theory, it is an exact extension not only for 
operator means but also for Atteia-Ra\"\i ssouli's ones. 
We give an example where our geometric mean can be defined 
even if their geometric one cannot. 
We show that our convex functional means satisfy monotonicity, 
semi-continuity, homogeneity, subadditivity, joint concavity, 
transformer inequality and normalization.
One of outstanding properties of these means appeares in those 
for constant functions, which suggests us to weights for 
operator means.