  For general mean-field backward stochastic differential equations (BSDEs) it
is well-known that we usually do not have the comparison theorem if the
coefficients depend on the law of $Z$-component of the solution process $(Y,
Z)$. A natural question is whether general mean-field BSDEs whose coefficients
depend on the law of $Z$ have the comparison theorem for some cases. In this
paper we establish the comparison theorems for one-dimensional mean-field BSDEs
whose coefficients also depend on the joint law of the solution process
$(Y,Z)$. With the help of Malliavin calculus and a BMO martingale argument, we
obtain two comparison theorems for different cases and a strong comparison
result. In particular, in this framework, we compare not only the first
component $Y$ of the solution $(Y,Z)$ for such mean-field BSDEs, but also the
second component $Z$.
