  In multiparameter quantum metrology, the weighted-arithmetic-mean error of
estimation is often used as a scalar cost function to be minimized during
design optimization. However, other types of mean error can reveal different
facets of permissible error combination. By introducing the weighted $f$-mean
of estimation error and quantum Fisher information, we derive various quantum
Cram\'er-Rao bounds on mean error in a very general form and also give their
refined versions with complex quantum Fisher information matrices. We show that
the geometric- and harmonic-mean quantum Cram\'er-Rao bounds can help to reveal
more forbidden region of estimation error for a complex signal in coherent
light accompanied by thermal background than just using the ordinary
arithmetic-mean version. Moreover, we show that the $f$-mean quantum Fisher
information can be considered as information-theoretic quantities to quantify
asymmetry and coherence as quantum resources.
