  In a four-dimensional spacetime, the DeWitt-Schwinger expansion of the
effective action associated with a massive quantum field reduces, after
renormalization and in the large mass limit, to a single term constructed from
the purely geometrical Gilkey-DeWitt coefficient $a_3$ and its metric variation
provides a good analytical approximation for the renormalized stress-energy
tensor of the quantum field. Here, from the general expression of this tensor,
we obtain analytically the renormalized stress-energy tensors of the massive
scalar field, the massive Dirac field and the Proca field in Kerr-Newman
spacetime. It should be noted that, even if, at first sight, the expressions
obtained are complicated, their structure is in fact rather simple, involving
naturally spacetime coordinates as well as the mass $M$, the charge $Q$ and the
rotation parameter $a$ of the Kerr-Newman black hole and permitting us to
recover rapidly the results already existing in the literature for the
Schwarzschild, Reissner-Nordstr\"om and Kerr black holes (and to correct them
in the latter case). In the absence of exact results in Kerr-Newman spacetime,
our approximate renormalized stress-energy tensors could be very helpful, in
particular to study the backreaction of massive quantum fields on this
spacetime or on its quasinormal modes.
