  Wiegerinck has shown that a separately subharmonic function need not be
subharmonic. Improving previous results of Lelong, of Avanissian, of Arsove and
of us, Armitage and Gardiner gave an almost sharp integrability condition which
ensures a separately subharmonic function to be subharmonic. Completing now our
recent counterparts to the cited results of Lelong, Avanissian and Arsove for
so called quasi-nearly subharmonic functions, we present a counterpart to the
cited result of Armitage and Gardiner for separately quasi-nearly subharmonic
functions. This counterpart enables us to slightly improve Armitage's and
Gardiner's original result, too. The method we use is a rather straightforward
and technical, but still by no means easy, modification of Armitage's and
Gardiner's argument combined with an old argument of Domar.
