  First, we give the definition for quasi-nearly subharmonic functions, now for
general, not necessarily nonnegative functions, unlike previously. We point out
that our function class incudes, among others, quasisubharmonic functions,
nearly subharmonic functions (in a slightly generalized sense) and almost
subharmonic functions. We also give some basic properties of quasi-nearly
subharmonic functions. Second, after recalling some of the existing
subharmonicity results of separately subharmonic functions, we give the
corresponding counterparts for separately quasi-nearly subharmonic functions,
thus improving previous results of ours, of Lelong, of Avanissian and of
Arsove. Third, we give two results concerning the subharmonicity of a function
subharmonic with respect to the first variable and harmonic with respect to the
second variable. The first result improves a result of Arsove, concerning the
case when the function has, in addition, locally a negative integrable
minorant. The second result improves a result of Ko{\l}odziej and
Thorbi\"ornson concerning the subharmonicity of a function subharmonic and
${\mathcal{C}}^2$ in the first variable and harmonic in the second.
