With the aid of our new version of a mean value type inequality, we improve our previous result of the subharmonicity of separately
subharmonic functions, and thus also the well-known related earlier results of Armitage and Gardiner and ours. Moreover, we give refinements, with concise proofs, to the basic classical results of Avanissian,
of Lelong, and of Arsove.
Keywords: Subharmonic, quasinearly subharmonic, separately quasinearly subharmonic function