Journal of Inequalities and Applications
Volume 2008 (2008), Article ID 149712, 15 pages
doi:10.1155/2008/149712
Research Article
Quasi-Nearly Subharmonicity and Separately Quasi-Nearly Subharmonic Functions
Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, 80101 Joensuu, Finland
Received 29 February 2008; Accepted 30 July 2008
Academic Editor: Shusen Ding
Copyright © 2008 Juhani Riihentaus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, Avanissian, 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 quasinearly subharmonic function. 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.