tailieunhanh - Báo cáo hóa học: " Research Article Quasi-Nearly Subharmonicity and Separately Quasi-Nearly Subharmonic Functions"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Quasi-Nearly Subharmonicity and Separately Quasi-Nearly Subharmonic Functions | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008 Article ID 149712 15 pages doi 2008 149712 Research Article Quasi-Nearly Subharmonicity and Separately Quasi-Nearly Subharmonic Functions Juhani Riihentaus Department of Physics and Mathematics University of Joensuu . Box 111 80101 Joensuu Finland Correspondence should be addressed to Juhani Riihentaus Received 29 February 2008 Accepted 30 July 2008 Recommended by Shusen Ding 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. 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. 1. Introduction . Previous results Solving a long standing problem Wiegerinck 1 Theorem page 770 see also Wiegerinck and Zeinstra 2 Theorem 1 page 246 showed that a separately subharmonic function need not be subharmonic. On the other hand Armitage and Gardiner 3 Theorem 1 page 256 showed that a separately subharmonic function u in a domain Q in Rm n m n 2 is subharmonic provided log u is locally integrable in Q where Ộ 0 x 0 x