No abstract

It is well known and important that if u ≥ 0 is subharmonic on a domain in R n and p > 0, then there is a constant C(n, p) ≥ 1 such that u(x) p ≤ C(n, p)MV (u p , B(x, r)) for each open ball B(x, r) ⊂ . The definition of a relatively new function class, quasi-nearly subharmonic functions, is based on such a generalized mean value inequality. It is pointed out that the obtained function class is natural. It has important and interesting properties and, at the same time, it is large: In addition to nonnegative subharmonic functions, it includes, among others, Hervé's nearly subharmonic functions, functions satisfying certain natural growth conditions, especially certain eigenfunctions, polyharmonic functions and generalizations of convex functions. Further, some of the basic properties of quasi-nearly subharmonic functions are stated in a unified form. Moreover, a characterization of quasi-nearly subharmonic functions with the aid of the quasihyperbolic metric and two weighted boundary limit results are given.

If u ≥ 0 is subharmonic on a domain Ω in R n and p > 0, then it is well-known thatWe recently showed that a similar result holds more generally for functions of the form ψ • u where ψ : R + → R + may be any surjective, concave function whose inverse ψ −1 satisfies the ∆ 2 -condition. Now we point out that this result can be extended slightly further. We also apply this extended result to the weighted boundary behavior and nonintegrability questions of subharmonic and superharmonic functions.

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.

On the extension in the Hardy classes and the Nevanlinna class Bulletin de la S. M. F., tome 112 (1984), p. 469-480 © Bulletin de la S. M. F., 1984, tous droits réservés. L'accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/