Quasinearly Subharmonic Functions Research Articles

On quasinearly subharmonic functions

We recall the definition of quasinearly subharmonic functions, point out that this function class includes, among others, subharmonic functions, quasisubharmonic functions, nearly subharmonic functions and essentially almost subharmonic functions. It is shown that the sum of two quasinearly subharmonic functions may not be quasinearly subharmonic. Moreover, we characterize the harmonicity via quasinearly subharmonicity.

Some Properties of Quasinearly Subharmonic Functions and Maximal Theorem for Bergman Type Spaces

Let denote the class of quasinearly subharmonic functions in unit ball . We provide, following result: if and if , then , where is the radial maximal function and , and . Also, we prove a maximal theorem for Bergman type spaces.

MEAN VALUE TYPE INEQUALITIES FOR QUASINEARLY SUBHARMONIC FUNCTIONS

AbstractThe mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

Quasi-Nearly Subharmonic Functions and Quasiconformal Mappings

We prove that the composition of a quasi-nearly subharmonic function and a quasiregular mapping of bounded multiplicity is quasi-nearly subharmonic. Also, we prove that if u ∘ f is quasi-nearly subharmonic for all quasi-nearly subharmonic u and f satisfying some additional conditions, then f is quasiconformal. Similar results are further established for the class of regularly oscillating functions.

Domination Conditions for Families of Quasinearly Subharmonic Functions

Domar has given a condition that ensures the existence of the largest subharmonic minorant of a given function. Later Rippon pointed out that a modification of Domar′s argument gives in fact a better result. Using our previous, rather general and flexible, modification of Domar′s original argument, we extend their results both to the subharmonic and quasinearly subharmonic settings.

Bi‐Lipschitz Mappings and Quasinearly Subharmonic Functions

After considering a variant of the generalized mean value inequality of quasinearly subharmonic functions, we consider certain invariance properties of quasinearly subharmonic functions. Kojić has shown that in the plane case both the class of quasinearly subharmonic functions and the class of regularly oscillating functions are invariant under conformal mappings. We give partial generalizations to her results by showing that in ℝn, n ≥ 2, these both classes are invariant under bi‐Lipschitz mappings.

Quasi-nearly subharmonic functions in locally uniformly homogeneous spaces

We define nonnegative quasi-nearly subharmonic functions on so called locally uniformly homogeneous spaces. We point out that this function class is rather general. It includes quasi-nearly subharmonic (thus also subharmonic, quasisubharmonic and nearly subharmonic) functions on domains of Euclidean spaces \({{\mathbb{R}}^n}\), n ≥ 2. In addition, quasi-nearly subharmonic functions with respect to various measures on domains of \({{\mathbb{R}}^n}\), n ≥ 2, are included. As examples we list the cases of the hyperbolic measure on the unit ball Bn of \({{\mathbb{R}}^n}\), the \({{\mathcal{M}}}\)-invariant measure on the unit ball B2n of \({{\mathbb{C}}^n}\), n ≥ 1, and the quasihyperbolic measure on any domain \({D\subset {\mathbb{R}}^n}\), \({D\ne {\mathbb{R}}^n}\). Moreover, we show that if u is a quasi-nearly subharmonic function on a locally uniformly homogeneous space and the space satisfies a mild additional condition, then also up is quasi-nearly subharmonic for all p > 0.

Subharmonic functions, generalizations, weighted boundary behavior, and separately subharmonic functions: A survey

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 includes, among others, quasisubharmonic functions, nearly subharmonic functions (in a slightly generalized sense) and almost subharmonic functions (essentially). In addition to the basic properties of quasi-nearly subharmonic functions, we list certain weighted boundary behavior properties, and a counterpart to Armitage’s and Gardiner’s result concerning the subharmonicity of a separately subharmonic function. We slightly sharpen our previous improvement to Armitage’s and Gardiner’s result, too. In addition, we recall our recent improvements to Kołodziej’s and Thorbiörnson’s result concerning the subharmonicity of a function subharmonic with respect to the first variable and harmonic with respect to the second.

Classes of Quasi-nearly Subharmonic Functions

It is well known and important that if u ≥ 0 is subharmonic on a domain Ω in ℝn and p > 0, then there is a constant C(n,p) ≥ 1 such that \(u(x)^p\leq C(n,p){\mathcal{MV}}(u^p,B(x,r))\) for each open ball B(x,r) ⊂ Ω. 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, Herve’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.

Quasi-Nearly Subharmonicity and Separately Quasi-Nearly Subharmonic Functions

Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, of Avanissian, of 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 quasi-nearly subharmonic functions. 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.

Quasi-nearly subharmonic functions and conformal mappings

If is a conformal mapping and u is a quasi-nearly subharmonic function, then u o ? is quasi-nearly subharmonic. A similar fact for "regularly oscillating" functions holds. .