subharmonic function
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2020 ◽  
Vol 54 (2) ◽  
pp. 188-192
Author(s):  
M.V. Zabolotskyi

For $\delta$-subharmonic in $\mathbb{R}^m$, $m\geq2$, function $u=u_1-u_2$ of finite positiveorder we found the asymptotical representation of the form\[u(x)=-I(x,u_1)+I(x,u_2) +O\left(V(|x|)\right),\ x\to\infty,\]where $I(x,u_i)=\int\limits_{|a-x|\leq|x|}K(x,a)d\mu_i(a)$, $K(x,a)=\ln\frac{|x|}{|x-a|}$ for $m=2$,$K(x,a)=|x-a|^{2-m}-|x|^{2-m}$ for $m\geq3,$$\mu_i$ is a Riesz measure of the subharmonic function $u_i,$ $V(r)=r^{\rho(r)},$ $\rho(r)$ is a proximate order of $u$.The obtained result generalizes one theorem of I.F. Krasichkov for entire functions.



Author(s):  
Simon Larson

Abstract Let $\Omega \subset {\mathbb{R}}^d $, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to{\mathbb{R}}$ be a non-negative subharmonic function. In this paper, we prove the inequality $$\begin{equation*} \frac{1}{|\Omega|}\int_{\Omega} f(x)\, \textrm{d}x \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\, \textrm{d}\sigma(x)\,. \end{equation*}$$Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset{\mathbb{R}}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then $$\begin{equation*} \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \end{equation*}$$Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al. [2].



2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Minghua Han ◽  
Jianguo Sun ◽  
Gaoying Xue

Abstract Our main aim in this paper is to obtain a new type of boundary integral behaviors of harmonic functions in a smooth cone. As an application, the least harmonic majorant of a nonnegative subharmonic function is also given.



2017 ◽  
Vol 101 (3-4) ◽  
pp. 590-607 ◽  
Author(s):  
T. Yu. Baiguskarov ◽  
B. N. Khabibullin ◽  
A. V. Khasanova


2016 ◽  
Vol 99 (3-4) ◽  
pp. 576-589 ◽  
Author(s):  
B. N. Khabibullin ◽  
T. Yu. Baiguskarov




2014 ◽  
Vol 114 (1) ◽  
pp. 86 ◽  
Author(s):  
A. Aytuna ◽  
A. Sadullaev

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among these different definitions. In section 3 we relate some of these notions to the linear topological type of the Fréchet space of analytic functions on the given manifold. In section 4 we look at some examples and show, for example, that the complement of the zero set of a Weierstrass polynomial possesses a continuous plurisubharmonic exhaustion function that is maximal off a compact subset.



Filomat ◽  
2007 ◽  
Vol 21 (2) ◽  
pp. 243-249 ◽  
Author(s):  
Vesna Kojic

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. .



2005 ◽  
Vol 2005 (3) ◽  
pp. 383-391
Author(s):  
Gundon Choi ◽  
Gabjin Yun

LetMbe a complete Riemannian manifold andNa complete noncompact Riemannian manifold. Letϕ:M→Nbe a surjective harmonic morphism. We prove that ifNadmits a subharmonic function with finite Dirichlet integral which is not harmonic, andϕhas finite energy, thenϕis a constant map. Similarly, iffis a subharmonic function onNwhich is not harmonic and such that|df|is bounded, and if∫M|dϕ|<∞, thenϕis a constant map. We also show that ifNm(m≥3)has at least two ends of infinite volume satisfying the Sobolev inequality or positivity of the first eigenvalue of the Laplacian, then there are no nonconstant surjective harmonic morphisms with finite energy. Forp-harmonic morphisms, similar results hold.



2004 ◽  
Vol 56 (2) ◽  
pp. 225-245 ◽  
Author(s):  
Gordon Blower ◽  
Thomas Ransford

AbstractThe norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for LebesgueLpspaces and the von Neumann-Schatten trace ideals. Banach spaces that areq-uniformly PL-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace idealscpare 2-uniformly PL-convex for 1 ≤p≤ 2.



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