Further result on Dirichlet-Sch type inequality and its application
- Liquan Wan^{1}Email author
https://doi.org/10.1186/s13660-017-1381-4
© The Author(s) 2017
Received: 4 March 2017
Accepted: 21 April 2017
Published: 8 May 2017
Abstract
In this paper we deal with a theoretical question raised in connection with the application of Dirichlet-Sch type inequality, obtained by Huang (Int. Math. J. 27(02):1650009, 2016), which has been already applied to obtain multiplicity results for boundary value problems in several recent papers. We also discuss a particular case of it in more detail. As an application, we deduce the least harmonic majorant and log-concavity of extended subharmonic functions.
Keywords
1 Introduction
Let Γ be the subset of the upper half unit sphere. The set \({\mathbf {R}}_{+}\times\Gamma\) in \({\mathbf {R}}^{n}\) is called a cone. We denote it by \(\mathfrak{C}_{n}(\Gamma)\), where \(\Gamma\subset{\mathbf {S}}_{1}\). The sets \(I\times\Gamma\) and \(I\times \partial{\Gamma}\) with an interval on R are denoted by \(\mathfrak{C}_{n}(\Gamma;I)\) and \(\mathfrak{S}_{n}(\Gamma;I)\), respectively. We denote \(\mathfrak {C}_{n}(\Gamma)\cap S_{R}\) and \(\mathfrak{S}_{n}(\Gamma; (0,+\infty))\) by \(\mathfrak {S}_{n}(\Gamma; R)\) and \(\mathfrak{S}_{n}(\Gamma)\), respectively.
Furthermore, we denote by dσ (resp. \(dS_{R}\)) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(\partial{\mathfrak{C}_{n}(\Gamma)}\) (resp. \(S_{R}\)) and by dw the elements of the Euclidean volume in \({\mathbf {R}}^{n}\).
Remark 1
A function \(g(t)\) on \((0,\infty)\) is \(\mathbb{A}_{d_{1},d_{2}}\)-convex if and only if \(g(t)t^{d_{2}}\) is a convex function of \(t^{d}\) \((d=d_{1}+d_{2})\) on \((0,\infty)\), or, equivalently, if and only if \(g(t)t^{-d_{1}}\) is a convex function of \(t^{-d}\) on \((0,\infty)\).
Remark 2
We will also consider the class of all continuous functions \(u(t,\Phi)\) \(((t,\Phi)\in\overline{\mathfrak{C}_{n}(\Gamma)})\) harmonic in \(\mathfrak{C}_{n}(\Gamma)\) with \(u^{+}(t,\Phi)\in \mathcal{A}_{\Gamma}\) \(((t,\Phi)\in\mathfrak{C}_{n}(\Gamma))\) and \(u^{+}(t,\Phi)\in\mathcal{B}_{\Gamma}\) \(((t,\Phi)\in\mathfrak {S}_{n}(\Gamma))\) is denoted by \(\mathcal{C}_{\Gamma}\) (see [6]).
In 2015, Jiang, Hou and Peixoto-de-Büyükkurt (see [7]) obtained the following result.
Theorem A
Recently, Wang, Huang and N. Yamini (see [8]) generalized Theorem A to the conical case.
Theorem B
The remainder of the paper is organized as follows: in Section 2, we shall give our main theorem; in Section 3, some necessary lemmas are given; in Section 4, we shall prove the main result.
2 Main result
In this section, we give the main result of this paper.
Our main aim is to give a least harmonic majorant of a nonnegative subharmonic function on \(\mathfrak{C}_{n}(\Gamma)\).
Theorem 1
3 Main lemmas
In order to prove our main result, we need the following lemmas.
Lemma 1
see [1]
Let u be a function subharmonic on \(\mathfrak{C}_{n}(\Gamma)\) satisfying (1.4). Then the limit \(\mathscr{U}_{u}\) \((-1<\mathscr{U}_{u}\leq1)\) exists.
Lemma 2
Proof
Lemma 3
Proof
Lemma 4
Proof
Lemma 5
Let u be a subharmonic function in \(\overline {\mathfrak{C}_{n}(\Gamma)}\) such that \(u'=u|\partial{\mathfrak{C}_{n}(\Gamma)}\) satisfies (1.5). Then \(\mathbb{PI}_{\mathfrak{C}_{n}(\Gamma)}[u'](P)\leq h(P)\) on \(\mathfrak{C}_{n}(\Gamma)\), where \(h(P)\) is the any harmonic majorant of u on \(\mathfrak{C}_{n}(\Gamma)\).
Proof
4 Proof of Theorem 1
Hence we see from (4.2) and (4.3) that (2.1) holds.
We see from Lemma 2 that \(-1\leq h^{\ast}(P)\leq\epsilon\) on \(\mathfrak{C}_{n}(\Gamma)\), which shows that \(h_{u}(P)\) is the least harmonic majorant in \(\mathfrak{C}_{n}(\Gamma)\). Theorem 1 is proved.
5 Conclusion
In this article, we dealt with a theoretical question raised in connection with the application of Dirichlet-Sch type inequality. Additionally, we discussed a particular case of it in more detail. As applications, we deduced the least harmonic majorant and log-concavity of extended subharmonic functions.
Declarations
Acknowledgements
I would like to thank the referees for their constructive suggestions and useful comments which resulted in an improved version of this paper. This work was supported by the 2015 Universities Philosophy Social Sciences Innovation Team of Henan Province: the Institutional Arrangements for Mixed Ownership Reform of Stateowned Enterprises in Henan Province (No. 2015-CXTD-09).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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