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# RETRACTED ARTICLE: Levin’s type boundary behaviors for functions harmonic and admitting certain lower bounds

*Boundary Value Problems*
**volume 2015**, Article number: 159 (2015)

## Abstract

In this paper, we prove Levin’s type boundary behaviors for functions harmonic and admitting certain lower bounds, which extend Pan, Qiao and Deng’s inequalities for analytic functions in a half-space.

## 1 Introduction and results

Let **R** and \({\mathbf{R}}_{+}\) be the set of all real numbers and the set of all positive real numbers, respectively. We denote by \({\mathbf{R}}^{n}\) (\(n\geq2\)) the *n*-dimensional Euclidean space. A point in \({\mathbf{R}}^{n}\) is denoted by \(P=(X,x_{n})\), \(X=(x_{1},x_{2},\ldots,x_{n-1})\). The Euclidean distance between two points *P* and *Q* in \({\mathbf{R}}^{n}\) is denoted by \(|P-Q|\). Also \(|P-O|\) with the origin *O* of \({\mathbf{R}}^{n}\) is simply denoted by \(|P|\). The boundary and the closure of a set *S* in \({\mathbf{R}}^{n}\) are denoted by *∂S* and *S̅*, respectively.

We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \({\mathbf{R}}^{n}\) which are related to Cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).

The unit sphere and the upper half-unit sphere in \({\mathbf{R}}^{n}\) are denoted by \({\mathbf{S}}^{n-1}\) and \({\mathbf{S}}^{n-1}_{+}\), respectively. For simplicity, a point \((1,\Theta)\) on \({\mathbf{S}}^{n-1}\) and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) for a set Ω, \(\Omega\subset{\mathbf{S}}^{n-1}\), are often identified with Θ and Ω, respectively. For two sets \(\Xi\subset{\mathbf{R}}_{+}\) and \(\Omega\subset{\mathbf{S}}^{n-1}\), the set \(\{(r,\Theta)\in{\mathbf{R}}^{n}; r\in\Xi,(1,\Theta)\in\Omega\}\) in \({\mathbf{R}}^{n}\) is simply denoted by \(\Xi\times\Omega\). In particular, the half-space \({\mathbf{R}}_{+}\times{\mathbf{S}}^{n-1}_{+}=\{(X,x_{n})\in{\mathbf{R}}^{n}; x_{n}>0\}\) will be denoted by \({T}_{n}\).

For \(P\in{\mathbf{R}}^{n}\) and \(r>0\), let \(B(P,r)\) denote the open ball with center at *P* and radius *r* in \({\mathbf{R}}^{n}\). \(S_{r}=\partial{B(O,r)}\). By \(C_{n}(\Omega)\), we denote the set \({\mathbf{R}}_{+}\times\Omega\) in \({\mathbf{R}}^{n}\) with the domain Ω on \({\mathbf{S}}^{n-1}\). We call it a cone. Then \(T_{n}\) is a special cone obtained by putting \(\Omega={\mathbf{S}}^{n-1}_{+}\). We denote the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\) with an interval on **R** by \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\). By \(S_{n}(\Omega; r)\) we denote \(C_{n}(\Omega)\cap S_{r}\). By \(S_{n}(\Omega)\) we denote \(S_{n}(\Omega; (0,+\infty))\) which is \(\partial{C_{n}(\Omega)}-\{O\}\).

We use the standard notations \(u^{+}=\max\{u,0\}\) and \(u^{-}=-\min\{u,0\}\). Further, we denote by \(w_{n}\) the surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{-1}\) of \({\mathbf{S}}^{n-1}\), by \({\partial}/{\partial n_{Q}}\) the differentiation at *Q* along the inward normal into \(C_{n}(\Omega)\), by \(dS_{r}\) the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(S_{r}\) and by *dw* the elements of the Euclidean volume in \({\mathbf{R}}^{n}\).

Let Ω be a domain on \({\mathbf{S}}^{n-1}\) with smooth boundary. Consider the Dirichlet problem

where \(\Lambda_{n}\) is the spherical part of the Laplace operator

We denote the least positive eigenvalue of this boundary value problem by *λ* and the normalized positive eigenfunction corresponding to *λ* by \(\varphi(\Theta)\),

In order to ensure the existence of *λ* and smooth \(\varphi(\Theta)\), we put a rather strong assumption on Ω: if \(n\geq3\), then Ω is a \(C^{2,\alpha}\)-domain (\(0<\alpha<1\)) on \({\mathbf{S}}^{n-1}\) surrounded by a finite number of mutually disjoint closed hypersurfaces for the definition of \(C^{2,\alpha}\)-domain. Then \(\varphi\in C^{2}(\overline{\Omega})\) and \({\partial\varphi}/{\partial n}>0\) on *∂*Ω (here and below, \({\partial}/{\partial n}\) denotes differentiation along the interior normal).

We note that each function \(r^{\aleph^{\pm}}\varphi(\Theta)\) is harmonic in \(C_{n}(\Omega)\), belongs to the class \(C^{2}(C_{n}(\Omega )\backslash\{O\})\) and vanishes on \(S_{n}(\Omega)\), where

In the sequel, for the sake of brevity, we shall write *χ* instead of \(\aleph^{+}-\aleph^{-}\). If \(\Omega={\mathbf{S}}^{n-1}_{+}\), then \(\aleph^{+}=1\), \(\aleph^{-}=1-n\) and \(\varphi(\Theta)=(2n w_{n}^{-1})^{1/2}\cos\theta_{1}\).

Let \(G_{\Omega}(P,Q)\) (\(P=(r,\Theta), Q=(t,\Phi)\in C_{n}(\Omega)\)) be the Green function of \(C_{n}(\Omega)\). Then the ordinary Poisson kernel relative to \(C_{n}(\Omega)\) is defined by

where \(Q\in S_{n}(\Omega)\), \(c_{n}= 2\pi\) if \(n=2\) and \(c_{n}= (n-2)w_{n}\) if \(n\geq3\).

The estimate we deal with has a long history which can be traced back to Levin’s type boundary behaviors for functions harmonic from below (see, for example, Levin [1], p.209).

### Theorem A

*Let*
\(A_{1}\)*be a constant*, \(u(z)\) (\(|z|=R\)) *be harmonic on*
\(T_{2}\)*and continuous on*
\({\partial T}_{2}\). *Suppose that*

*and*

*Then*

*where*
\(z=Re^{i\alpha}\in{T}_{2}\)*and*
\(A_{2}\)*is a constant independent of*
\(A_{1}\), *R*, *α**and the function*
\(u(z)\).

Recently, Pan *et al.* [2] considered Theorem A in the *n*-dimensional case and obtained the following result.

### Theorem B

*Let*
\(A_{3}\)*be a constant*, \(u(P)\) (\(|P|=R\)) *be harmonic on*
\(T_{n}\)*and continuous on*
\(\overline{T}_{n}\). *If*

*and*

*then*

*where*
\(P\in T_{n}\)*and*
\(A_{4}\)*is a constant independent of*
\(A_{3}\), *R*, \(\theta_{1}\)*and the function*
\(u(P)\).

Now we have the following.

### Theorem 1

*Let**K**be a constant*, \(u(P)\) (\(P=(R,\Theta)\)) *be harmonic on*
\(C_{n}(\Omega)\)*and continuous on* ̅. *If*

*and*

*then*

*where*
\(P\in C_{n}(\Omega)\), \(\rho(R)\)*is nondecreasing in*
\([1,+\infty)\)*and**M**is a constant independent of**K*, *R*, \(\varphi(\theta)\)*and the function*
\(u(P)\).

By taking \(\rho(R)\equiv\rho\), we obtain the following corollary, which generalizes Theorem B to the conical case.

### Corollary

*Let**K**be a constant*, \(u(P)\) (\(P=(R,\Theta)\)) *be harmonic on*
\(C_{n}(\Omega)\)*and continuous on* ̅. *If*

*and*

*then*

*where*
\(P\in C_{n}(\Omega)\), *M**is a constant independent of**K*, *R*, \(\varphi(\theta)\)*and the function*
\(u(P)\).

### Remark

(see [2])

From corollary, we know that conditions (1.1) and (1.2) may be replaced with weaker conditions

and

respectively.

## 2 Lemma

Throughout this paper, let *M* denote various constants independent of the variables in question, which may be different from line to line.

### Lemma 1

*for any*
\(P=(r,\Theta)\in C_{n}(\Omega)\)*and any*
\(Q=(t,\Phi)\in S_{n}(\Omega)\)*satisfying*
\(0<\frac{t}{r}\leq\frac{4}{5}\);

*for any*
\(P=(r,\Theta)\in C_{n}(\Omega)\)*and any*
\(Q=(t,\Phi)\in S_{n}(\Omega; (\frac{4}{5}r,\frac{5}{4}r))\).

*Let*
\(G_{\Omega,R}(P,Q)\)*be the Green function of*
\(C_{n}(\Omega,(0,R))\). *Then*

*where*
\(P=(r,\Theta)\in C_{n}(\Omega)\)*and*
\(Q=(R,\Phi)\in S_{n}(\Omega;R)\).

## 3 Proof of theorem

Applied Carleman’s formula (see [6–8]) to \(u=u^{+}-u^{-}\) gives

It immediately follows from (1.3) that

and

Notice that

Hence from (3.1), (3.2), (3.3) and (3.4) we have

and

And (3.6) gives

Thus

By the Riesz decomposition theorem (see [7]), for any \(P=(r,\Theta)\in C_{n}(\Omega;(0,R))\), we have

Now we distinguish three cases.

*Case* 1. \(P=(r,\Theta)\in C_{n}(\Omega;(\frac {5}{4},\infty))\) and \(R=\frac{5}{4}r\).

Since \(-u(x)\leq u^{-}(x)\), we obtain

from (3.8), where

Then from (2.1) and (3.7) we have

and

By (2.2), we consider the inequality

where

and

We first have

from (3.7). Next, we shall estimate \(I_{32}(P)\). Take a sufficiently small positive number *k* such that

for any \(P=(r,\Theta)\in\Pi(k)\), where

and divide \(C_{n}(\Omega)\) into two sets \(\Pi(k)\) and \(C_{n}(\Omega)-\Pi(k)\).

If \(P=(r,\Theta)\in C_{n}(\Omega)-\Pi(k)\), then there exists a positive \(k'\) such that \(|P-Q|\geq{k}'r\) for any \(Q\in S_{n}(\Omega)\), and hence

which is similar to the estimate of \(I_{31}(P)\).

We shall consider the case \(P=(r,\Theta)\in\Pi(k)\). Now put

where

Since

we have

where \(i(P)\) is a positive integer satisfying \(2^{i(P)-1}\delta(P)\leq\frac{r}{2}<2^{i(P)}\delta(P)\).

Since \(r\varphi(\Theta)\leq M\delta(P)\) (\(P=(r,\Theta)\in C_{n}(\Omega)\)), similar to the estimate of \(I_{31}(P)\) we obtain

for \(i=0,1,2,\ldots,i(P)\).

So

From (3.12), (3.13), (3.14) and (3.15) we see that

On the other hand, we have from (2.3) and (3.5) that

We thus obtain from (3.10), (3.11), (3.16) and (3.17) that

*Case* 2. \(P=(r,\Theta)\in C_{n}(\Omega;(\frac{4}{5},\frac{5}{4}])\) and \(R=\frac{5}{4}r\).

Equation (3.8) gives that \(-u(P)= I_{1}(P)+I_{5}(P)+I_{4}(P)\), where \(I_{1}(P)\) and \(I_{4}(P)\) are defined in Case 1 and

Similar to the estimate of \(I_{3}(P)\) in Case 1 we have

which together with (3.10) and (3.17) gives (3.18).

*Case* 3. \(P=(r,\Theta)\in C_{n}(\Omega;(0,\frac{4}{5}])\).

It is evident from (1.4) that we have \(-u\leq K\), which also gives (3.18).

From (3.18) we finally have

which is the conclusion of Theorem 1.

## Change history

### 14 April 2020

A Correction to this paper has been published: https://doi.org/10.1186/s13661-020-01376-7

## References

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**40**(1), 1-7 (2014)Azarin, V: Generalization of a theorem of Hayman on subharmonic functions in an

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*n*-dimensional cone. Front. Math. China**8**(4), 891-905 (2013)Qiao, L: Integral representations for harmonic functions of infinite order in a cone. Results Math.

**61**(1), 63-74 (2012)Rashkovskii, A, Ronkin, L: Subharmonic functions of finite order in a cone. I. General theory. Teor. Funkc. Funkc. Anal. Ih Prilozh.

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## Acknowledgements

This paper was written while the corresponding author was at the Department of Mathematics of the University of Ioannina, as a visiting professor.

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### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors read and approved the final manuscript.

The Editors-in-Chief have retracted this article because it significantly overlaps with a previously published article by Pang et al. The article also shows evidence of authorship manipulation. The identity of the corresponding author could not be verified; the University of Ioannina have confirmed that Mohamed Vetro has not been affiliated with their institution. The authors have not responded to correspondence regarding this retraction.

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Li, Z., Vetro, M. RETRACTED ARTICLE: Levin’s type boundary behaviors for functions harmonic and admitting certain lower bounds.
*Bound Value Probl* **2015**, 159 (2015). https://doi.org/10.1186/s13661-015-0421-6

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DOI: https://doi.org/10.1186/s13661-015-0421-6