Levin's type boundary behaviors for functions harmonic and admitting certain lower bounds (Q283589)

For a domain \(\Omega\subset {\mathbb S}^{n-1}\), \(n\geq 2\), denote \(C_n(\Omega)={\mathbb R}_+\times \Omega\). Also, denote by \((\lambda,\varphi)\) the normalized first pair of eigenvalue/eigenfunction of the spherical part of the operator corresponding to the Laplace operator. The main result of the paper establishes the following: If \(u\) is a harmonic function on \(C_n(\Omega)\) and continuous on \(\overline{C_n(\Omega)}\) such that (i) \(u(P)\leq KR^{\rho(P)}\) for \(P=(R,\theta)\in C_n(\Omega\times[1,\infty))\), where \(K\) is a positive constant, \(\rho\) is an increasing function on \([1,\infty)\) and \(\rho>\frac{-n+2+\sqrt{(n-2)^2+4\lambda}}{2}\); (ii) \(u(P)\geq -K\) for \(P=(R,\theta)\in \overline{C_n(\Omega)}\), \(R\leq 1\); then \[ u(P)\leq -KM(1+\rho(R) R^{\rho(R)})\varphi^{1-n}(\theta)\quad\text{ in }C_n(\Omega), \] where \(M>0\) is a constant independent on \(K, R, \varphi(\theta)\). Editorial remark: According to the retraction notice, ``the Editors-in-Chief have retracted this article because it significantly overlaps with a previously published article by [\textit{S. Pang} and \textit{B. Ychussie}, J. Inequal. Appl. 2015, Paper No. 108, 9 p. (2015; Zbl 1311.31003)]. 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.''