Convexity of means and growth of subharmonic functions in strips (Q1083589)

Let \(\Omega =\{(x,y):\) \(x\in {\mathbb{R}}^ n\), \(y\in (-\pi /2,\pi /2)\}\) where \(n\geq 1\), let u be subharmonic in \(\Omega\), and let L(u;r) denote a certain weighted integral of u over the surface \(\{\) (x,y)\(\in \Omega:| x| =r\}\). This paper presents convexity and growth properties for L(u;r) which are analogous to earlier results of the author for means of subharmonic functions in an n-dimensional cone [Ark. Mat. 21, 29-43 (1983; Zbl 0522.31005)]. For example, Theorem 1: If \(0<\lambda <1\) and \[ u(x,\pm \pi /2)\leq \cos (\pi \lambda /2)u(x,0)\quad (x\in {\mathbb{R}}^ n), \] then L(u;r) is convex with respect to the family \(A\Phi (\lambda r)+B\Psi (\lambda r)\), where \(\Phi\), \(\Psi\) are constructed from Bessel functions of order (n-2)/2. Further, if u is non- negative, the same convexity property is shared by \(\{L(u^{\alpha};r)\}^{1/\alpha}\) for any \(\alpha >1\). A brief mention is also made of corresponding results for the infinite cylinder \(\{\) (x,y): \(| x| <1\), \(y\in {\mathbb{R}}\}.\) Remark: The reviewer [Ark. Mat. 24, 175-190 (1986)] has recently introduced a mean for subharmonic functions in strips which unifies the above L(u;r) with means studied by a number of other authors.