Estimates for convex integral means of harmonic functions (Q2921038)

Let \(f\) be an integrable function on the unit sphere \(S\) in \(\mathbb{R}^{n}\), let \(g\) be its cap-symmetric decreasing rearrangement, and let \(u,v\) be the Poisson integrals of \(f,g\), respectively, in the unit ball \(B\). Also, let \(M(h,r)\) denote the mean value of a function \(h\) over the sphere \(\{|x|=r\}\). The author shows that, if \(\Phi :\mathbb{R}\rightarrow \mathbb{R}\) is convex, then \(M(\Phi \circ u,\cdot )\leq M(\Phi \circ v,\cdot )\) on \((0,1)\), and discusses when equality occurs. This generalizes earlier results in the plane due to \textit{R. M. Gabriel} [Proceedings L. M. S. (2) 34, 305--313 (1932; JFM 58.0511.02 and Zbl 0005.29402)] and \textit{A. Baernstein} [Indiana Univ. Math. J. 27, 833--852 (1978; Zbl 0372.42007)]. Next, let \(u\) be a harmonic function on \(B\) satisfying \(|u|<1\) and \(u(0)=0\), and let \(U\) be the Poisson integral of the function \((x_{1},\dots,x_{n})\mapsto \mathrm{sgn}(x_{1})\) on \(S\). The second main theorem says that, if \(\Phi :(-1,1)\rightarrow \mathbb{R}\) is convex, then \(M(\Phi \circ u,\cdot )\leq M(\Phi \circ U,\cdot )\) on \((0,1)\), and again discusses the case of equality. This result implies some known inequalities.