An inequality for sums of subharmonic and superharmonic functions

Abstract. We prove a general inequality for the distributional Laplacian of a sum of a subharmonic and a superharmonic function postcomposed with a convex function of linear growth. We use this inequality to show that convex functions of linear growth operate by means of postcomposition on the class of sums of subharmonic and superharmonic functions. Let u be a sum of a subharmonic and a superharmonic function in an open set Ω in R n, n ≥ 2, and let f: R → R be a convex function of linear growth. In this paper we consider the problem of finding lower bounds for the distributional Laplacian ∆(f ◦ u) of the composite function f ◦ u in terms of the Laplacian ∆u.