Convex functions and subharmonic functions (Q1917795)

Let \(\Omega\) be a domain in \(\mathbb{R}^n\), \(n\geq 2\); \(\varphi: (a,b)\to \mathbb{R}\) convex, \(h:\Omega\to (a,b)\) harmonic, \(u:\Omega\to [a,b)\) subharmonic. Then we know \(\varphi\circ h\) is subharmonic in \(\Omega\); so is \(\varphi\circ u\) if \(\varphi\) is increasing also. Here these results are generalized to the case of several variables; for this, the author first proves a form of the 2-dimensional Jensen's inequality. It is also pointed out how these results can be obtained when \(\Omega\) is a harmonic space in the context of the axiomatic potential theory.