In the literature, there is an inequality called the Alexandrov-Bol inequality which is frequently used in partial differential equations. Here we just recall its statement without any proof.
Theorem. Let be a good domain in . Assume be a positive function satisfying the elliptic inequality
in . Then it holds
An analytic proof was given by C. Bandle aroud 1975 when she assumed to be real analytic. The above version was due to Suzuki in an elegant paper published in the Ann. Inst. H. Poincare in 1992 [here]. The proof is mainly depended on the isoperimetric inequality for the flat Riemannian surfaces. We refer the reader to the paper by Suzuki for the proof.