Boundary Harnack inequality

The Boundary Harnack Inequality is a name given (confusingly) to two related results for nonnegative functions $u$ which are solutions of elliptic equations. The first result, also known as Carleson's estimate, says that for non-negative solutions, their values in a neighborhood of the boundary are bounded in terms of the value at some interior point. The second result, also known as the boundary comparison estimate, says that two non-negative solutions which are zero on some portion of the boundary, have a Holder continuous ratio with respect to each other in some neighborhood of the boundary.