The laplacian comparison theorem says that if a $n$-dimensional Riemannian manifold has nonnegative Ricci curvature, then the distance function to any point satisfies $\Delta d\leq\frac{n-1}{d}$. There is also a hessian comparison theorem, and versions for different curvature bounds.

These are covered in several books on differential geometry, but I haven't been able to find any information on their history. Where were they first established?