Mathematical analysis is concerned with the study of infinite processes, and the differential calculus of Newton and Leibniz lies at its heart. It provided the foundation and the language for Newtonian mechanics and the whole of mathematical physics. Over the past three centuries it has permeated much of mathematics and science.
Associated with the limiting process there are many technically difficult “estimates” or inequalities, of a combinatorial or algebraic nature, which prepare the ground and justify passing to the limit. Such estimates are often extremely hard since they address some subtle and important aspect of the problem at hand. Establishing this becomes a key step, opening the door to a wide variety of applications.