Perturbation theory is a practically and conceptually important tool of physics, especially in quantum mechanics. It represents an ultimate method in the quantum chemical description of many-electron systems like atoms or molecules, where the Rayleigh-Schrödinger formulation is most frequently applied. In spite of 100 years of research, the sufficient and necessary conditions for the convergence of this infinite series is yet unknown. Even worse, the perturbation series is quite often divergent in many important applications.

After a short review on perturbation theory, we discuss some possibilities to accelerate convergence as well as some methods of resumming divergent series by means of complex analysis techniques. The latter include analytic continuation and the numerical solution of the inverse boundary problem, where the boundary values are sought for a partial differential equation on the complex plane in the knowledge of the solution of that equation in certain domain.

In this talk we consider two classes of functional inequalities and the related functional equation. Under natural but general circumstances we show that if the solution set of the functional equation forms a 2-dimensional Beckenbach family then the continuous solutions of the functional inequality are exactly those functions which are convex with respect to the Beckenbach family. Several concrete examples and applications will be provided.