  The well known Jensen inequality, holds true for every convex functions.
However, we found that it is possible to apply it to some problems related to
nonconvex functions for which Jensen's inequality holds true locally. Having
considered a set of such functions, we noted some general patterns. We show
that the key point, which provides Jensen's inequality holds true locally, is
that the plot of function should be situated at only one side from the local
base curve defined compatible with conditional variables. Moreover, we have
achieved even more general result. It turned out that the graph of the function
can be located on either sides of the local base curve, with the conditions.
This result allows one to prove easily difficult types of inequalities, and on
the other hand to broaden applications in physics, economy, and information
theory. On the basis of the conducted analysis of different sources it is
possible to claim, that our method is applicable to about three fourths of
studied inequalities related to Jensen's inequality.
