Two-side bound of a root of an equation containing complete elliptic integrals

We prove uniqueness of root of an equation arising in a problem of geometric control theory. The problem consists of description of singularity of the sub-Riemannian sphere on the Engel group near abnormal length minimizer.
During the proof, several new inequalities for complete elliptic integrals were obtained. For example, we proved that the function $K(k) E(k)$ is increasing at the segment $[0, 1)$; this fact was not noticed before in literature.
The method of investigation developed and the results obtained can be useful both for the study of elliptic integrals and for solving problems were such integrals arise (e.g. in problems of sub-Riemannian geometry). (In Russian).