  Motivated by prominent problems like the Hilali conjecture Yamaguchi--Yokura
recently proposed certain estimates on the relations of the dimensions of
rational homotopy and rational cohomology groups of fibre, base and total
spaces in a fibration of rationally elliptic spaces.
  In this article we prove these estimates in the category of formal elliptic
spaces and, in general, whenever the total space in addition has positive Euler
characteristic or has the rational homotopy type of a homogeneous manifold
(respectively of a known example) of positive sectional curvature.
Additionally, we provide general estimates approximating the conjectured ones.
  Moreover, we suggest to study families of rationally elliptic spaces under
certain asymptotics, and we discuss the conjectured estimates from this
perspective for two-stage spaces.
