  For mappings of finite distortion actively investigated last 15--20 years,
problems of a so-called lower order are discussed. It is proved that, mappings
with finite length distortion $f:D\rightarrow {\Bbb R}^n,$ $n\ge 2,$ which have
locally integrable other dilatation in degree $\alpha>n-1,$ and have a finite
asymptotic value are of a uniformly lower bounded lower order.
