  Gronwall conjecture states that a planar 3-web which admits more than one
distinct linearization is locally equivalent to an algebraic web. We give a
partial answer to the conjecture in the affirmative for the class of planar
3-webs with the web curvature that vanishes to order three at a point. The
differential relation on the third order jet of web curvature provides an
explicit criterion for unique linearization.
