Let Gbe an open set in the complex plane   not containing a fixed point a. Let Q be some exceptional set on   containing the point a. Let u be a function subharmonic in G, and λ: (0, + ∞) → [- ∞, + ∞) be a function which, in some generalized sense, is concave in respect to logarithm of the independent variable x. Under certain natural conditions on G aQu, λ there is proved that if   then   in G. This is a local contour-and-solid problem for subharmonic functions. Earlier the author obtained some results in this problem, see [5], [6]. In this paper there are established new results.