**Extremal problem for invariant subspaces in the weighted Bergman spaces A ^{p}_{α}, α>-1, p>0**

We find an explicit formula for the solution of an extremal problem for the invariant subspaces in the weighted Bergman spaces A

^{2}

_{α}, -1<α<∞ consisting of functions having one zero of order at least k, k=0,1,…. A consequence of this formula is the failure of the Beurling-type theorem for α=4. We also consider the analogous extremal problem for A

^{p}

_{α}, 0<p<∞.

The talk is based on joint work with M. T. Nowak, M. Wołoszkiewicz-Cyll.