In this paper a class of quasiconformal mappingsf=f(z) which are conformal in a neighbourhood of infinity is considered. Besides, their dilatationpis restricted in the mean as in [2], [7], [10] and [11]. For this class an area theorem will be derived which implies the area theorem in [5] and, on the other hand, improves an inequality arising from the solution of an extremal problem for the coefficientαin the series expansionf(z) =z+α1/z+α2/z2+… at infinity. Finally, a generalization connected with other extremal problems is investigated.