On mixed problems for extremal decompositions

The author has solved the problem of the extremal decomposition of a finite bordered Riemann surface into doubly-connected domains of modules M-j, j = 1,..., L, maximizing Sigma (L)(j=1) a(j)(2)M(j) with a(j), j = 1,...,L, no n-negative numbers not all zero. Renelt considered a different type of extr emal decomposition, minimizing Sigma (L)(j=1) b(j)(2)M(j)(-1) with b(j), j = 1,...,L, positive numbers. The present paper shows that both problems are special cases of a more general problem.