Let G be a domain in the z-plane and a_1, a_2,…, a_n be n distinct points of G. Let G_1, G_2,…, G_n be a system of sub-domains of G, such that a_i ∈ G_i (i=1, 2,…, n) and that G_i G_i=0(i≠j). We denote the mapping radius of G_i with respect to a_i by R(a_i, G_i), Let x_1, x_2,…,x_n be a set of positive numbers and write Our problem is to investigate the properties of extremal domains G_1, G_2,…, G_n for the quantity I_G(a_1,…,a_n; x_1,…x_n). In the case of G being a circle or the whole plane, the problem was solved by Kolbina and others, for n =2. The aim of present note is to suggest a method for attacking the problem in general.The expression P(z)dz~2 is said to be a positive quadratic differential in the domain G, if it is positive or zero on the boundary of G, the function P(z) is to be assumed meromorphic on the closure G of G. For simplicity, we assume that the boundary of G consists only of analytic curves. Our main theorem isTheorem 1. Let g_1, g_2,…,g_n be a sy stem of non-overlapplng domains in G such that a_ν∈ g_ν, ν=1, 2,…, n. The necessary and sufficient condition for the truth of the equation is that the boundary curves of g_1, g_2, …, g_n, are all the trajectories with end points being the zeros of a positive quadratic differential P(z)dz~2 in G. Where P(z)dz~2 satisfies the following conditions:(1) P(z) is regular in G with the exception of the double poles a_1, a_2, …, a_n, whereat we have the Laurent expension(2) P(z) has at least one zero in each closed boundary curve of G, unless n=1 and G is simply connected,(3) the equation holds good for any pair of zeros z', z" of P(z). Under these conditions, one branch of the function is regular and schlicht in g_ν, and maps g_ν onto unit circle, and transforms the point a_ν into ζ=0. For the case that G is the circle |z| < R, we have the following Theorem 2. Let a_1, a_2,…, a_n be n distinct points in the circle |z| < R. Let b_1, b_2,…, b_(2(n-1)) be 2(n-1) points in the closed circle |z| ≤ R. Let the function satisfy the following conditions(1)there is at least one b_ν_o such that |b_ν_o| = R,Then the domains g_1 g_2,…, g_n having the boundary curves form a system of extremal domains for the quantity I_G(a_1, a_2,…, a_n; x_1, x_2,…, x_n), and the function maps g_ν onto |ζ|<1.Similar results can be obtained for the case G being a plane.We conclude this note by proving the following.Theorem 3. The necessary and suffident condition that the system g_1, g_2, …, g_n become extremal domains for the quantity I_G(a_1,…,a_n; x_1,…,x_n) is that the boundaries of these domains are identical with the level curves H(z)=0 of the function H(z), where H(z) is harmonic in G, two-valued, and satisfies the following conditions:(1) the modulus |H(z)| is single-valued and vanishes on the boundary of G,(2) it possesses only a finite number of branch points of order two in the closure G of G, and the pair of equations H(a)= 0 and lim |grad H(z)| = 0 hold at any branch point a,(3) On each dosed boundary curve of G, there exists at least one branch point of H(z),(4) H(z) is harmonic on the closure G of G with the exception of branch points and logrithmic poles a_1, a_2 ,…, a_n , with the residues ν=1, 2, …, n respectively.Under these conditions we haveAs a corollary of this theorem we have the inequality: where H(z, ζ) denotes the regular part of the Green function G(z,ζ) of G at z=ζ, i.e. H(z, ζ) = log |z-ζ1+ G(z, ζ).