關於區域的映照半徑

<正> §1.設α_1,α_2,…,α_n是z平面上n個相異的點,G_1,G_2,…,G_n是z平面上n個不互相重叠的有限區域,α_k屬於G_k,記G_k對於α_k的映照半徑為

THE PRODUCT OF THE MAPPING RADII OF NONOVERLAPPING DOMAINS

Let a_1, a_2, …, a_n be n distinct finite points in the z-plane; G_1, G_2 G_n being a system of non-overlapping domains in the z-plane such that a_i ∈ G_i (i=1, 2…n), we denote the mapping radius of G_i with respect to a_i by R(a_i, G_i). Golousin invesfigated the extremal domains G_1,…, G_n, for which the product becomes maximum. He obtained a system of differential equations for the extremal functions corresponding to the extremal domains. Although Golousin's result has been simplified by the authort, the problem of determination of the extremal domains still leaves open. The first aim of present note is to establish the following:Theorem 1. Let P(z) be any polynomial such that (i) its degree is equal to or less than 2n-3, (ii) and (iii) for the roots z_1, z_2,…, z_m of P(z)=0, (The existence of P(z) is known.) Then the domains with the boundaries defined by the equation are extremal for the product .Further, the corresponding mapping functions are the inverse functions ofCorollary. If a_k=ae~(2πki/n)(k=1, 2,..., n), then In this case the ,boundary of the extremal domains are n rays ρexp{2π(ν+1/2)/n i+iarga} (0≤ρ≤∞, ν=1, 2,…, n) issuing from the origin.The proofs for the above results are based on the method of extremal length due to Ahlfors and Beurling.By the same method, we can investigate the problem proposed by Grotzsch and Lavrentieff. The problem is equivalent, as shown by Golousin to the statement: Given n points a_1,… a_n in the z-plane, let G be any region which contains z=0 but does not the points a_1,…,a_n and ∞. Determine the upper bound of the mapping radius of G with respect to z=0.The extremal domain G=G(0; a_1, a_2,…a_n) is given by the following:Theorem 2.Let (z) be any polynomial such that (i) the degree m is equal to or less than n-1, (ii)and (iii) for the roots a_n+1, a_n+2,…, a_(n+m)of (z)=O, Then the curve defined by the equation gives the boundary of an extremal domain G(0; a_1,…,a_n). The inverse function of the corresponding extremal function is As a corollary of this theorem, we have R(0,G)≤4~1/n)|a| for a_k=ae~((2πk/n)i), k=1, 2,…, n.