Recently many important results on rings and quasiconformal mappings in space have been obtained by B. V. Šabat [9], F. W. Gehring [3], J. Väisälä [11] and others. The modulus of a ring in space can be defined in three apparently different but essentially equivalent ways. (See Gehring [4]). In the theory of quasiconformal mappings in space, some properties for moduli of rings in space play an important role, because the method by means of moduli acts also as a substitute in space for the Riemann mapping theorem which does not hold in space.
Recently J.Dufresnoy [3], M.Tsuji [8] and Z. Yύjόbό [9] have proved a generalization of Ahlfors' discs theorem [1] by use of Ahlfors* theory [2] of covering surfaces. On the other hand, A. Pfluger [6] and Y. Juve [4] have obtained an extension of Koebe's distortion theorem to some univalent pseudo-regular functions.From the results mentioned above, we are motivated to write this paper. First, in 1, we define the functions which are called pseudo-analytic (K) and {K} by following S.Kakutani and A. Pfluger. In 3, by the method due to Z. Yύjόbό and by the aid of lemmas which are stated in 2, we establish a theorem for our function which is pseudo-analytic {K} corresponding to the above Dufresnoy-Tsuji-Yύjόbό's theorem for the analytic function. Further, as its application, an extension of Bloch's theorem is proved in 4.
Let w = f(z) be a quasiconformal mapping, in the sense of Pfluger [5]-Ahlfors [1], with maximal dilatation K, which will be simply referred to a K-QC mapping. As is well known, any K-QC mapping w = f(z) of Im z > 0 onto Im w > 0 can be extended to a homeomorphism from Im z ≧ 0 onto Im w ≧ 0 and hence it transforms any set of logarithmic capacity zero on Im z = 0 into a set with the same property on Im w = 0.
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