单叶调和映照的反函数

The Inverse Functions of Univalent Harmonic Mappings

设是在一个单连通区域上的单叶调和映照，我们证明了反函数ｚ＝ｆ－１（）也是调和映照的充要条件是ｆ为下面三类函数之一：（ｉ）单叶共形映照；（ｉｉ）仿射交换映照；（ｉｉｉ）具有形式ｆ（ｚ）＝Ａ［ａｚ＋β＋ｌｏｇ（１－ｅ－ａｚ－β）－ｌｏｇ（１－ｅ－ａｚ－β）］＋Ｂ的调和映照，其中Ａ，Ｂ，α和β是常数且满足条件Ｒ（ａｚ＋β）＞０，Ｚ∈Ｄ．

Let f(z) = h(z) + g(z) be harmonic univalent in a simply connected domain D C. It is proved that the inverse mapping Z = f-1(W) is also harmonic if and only if f is any one of the following three kinds of fUnctions: (i) one-to-one conformal mapping; (ii) affine transformation; (iii) function of the form where A, B, α and βare constants with the later two subject to the condition R(az+β) > 0, z ∈ D.