用超几何函数定义的解析函数的性质

Properties of Analytic Functions Defined by Hypergeometric Functions

引进单位圆盘内解析函数类 Ｔ（λ，Ａ，Ｂ）（λ≥ ０，－ １≤ Ａ＜ Ｂ ≤１，Ｂ ≥ ０）．给定 Ｆ１（ｚ）＝ｚＦ（ａ，ｂ；ｃ；ｚ），这里Ｆ（ａ，ｂ；ｃ；ｚ）是超几何函数，我们对ａ，ｂ，ｃ，Ａ；Ｂ，λ，μ确定条件，使得函数（１－μ）Ｆ１（ｚ）＋ μｚＦ１’（ｚ）（μ ≥ ０）属于类 Ｔ（λ，Ａ，Ｂ）．

A class T(λ, A, B) (λ ≥ 0, - 1 ≤ A < B ≤l and B ≥ 0) of analytic functions in the unit disk is introduced. Given the hypergeometric function F(a , b; c; z) and F1 (z) = zF(a , b; c; z) , we determine conditions on a, b, c, A, B I λ, to guarantee that (1 - )F1 (z) + zF1' (z) (30) will be in the class T(X, A, B).