H-transformations and their applications to families of univalent and multivalent functions.

在本课题中，我们着重研究了h变换的各种性质及其应用。h变换是由h函数定义的积分变换。saiigo试图将映射、有界性、满性、值域、逆变换等各种性质描述为作用于各种函数空间的映射，从h变换可以理解为超几何函数的推广这一观点出发。他还发展了若干h变换的变体以及与之相关的积分变换。随后，他成功地系统地计算了由若干变量的h函数积定义核的积分变换的分数积分。Owa定义了推广星形函数和凸型函数的一元函数族。从基本星形函数和凸型函数的概念是基于一元函数的解析性的观点出发来定义这些族。他发现了包涵体的各种性质和函数系数的估计。齐藤发现了关于一类解析函数的辐角系数的一个新的不等式。吉田关注的是与无限维空间相关的弗兰克尔引理。福岛对多谐映射的解析延拓问题很感兴趣。

In this project we focused our attention upon the research to find the various properties about the H-transformations and their applications. H-transformations are integral transformations defined by H-functions. Saigo attempted his main efforts to describing the various properties such as mappingness, boundedness, surjectivity, range and inverse transformations as the mapping acting on various function spaces, from the view point of that the H-transformations can be understood as the generalization of the hypergeometric functions. He also developed several variations of H-transformations and integral transformations related with it. Subsequently, he succeeded the systematic calculation of fractional integrals for the integral transformations with the kernels defined by the product of H-functions of several variables.Owa defined the families of univalent functions which generalise the star-shaped functions and convex-type functions. These families are defined from the view point from that the notions of the basic star-shaped functions and convex-type functions are based on the analyticity of the univalent functions. He found the various properties of inclusions and estimates of the coefficients of the functions.Saito found a new inequality about the coefficients for arguments of some family of analytic functions. Yoshida concerned with Frenkel's lemma associated with an infinite dimensional space. Fukushima was interested in analytic continuation problems for multi-harmonic mappings.