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

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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.