A Study on Applications of Fractional Calculus to Univalent and Multivalent Functions

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In this project, placing Fractional Calculus and its Applications to the center of the investigation, we studied jointly uniting individual storages of their research efforts. 1) Fractional Calculus : Fractional calculus operators involving Appell's function FィイD23ィエD2 as kernels were defined and analyzed, which are generalizations of that involving Gauss's hypergeometric function proposed by the head investigator in 1978. Further, we studied multidimensionability of fractional calculus operators. Two kinds of multidimensional extensions of fractional calculus operators of Riemann-Liouville and Weyl were already proposed and discussed by us. Here we defined and investigated what involves Gauss's function. 2) H-transform and its application : As an extension of integral transforms involving various special functions) the transform with H-function as an integral kernel, H-transform, was studied. In the theory on the space LィイD2υ,2ィエD2, mapping properties, boundedness, injection and repre … More sentation property of the H-transform were first investigated, and, then, that the properties are extendable on the space LィイD2υ,γィエD2 for any ィイD2γィエD2 is shown. At the moment, precise properties of the H-function should be examined and used. The invertibility of the H-transform on LィイD2υ,γィエD2 was also studied. Such general results on the H-transform may produce mapping theorems of integral transforms involving various concrete special functions. Here, we investigated the Meijer transform, Lommel-Maitland transform, Hardy-Titchmarsh transform, and we obtained distinct results severally. 3) Fractional Integral and Differential Equations : Relating to fractional calculus, we studied the Abel-Volterra type integral equations and investigated asymptotics near zero of the solutions. 4) Application of Fractional Calculus to Theory of Geometric Functions: Distortion theorems, starlikeness, convexity of analytic functions by using the fractional integral operator involving G-function as a kernel defined by V.S. Kiryakova are studied. The results deduce theorems for the Hohlov operator, Riemann-Liouville fractional calculus operator, and fractional calculus operators involving the Gauss function and the Appell function FィイD23ィエD2 due to the present head investigator.For such investigations, the grant-in-aid was effectively used mainly for the trip expensises of the discussions of members of the project and of the presentations of them at meetings in foreign countries. Less