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

分数阶微积分在单价和多价函数中的应用研究

基本信息

批准号：
09640238
负责人：
SAIGO Megumi
金额：
$ 1.47万
依托单位：
FUKUOKA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640238/
关键词：
fractional calculus univalent and multivarlent functions Gauss hypergeometric function H-function H-transform Appell function integral transforms Abel-Volterra integral equation 単葉関数 p-葉関数星型関数凸型関数 Gauss超幾何関数多葉関数 Appell超幾何関数

项目摘要

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

在这个项目中，我们将分数阶微积分及其应用作为调查的中心，共同研究他们的研究成果的个人存储。1)分数阶微积分：定义并分析了以Appell函数F \ \ D23 \ \ D2为核的分数阶微积分算子，这是1978年首席研究员提出的涉及高斯超几何函数的分数阶微积分算子的推广。进一步研究了分数阶微积分算子的多维性。我们已经提出并讨论了Riemann-Liouville和Weyl分数阶微积分算子的两种多维扩展。这里我们定义并研究了涉及高斯函数的内容。2) h变换及其应用：作为涉及各种特殊函数的积分变换的扩展，研究了以h函数为积分核的变换h变换。本文首先研究了h变换的映射性质、有界性、注入性和表示性，然后证明了这些性质在空间上是可扩展的，适用于任何空间。此时，应该检查和使用h函数的精确性质。同时研究了L′′γ′′D2上h变换的可逆性。这种关于h变换的一般结果可以产生涉及各种具体特殊函数的积分变换的映射定理。本文研究了Meijer变换、Lommel-Maitland变换、Hardy-Titchmarsh变换，分别得到了不同的结果。3)分数阶积分与微分方程：与分数阶微积分相关，研究了Abel-Volterra型积分方程的解的近零渐近性。4)分数阶微积分在几何函数理论中的应用：利用V.S. Kiryakova定义的以g函数为核的分数阶积分算子，研究了解析函数的畸变定理、星形和凸性。结果推导出了Hohlov算子、Riemann-Liouville分数阶微积分算子以及涉及高斯函数和Appell函数的分数阶微积分算子的定理。对于这种调查，赠款主要有效地用于项目成员讨论和在外国会议上作报告的旅费。少