Megumi Saigo

H-Transforms : Theory and Applications

Definition, Representations and Expansions of the H-Function. Properties of the H-Function H-Transform on the Space Ln,2. H-Transform on the Space L n,t Modified H-Transforms on the Space L n,t. G-Transform and Modified G-Transforms on the Space L n,t. Hypergeometric Type Integral Transforms on the Space L n,t. Bessel Type Integral Transforms on the Space L n,t. Bibliography Subject Index Author Index Symbol Index.

Some inclusion properties of a certain family of integral operators

Abstract The authors introduce several new subclasses of analytic functions, which are defined by means of a general integral operator I λ,μ , and investigate various inclusion properties of these subclasses. Many interesting applications involving these and other families of integral operators are also considered.

GENERALIZED MITTAG-LEER FUNCTION AND GENERALIZED FRACTIONAL CALCULUS OPERATORS

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mittag-Leffler function E γ ρ,μ(z) in the kernels. Similar assertions are presented for the integral operators containing the Mittag-Leffler and Kummer functions, E ρ,μ(z) and Φ(γ, μ; z), in the kernels, and applications are given to obtain solutions in closed form of the integral equations of the first kind.

A class of distortion theorems involving certain operators of fractional calculus

Abstract The object of the present paper is to investigate a general class of fractional integral operators involving the Gauss hypergeometric function. Several interesting distortion theorems for various subclasses of analytic and univalent functions are proved in terms of these operators of fractional calculus. Some special cases of the results presented here are also indicated.

On mittag-leffler type function, fractional calculas operators and solutions of integral equations

The special entire function of the form with is introduced, where α>0, m>0 and α(im+1)+1≠ 0,−1, −2,.....for i=0,1,2,....For m = 1, Eα1,l(z) coincides with the Mittag-Leffler function Eα,α+1 ,with exactness to the constant multiplier γ(αl+1)The connections of Eα,m,l(z) with the Riemann-Liouville fractional integrals and derivatives are investigated and their applications to solving the linear Abel-Volterra integral equations are given.

Multiplication of fractional calculus operators and boundary value problems involving the Euler-Darboux equation☆

Abstract With a view to presenting solutions of various boundary value problems involving the celebrated Euler-Darboux equation, the authors consider the multiplication of certain classes of operators of fractional calculus defined in terms of the Gaussian hypergeometric function. The fractional calculus operators studied here incorporate, as their special cases, both the Riemann-Liouville and Erdelyi-Kober operators, and are appropriately restricted in order to yield explicit solutions of some of the aforementioned boundary value problems in terms of Appell functions and Kampe de Feriet functions of two variables.

On Mittag-Leffler type function and applications

This is the continuation of the paper [4] which was devoted to introduce a special entire function named a Mittag -Leffler type function E α,m,l (z) to discuss its connections with the Riemnn Loiuville fractional integral and derivatives and to solve the linear Abel-Volterra integral equation. Here we construct explicit soliutions of special differential equations of fractional order and as their consequences, of ordinary differential equations. Examples are also exhibited.

Some characterization theorems for starlike and convex functions involving a certain fractional integral operator

A rotatable vacuum spindle supports a substrate thereon. Means coat the substrate on the non-supported face. Means rotate the spindle and substrate thereon at a speed whereby to distribute the coating material. Means disposed about the spindle direct an annular fluid stream outwardly and against the supported face of the substrate whereby to prevent creep of the coating material onto said supported face.

Integral transforms with fox's H-function in spaces of summable functions

Integral transforms involving Foxs H-functions as kernels are studied on the space Λν,2 of functions f such that Mapping properties such as the boundedness, the repesentation and the range of these H-transforms are given.

Geometric Properties of Convolution Operators Defined by Gaussian Hypergeometric Functions

Let {\cal A} be the class of normalized analytic functions in the unit disk {\cal U} and define the class {\cal P}_{\gamma}(\beta) = \left\{f \in {\cal A} \vert \exists \varphi \in {\bf R} \vert \hbox{Re} \bigg\{ e^{i\varphi} \left((1-\gamma) \,{\,f(z) \over z} + \gamma f^{\prime}(z) - \beta}\right)\bigg\} \gt 0, \; z \in {\cal U} \bigg \}. For a function f \in {\cal P}_{\gamma}(\beta) and the Gaussian hypergeometric function _{2}F_{1} (a,b;c;z) , we investigate the convexity and starlikeness for the convolution operator H_{a,b,c} defined by H_{a,b,c} (f)(z) = z \; {_{2} F_{1}} (a,b;c;z) \ast f(z).