Certain Subclass of Harmonic Univalent Functions Associated With Fractional Calculus Operator

In this paper, a new subclass of complex-valued harmonic univalent functions f(z)=h(z)+g(z)¯¯¯¯¯¯¯¯¯ in the open disk U={z:z∈C and |z|<1} defined by using a linear operator. We investigate coefficient condition, extreme points, distortion and convex combination for this subclass.

In this paper, using Differential operator a certain subclass of harmonic univalent functions in the unit disc U = {z ∈ C : |z| < 1} is investigated. Some properties such as coefficient bounds, convex combination, extreme points and convolution conditions of this class are obtained. The newly introduced subclass of harmonic univalent functions is more general than the subclasses earlier found in the literature.

Let SH be the class of functions f = h+g that are harmonic univalent and sense-preserving in the open unit disk U = { z : |z| < 1} for which f (0) = f'(0)-1=0. In this paper, we introduce and study a subclass H( α, β) of the class SH and the subclass NH( α, β) with negative coefficients. We obtain basic results involving sufficient coefficient conditions for a function in the subclass H( α, β) and we show that these conditions are also necessary for negative coefficients, distortion bounds, extreme points, convolution and convex combinations. In this paper an attempt has also been made to discuss some results that uncover some of the connections of hypergeometric functions with a subclass of harmonic univalent functions.

With the development of factional calculus theory and applications in different fields in recent years, the rational approximation problem of fractional calculus operator has become a hot spot of research. In the early 1950s and 1960s, Carlson and Halijak proposed regular Newton iterating method to implement rational approximation of the one-nth calculus operator. Carlson regular Newton iterating method has a great sense of innovation for the rational approximation of fractional calculus operator, however, it has been used only for certain calculus operators. The aim of this paper is to achieve rational approximation of arbitrary order fractional calculus operator. The realization is achieved via the generalization of Carlson regular Newton iterating method. To construct a rational function sequence which is convergent to irrational fractional calculus operator function, the rational approximation problem of fractional calculus operator is transformed into the algebra iterating solution of arithmetic root of binomial equation. To speed up the convergence, the pre-distortion function is introduced. And the Newton iterating formula is used to solve arithmetic root. Then the approximated rational impedance function of arbitrary order fractional calculus operator is obtained. For nine different operational orders with n changing from 2 to 5, the impedance functions are calculated respectively through choosing eight different initial impedances for a certain operational order. Considering fractional order operation characteristics of the impedance function and the physical realization of network synthesis, the impedance function should satisfy these basic properties simultaneously: computational rationality, positive reality principle and operational validity. In other words, there exists only rational computation of operational variable s in the expression of impedance function. All the zeros and poles of impedance function are located on the negative real axis of s complex plane or the left-half plane of s complex plane in conjugate pairs. The frequency-domain characteristics of impedance function approximate to those of ideal fractional calculus operator over a certain frequency range. Given suitable initial impedance and for an arbitrary operational order, it is proved that the impedance function could meet all properties above by studying the zero-pole distribution and analyzing frequency-domain characteristics of the impedance function. Therefore, the impedance function could take on operational performance of the ideal fractional calculus operator and achieve the physical realization. It is of great effectiveness in the generalization of this kind of method in both theory and experiment. The results educed in this paper are the basis for further theoretic research and engineering application in constructing the arbitrary order fractional circuits and systems.

The purpose of the present paper is to introduce a new subclass of harmonic univalent functions associated with a $q$-Ruscheweyh derivative operator. A necessary and sufficient convolution condition for the functions to be in this class is obtained. Using this necessary and sufficient coefficient condition, results based on the extreme points, convexity and compactness for this class are also obtained.

We investigate some properties of a new subclass of harmonic univalent functions defined by multiplier transformation. We obtain coefficient conditions, extreme points, distortion bounds, convolution, and convex combination for the above subclass of harmonic functions.

In recent years, various operators of fractional calculus (that is, calculus of integrals and derivatives of arbitrary real or complex orders) have been investigated and applied in many remarkably diverse fields. The main object of this paper is to consider some miscellaneous properties and applications which are associated with several fractional differintegral operators. We first investigate, in a systematic and unified manner, various families of series identities which emerged in connection with some of these fractional differintegral formulas. By using such operators of fractional calculus, a number of integral formulas as well as fractional differintegral formulas involving inverse hyperbolic functions are also evaluate.

We define and investigate a special class of Salagean-type harmonic univalent functions in the open unit disk. We obtain coefficient conditions, extreme points, distortion bounds, convex combinations for the above class of harmonic univalent functions. 2000 Mathematics Subject Classification. Primary 30C45 ; Secondary 30C50

Some operators of fractional calculus and their applications involving a novel class of analytic functions

For , let denote the class of sense preserving harmonic mappings in the unit disk satisfying . The main aim of this paper is to study some basic properties such as coefficient bounds, growth estimates and convolution for functions in the class . We end the paper with an application, and construct harmonic univalent polynomials belonging to .

A certain subclass of analytic functions associated with operators of fractional calculus

The objective of this paper is to explore and establish certain generalized fractional differentiation and integration involving multivariate generalized Mittag-Leffler function introduced by Salim and Faraz. The generalization of Saigo fractional calculus operators that is Marichev-Saigo-Maeda Fractional Calculus Operator is utilized to establish some formulas related to generalized multivariate M-L function. In conclusion, we depict some key characteristics of generalized fractional integrals and differentiation pertaining to Generalized Multivariate Mittag- Leffler Function.

In this paper we introduce a new subclass of multivalent analytic functions defined by fractional calculus operator. Such results as subordination and superordination properties, convolution properties, inequality properties and other interesting properties of this subclass are proved.

This project aimed to gather together the latest developments in research concerning complex-valued functions from the perspective of geometric function theory. Scholars contributions weresought on topics including, but not limited to: new classes of univalent and bi-univalent functions; studies regarding coefficient estimates including the FeketeSzego functional, Hankel determinants, and Toeplitz matrices; applications of different types of operators in geometric function theory including differential, integral, fractional, or quantum calculus operators; differential subordination and superordination theories in their classical form and also concerning their recent extensionsstrong and fuzzy differential subordination and superordination theories; applications of different hypergeometric functions and orthogonal polynomials in geometric function theory. The presentation of new results obtained by using any other techniques which can be applied in the field of complex analysis were also welcomed. Hopefully, through this project, new lines of research associatedwithgeometric function theory have been highlighted and willserve toboost development inthis field.

The objective of this paper is to present several results in theory of analytic functions (which are univalent, starlike- or prestarlike- and convex) by associating certain fractional calculus operators. A linear operator is fruitfully employed in our investigations, and applications to Wright's generalized hypergeometric functions are also discussed.