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The investigation of operators is significant in mathematics, particularly in geometric function theory. Operators have generated a considerable amount of attention in the theory of geometric functions because of their significance in generalising and preserving subclasses of analytic univalent functions. Many mathematicians have discussed operators and the new generalisation in various articles. Research on operators has been conducted since 1915. Since then, mathematicians are still very interested in discussing operators with many properties of analytic univalent functions. Hence, this paper introduces a new generalised operator D λ , α s , m , k f (z) of analytic functions in the open unit disc using the generalised Srivastava-Attiya operator, generalised Al-Oboudi operator, and binomial series. Based on this new operator, some subclasses are defined using differential subordinations method. Furthermore, the inclusion properties are obtained for functions in these subclasses. [ABSTRACT FROM AUTHOR]

The inequality of finding the upper bounds for the nonlinear functional |a3 - µa22| of the Taylor-Mclaurin series is popularly famous as the Fekete-Szegö inequality. This inequality has a rich history in the geometric function theory. Its source by Fekete-Szegö in 1933. In this paper, through the advantage of applying concepts quasi-subordinate, Sakaguchi functions and exponential functions, the authors construct a new subclass Ne,q(s, b, λ) of Non-Bazilević functions. As well as, obtained the first and two Taylor-Maclaurin coefficient estimates, sharp upper bounds of the Fekete-Szegö inequality and majorization for functions which belong to this subclass. [ABSTRACT FROM AUTHOR]

Let B(α) denote the class of functions f(z) which are analytic with an open unit disk U = {z : z ε < 1} and satisfy Re{αz²f″(z)g/(z)+zf′(z)/g(z)}>0 for some 0 ≤ α < 1, where g(z) is a normalized starlike function. This paper focused on attaining the sharp bound of the third Hankel determinant, H3(1) by the help of some related lemmas, application of triangle inequalities and maximization of the function. The results obtained leads to development of Hankel determinant and more generally to the geometric function theory. [ABSTRACT FROM AUTHOR]

The study of 300 years old history of quantum calculus or q−calculus or q−disease, since the Bernoulli and Euler, is often considered to be one of the most difficult subjects to engage in mathematics. Nowadays there is a rapid growth of activities in the area of q−calculus due to its applications in various fields such as mathematics, mechanics, and physics. The history of study of q−calculus may be illustrated by its wide variety of applications in quantum mechanics, analytic number theory, theta functions, hypergeometric functions, theory of finite differences, gamma function theory, Bernoulli and Euler polynomials, Mock theta functions, combinatorics, umbral calculus, multiple hypergeometric functions, Sobolev spaces, operator theory, and more recently in the theory of analytic and harmonic univalent functions. In q−calculus, we are generally interested in q−analogues that arise naturally, rather than in arbitrarily contriving q−analogues of known results. While focusing on excitement and romance with development of q−calculus and its applications in certain fields of mathematical sciences and physics, we will also look at q−analogues of some of the recent results in geometric function theory and, in particular, theory of analytic and harmonic univalent functions in the unit disc. [ABSTRACT FROM AUTHOR]

Recently, the authors have shown that a certain combinatorial identity in terms of generators of quaternions is related to a particular sequence of rational numbers (Vietoris' number sequence). This sequence appeared for the first time in a theorem by Vietoris (1958) and plays an important role in harmonic analysis and in the theory of stable holomorphic functions in the unit disc. We present a generalization of that combinatorial identity involving an arbitrary number of generators of a Clifford algebra. The result reveals new insights in combinatorial phenomena in the context of hypercomplex function theory. [ABSTRACT FROM AUTHOR]

Let S*LC represent a new subclass of analytic functions f defined in the open unit disk in the complex plane such that zf'(z)f(z) lies in the interior of the domain bounded by the limacon given by the equation (4x2+4y2-8x-5)2+8(4x2+4y2-12x-3)=0. In this paper, we investigate the geometric properties of functions in this class. [ABSTRACT FROM AUTHOR]

Let R(α,δ) denote the subclass of normalized analytic functions in the open unit disk D={z∈ℝ:|z|<1} satisfying the condition Reeiδf'(z)>α for some 0≤α<1 and cos(δ)-α>0. For f∈R(α,δ), the main aim of this paper is to obtain the upper bounds for the third Hankel determinant, |H3(1)|≤|a3|‖a2a4-a23|+|a4|‖a2a3-a4|+|a5|‖a3-a22|. [ABSTRACT FROM AUTHOR]

The theory of functional operator has been employed in several areas of mathematics, especially in the geometric function theory in the open unit disk. By utilizing the Moment-Generating function, we define a new fractional linear operator in the open unit disk. We discuss some geometric properties of this operator on some subclasses by applying the concept Jack's lemma. [ABSTRACT FROM AUTHOR]

This paper establishes approximate solution for non-linear iterative fractional differential equations: dγ v(s)/dsγ = X (s,v,v(v)), where γ ϵ (0; 1]; s γ I := [0; 1]. Our method is based on some convergence tools for analytic solution in a connected region. We show that the suggested solution is unique and convergent by some well known geometric functions. [ABSTRACT FROM AUTHOR]

In this paper, firstly a new integral equality is proved and based on this equality some new integral inequalities of Hadamard type are given for differentiable Geometric convex functions. [ABSTRACT FROM AUTHOR]

In this paper we get some results in the regular quaternion function theory by means of a radial intergration method. The general solution of Stokes equations for a viscous incompressible fluid is obtained, expressed in terms of two regular quaternion functions. It is shown that this quaternion representation in the case of two dimensional flows goes into the general solution expressed in terms of two functions of a complex variable. An application of the quaternion representation for a boundary value problem in a ball is shown. [ABSTRACT FROM AUTHOR]

Geometric function theory has a wide scope and for each problems pointed out in this theory involve function representative and translated into geometric aspect. The theory inclines towards the concept of univalency and analyticity. Riemann Mapping Theorem plays an important role in combining both concepts. This combination inteprets the form of a domain where the functions being defined. Here the basic idea of geometric function theory and the development of the theory are discussed briefly. [ABSTRACT FROM AUTHOR]

We introduce a class of slice regular functions of several Clifford variables. Our approach to the definition of slice functions is based on the concept of stem functions of several variables and on the introduction on real Clifford algebras of a family of commuting complex structures. The class of slice regular functions include, in particular, the family of (ordered) polynomials in several Clifford variables. We prove some basic properties of slice and slice regular functions and give examples to illustrate this function theory. In particular, we give integral representation formulas for slice regular functions and a Hartogs type extension result. [ABSTRACT FROM AUTHOR]

In the present paper, the outher obtaoin the sharp Fekete-Szego Inequalities for new class defiend on the open unit disk. The outher also obtain some properties of this class which defined by linear operators. [ABSTRACT FROM AUTHOR]

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this contribution, we establish a Cauchy-Kovalevskaya extension theorem for discrete monogenic functions defined on the grid Zhm of m-tuples of integer multiples of a variable mesh width h. Convergence to the continuous case is investigated. As illustrative examples we explicitly construct the Cauchy-Kovalevskaya extensions of the discrete delta function and of a discretized exponential. [ABSTRACT FROM AUTHOR]

The Hartog’s type phenomena in several complex variables are best understood in term of the Dolbeault sequence. A lot of attention was paid in the last decades to its analogue in the function theory of several Clifford variables. The first operator in this resolution is the Dirac operator in several variables. The complete description is known in the stable case in dimension 4. In the nonstable case, other tools like the Penrose transform must be used, which gives an algebraic description of the resolution. The next step is an explicite analytic construction of the operators in the resolution, which is to some extent shown here. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the quantum Gross Laplacian denoted ΔQG is a rotation-invariant operator. For this purpose, we use the Schwartz-Grothendieck kernel theorem and the characterization theorem of rotation-invariant distributions and operators. [ABSTRACT FROM AUTHOR]

In this paper we consider generic primal-dual methods for solving semidefinite programming problems based on a new class of kernel function defined on the positive definite cone. By using some appealing and mild conditions of the new class, we prove by simple analysis that the new class based large update primal-dual interior point methods enjoy an [formula] iteration bound to solve semidefinite programming problems with special choice of parameters. [ABSTRACT FROM AUTHOR]

Using a ladder-rainbow kernel previously established for light quark hadron physics, we explore the extension to masses and electroweak decay constants of ground state pseudoscalar and vector quarkonia and heavy-light mesons in the c- and b-quark regions. We make a systematic study of the effectiveness of a constituent mass concept as a replacement for a heavy quark dressed propagator for such states. The difference between vector and axial vector current correlators is explored within the same model to provide an estimate of the four quark chiral condensate and the leading distance scale for the onset of non-perturbative phenomena in QCD. [ABSTRACT FROM AUTHOR]

In this paper we present applications of a special class of homogeneous monogenic polynomials constructed, in the framework of hypercomplex function theory, in order to be an Appell set of polynomials. In particular, we derive important properties of an associated exponential function from R3 to R3 and propose a generalization to Rn+1. [ABSTRACT FROM AUTHOR]

The Dolbeault sequence is a fundamental tool for many problems in the function theory of several complex variables. A lot of attention was paid in the last decades to its analogue in the function theory of several Clifford variables. The first operator in this resolution is the Dirac operator in several variables. The complete description is known in dimension 4 (i.e., in the case of quaternionic variables, see [1, 6, 4]). Much less is known in higher dimensions. The case of three variables was described completely (see [18]). The full description of the complex for all dimensions is not known at present. Even the case of the stable range (i.e., when the number of variables is less or equal to the half of dimension) is still not fully understood. There are two different approaches to the stable range case, one based on classical algebraic geometry (the Hilbert syzygy theory, see [8]), the other one on representation theory (differential invariants in certain parabolic geometries, see [14, 20]). Differential operators in these resolutions are acting on vector-valued functions. Such spaces of functions are quite complicated in general and the first problem in the description of the resolution is to understand their dimensions. Both the approaches mentioned above suggest an answer to this question, although such answers look quite different. The aim of the paper is to compare these two results and to show that they lead to complicated combinatorial identities. [ABSTRACT FROM AUTHOR]

The initial analyses of the next-to-leading logarithmic corrections to the BFKL kernel were very discouraging. Encouraged by the success of new methods in the analysis of the BFKL equation at full NLL accuracy we demonstrate in this talk how some of the initial conclusions were based on a breakdown of the tools used in the analysis rather than the framework itself. © 2005 American Institute of Physics [ABSTRACT FROM AUTHOR]

In this paper we report multifrequency single pulse polarization observations of the PSR B0329+54 normal mode from the Giant Meterwave Radio Telescope (GMRT) at 325 and 610 MHz and the Effelsberg Observatory at 2695 MHz. Towards the central part of the polarization position-angle traverse there is an unusual “arc”-like structure, which is comprised of a broadband “kink” and a frequency-dependent “spot.” The features are intimately connected with the intensity dependence of the core component: the stronger emission arrives earlier and its linear polarization is displaced farther along the “kink”. Moreover, at high intensities, the circular polarization is LH/RH antisymmetric; the nearly complete RH circular is characteristic of the weaker, later core subpulses. We find that the “kink” emission is associated with the extraordinary (X) propagation mode, and hence propagation effects do not appear capable of producing the core component's broadband, intensity-dependent emission. Rather, the overall evidence points to a largely geometric interpretation in which the “kink” provides a rare glimpse of the accelerating cascade or height-dependent amplifier responsible for the core radiation. [ABSTRACT FROM AUTHOR]

The angular and radial Mathieu functions, or elliptic-cylinder functions, are solutions of the two ordinary differential equations of the second order with variable coefficients and were originally proposed by È. Mathieu in 1868 for finding the modes in en elliptic membrane. Since then, they have found numerous and important applications to the problems in physics and engineering science thus generating a vast research literature about them. It's appeared recently that there exists a theory of functions with quaternionic values and of two real variables which is determined by a Cauchy-Riemann-type operator with quaternionic, variable coefficients and which is intimately related to the Mathieu equations. In the work, it is explained all the above and some basic facts of the arising quaternionic function theory. [ABSTRACT FROM AUTHOR]