Harmonic function is one of an important branches of complex analysis. The first study of complex valued, harmonic mappings defined on a domain D was given by ClunIe and SheIl-Small [1]. Harmonic functions have been studied by different researchers such as Silverman [6]. In the present paper, a new class of harmonic univalent functions will be introduced. Various properties of functions belong to this class which include coefficient bounds, growth bounds, a closure property, extreme points, neighborhood and a convex combination will be obtained.

The division algorithm for multivariate polynomials over fields has been introduced not very long ago, in connection with algorithmic and computational problems in the rings since the work of Buchberger. The well-known fact is a term order must be a well-ordering and the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not leading term, but is freely chosen. In this paper, we will show the original geometric approach to the classification of all possible orders on the ring of polynomials in two variables is given. The connection between this classification and the well-known classification of Robiano is exposed in details.

The theory of operators is an important subject in analytic function theory, geometric function theory and univalent function theory. It also is an important subject in applied sciences. It is still an active field of research and various types of problems, which can be solved by generalising operators. Recently, many researchers have shown great interests in the study of differential operators in the theory of univalent functions and various subclasses of analytic functions defined in the open unit disc. In this paper a generalised derivative operator will be used to derive some results concerning the subordination and superordination of analytic function in the open unit disc.

The Holling type II harvesting term has the property that it is small for small population values, but grows monotonically with population growth, eventually saturating, at a constant value for very large populations. We consider here a population evolving according to a logistic rate, but harvested (predated) subject to a Holling type II harvesting term that varies slowly with time, possibly due to slow environmental variation. Application of a multitiming method gives us an approximation to the population at any time in two cases- survival to a slowly varying limit, and extinguishment to zero. The situation where there is a transition from survival to extinction is also analyzed, using a matched expansions approach. A uniformly valid approximate expression for the population, valid for all times is obtained. These results are shown to agree well with the results of numerical calculations.

The object of the present paper is to derive some results on differential subordination associated with a generalized derivative operator for certain normalized analytic functions in the open unit disc. The authors establish sandwich type theorems. These results extend many previously known results.

The object of the present paper is to introduce a new subclass of p-valent starlike functions with negative coefficients in the open unit disc which is defiined by a generalized derivative operator. We obtain coefficient inequalities, growth and distortion theorems and extreme points for the subclass of p-valent functions.

The aim of this paper is to introduce and study a new class of Bazilevič p-valent function of order β by using the subordination concept between this function and a generalized derivative operator. Some interesting properties are also obtained

The theory of analytic functions and more specific p-valent functions, is one of the most fascinating topics in one complex variable. There are many remarkable theorems dealing with extremal problems for a class of p-valent functions on the unit disk U. Recently, many researchers have shown great interests in the study of differential operator. The objective of this paper is to define a new generalized derivative operator of p-valent analytic functions of fractional power in the open unit disk U. This operator generalized some well-known operators studied earlier, we mention some of them in the present paper. Motivation by the generalized derivative operator; we introduce and investigate two new subclasses of starlike p-valent analytic functions of fractional power with positive coefficients and starlike p-valent analytic functions of fractional power with negative coefficients, respectively. In addition, a sufficient condition will be obtained. Some corollaries are also pointed out. Moreover, we determine the extreme points of functions belong to the class.

The aim of the article is to study a sufficient condition for the classes of starlike and convex harmonic univalent functions with respect to k- symmetric points. The necessary condition for f to be in the classes will be studied. In addition, growth bounds for functions will also be given.

In this paper, we study and introduce the majorization properties of a new class of analytic p-valent functions of complex order defined by the generalized hypergeometric function. Some known consequences of our main result will be given. Moreover, we investigate the coefficient

estimates for this class.