Harmonic function is one of an important branches of complex analysis. The first study of complex – valued, harmonic mappings defined on a domain ꓓ ⊂ C 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.

We first present purely combinatorial proofs of two facts: the well-known fact that a monomial ordering must be a well ordering, and the fact (obtained earlier by Buchberger, but not widely known) that the division procedure in the ring of multivariate polynomials over a field terminates even if the division term is not the leading term, but is freely chosen. The latter is then used to introduce a previously unnoted, seemingly weaker, criterion for an ideal basis to be Gröbner, and to suggest a new heuristic approach to Gröbner basis computations.

Division in the ring of multivariate polynomials is usually not a part of the standard university math curriculum. However, the algorithm is elementary and it has very important consequences for algebraic computations. In this paper, the algorithm is explained and illustrated with some examples, and the importance of the choice of monomial ordering is stressed. The notion of Gr¨obner basis is introduced and explained on examples. The paper can be used by math students and teachers as a brief description of this very important topic and introduction for reading more detailed textbook