Two New Subclasses Of The Function

The theory of complex function is a basic branch of mathematics,it is the study of complex function, especially, complex analytic function and the mathematical theory of meromorphic function. In the 19th century, the related research which was done by cauchy become the basis of the theory of complex functions, through constant development, it becomes an important part of mathematics because of its perfect theory and superb skills. Its theory is applied to many branches in the field of mathematics and influence their development, such as the integral equation, differential equation, probability theory and theory of numbers and so on, at the same time it is often as has a indispensable tool to be used in practical problems, such as fluid mechanics, aviation mechanics, thermodynamics, electricity, etc.We are well aware of analytic function’s impartance which embody obviously in this theory, and in a long time ago a large number of mathematicion at home and abroad were attracted to study it. The well-known Riemann Mapping Theorem sustain powerfully the argument that schlicht analytic functions possess indubitably places in the theory of functions of a complex variable. The researchers explore further the argument and get tremendous achievement which contain area theorem,deviation theorem,koebe’s distortion theorem and so on.In recent years, many complex analysis workers are targeting the p-valent of analytic functions,especially,broadening the thinking from A1 to Ap.The theory of the p-valent function is more than just a general theory of the simple function.This research and extension process is miscellaneous. In 1955,Hayman became the first successful precise inequality scholar. In the process of studying multi-valent function function, subordination and Hadamard product are applied in the following work by the mathematicians.Janowski and the other experts get a series of the subclass of analytic function and some classical qualities which comprise distortion theorem,estimate of coefficient,inclusion relationship, and convolution quality. Inspired by[1] to [7],according to the relevant knowledge,Gp,k(λ,A,B), Qp,k(λ,A,B)are introduced, then we explore some of the relevant properties of Gpk(λ,A,B),Qp,k(λ,A,B).There are two main parts in this article:The first part introduces the concept of p-valent starlike functions, differential subordination and Hadamard convolution, etc.,and also prove relevant lemma which is used in this article,then define G p,k(A,A,B), Q p,k{λ,A,B) according to inequality coefficient.The second part discusses some properties of the two classes of functions detaily, such as distortion theorems, inclusion relations, convolution properties and integral transformation.