Some Properties Of Multi - Leaf Analytic Functions Associated With Generalized Fractional Micro - Integral Operators

Geometric function theory is one of the oldest branch of mathematical research and full of vitality, it is a classical research field, has attracted the attention of mathematicians. Its theory and method can not only solve difficult problems in the topology, differential equations, differential geometry, analytic function theory and many other research fields, but also applied to many field of natural science such as physics, aerodynamics, etc. Univalent function is one of the important research contents of geometric function theory. Their theoretical studies include the area theorem of univalent function, deviation theorem, the growth theorem, subordination chain, coefficient estimates, differential subordination and Briot-Bouquet differential equations,etc...Since the seventies and eighties of the last century, with the development of differential subordination theory, the studying of geometric function theory also rejuvenated. Many scholars, combine convolution operator with fraction calculus operators and univalent function theorem and they have got a lot of important results, such as Sanford S.Miller and Petru T.Mocanu[1].Recently, some scholars began from the research field of univalent functions to expand the research field of multi leaf function, namely the study of function space from A1 expanding to Ap. Scholars use Hadamard product and constructe many new operators in space, such as Φp（η,λ）（z）[2], Φp（a,c;z）[3], Noor integral operator, and so on. Many interesting conclusions are obtained by studying the properties of operators.Inspired by the above, we define a new operator Ω_z^{（λ,p）}. Making use of operatorΩ（λ,p） and differential subordination, S_p^λ（η;A,B） is introduced. We will investigate the inclusion relationships of S_p^λ（η;A, B） and some useful properties of Ω_z^{（λ,p）}.The following is the structure and main content of this paper:The first part is the introduction, which mainly introduces the definition of the subordination, Hadamard convolution, Gauss hypergeometric function and other preliminary knowledge, and gives some important definitions and the relevant lemmas used in this paper.The second part is the inclusion relationships of S_p^λ （η; A, B） and some properties of the operator Ω_z^{（λ·p）}.