And Noor Integral Operator About The Nature Of Some Of The Leafy Analytic Function Subclass

Complex analyasis is the mathematics theory of studying on the complex functions,especially on meromorphic functions and analytic functions.It is one branch of mathematics with ancient and rich life, which is a classical research field and has attracted great attention of many mathematicians.The theories and methods of it is not only to solve many problems in differential equation analytic number theory、differential geometry、topology,but also more gnerally used in many natural science areas, such as theoretical physics, aerodynamics, etc.Univalent functions and subordinate principle are the important contents of geometric function theory which include the theoretical study of area theorem、growth theorem、distortion theorem、coefficient estimation subordination chain、differential equation and differential subordination,etc.Many scholars, such as Miller and Mocanu, have done a lot of work in this field.The cornerstone of Geometric Function Theory is the theory of univalent functions,which was initiated by Koebe.His paper published in1907is generally regardd as giving the first actual result in the theory of Univalent Functions.Later Gronwall’s proof of the area theorem in1914and Biberbach’s estimate for the second coefficient of a normalized univalent function in1916and its consequences proved the importance of the subject in its own right.A function f that attains every value at most p times and some values exactly p times in a domain D is said to be p-valent or is called multivalent of order p,provided p>1.Taking p=1,we have univalent functions because in this case f assumes no value more than once in D.The theory of p-valent Functions is much more than just a generalization of Univalent Function Theory. The extension of any result from univalent to p-valent may be trivial or extremely difficult or perhaps false.There are several results that generalize classical results on univalent functions.The first sccess in obtaining sharp inequalities for multivalent functions was attained by Hayman in1955.Apart from this, researchers have introduced subclasses of multivalent analytic functions which are multivalently starlike with respect to origin,multivalently convex and multivalently close-to-convex in the open unit disk. Further, the concept of functions of bounded boundary rotation which generalizes convex functions originates from Lowner in1917and later Paatero has made a thorough study of this class. Pinchuk,Brannan,Kirwan,Padmanabhan and Parvatham,Moulis, Coonce,Noor,Noor et al. and several other mathematicians have considered the class of functions of bounded boundary and the class of functions of bounded radius rotation.Noor also studied these classes by using Noor integral operator,Ruscheweyh derivative operator,generalized Bernardi integral and Jim-Kim-Srivastava operator respectively. In the research presented here we use the techniques of convolution(Hadamard product) and the differential subordination.Janowski used the concept of subordination and explored many interesting properties of subclasses of analytic functions.In1973,using convolution technique,Ruscheweyh and Sheil-Small proved Polya-Schoenberg conjecture, they showed that the classes of starlike functions,convex functions and close-to-convex functions are invariant under convolution with convex function.Ruscheweyh, Duren as well as Goodman,developed these concepts.Using the techniques developed in Ruscheweyh,several authors have proved that many other analytic classes are closed under convolution with convex (and some other related) functions. Apart from this,certain linear operators such as Carlson-Shaffer operator, Ruscheweyh derivative, Noor integral operator have been defined in the open unit disk by using the Hadamard product or convolution. Using these operators, several interesting subclasses of analytic functions are introduced and their classical properties such as coefficient estmates, distortion theorems and covering results are studied systematically.The whole thesis comprises five chapters.The chapter vise distribution is given as below:In chapter1,we introduce some basic and classical concepts of p-valent analytic functions, p-valent starlike function, p-valent convex function, p-valent strongly starlike function, Noor integral operator,which supply a necessary environment for the investigation of the work presented in this thesis.Two subclasses cp(h) and κ(h) of multivalent analytic functions are also defined. They perform a key role for the description of our main results.The two main tools Hadamard convolution and differential subordination are concisely discussed here.In chapter2,we list some relative lemmas,which makes preparation for the third and the forth chapter.In chapter3,we mainly discusses the sufficient condition for the classφ(1+z/1-z)In chapter4,we mainly investigate argument properties and their consequences in relation with Noor integral operator.In chapter5,we mainly discusses the growth theorems, distortion theorems and covering theorems for the subclsses φ(h) and κ(h),as well as relative lemmas.