Some Properties Of Analytic Functions

Let h ( r,s,t) be a complex function defined in a domain of D ? C3.In 1978, making use of determining conditions on h ( r,s,t),Miller and Moconu defined a classΨn (a).They obtained a theorem that can be used to obtain functions with positive real part.Applications of the results to differential equations,harmonic functions and univalent function theory were also given[1].In the first part of this article,we define a class Hn (a) by making use of determining conditions on h ( r,s,t).A theorem that can be used to obtain functions with negative real part can be obtained.Application of the result to differential equations is given.In 1993,Nunokawa obtained a important theorem about argument estimate[2].In 2000, Nunokawa,Owa etc made a generalization of the theorem[3].Recently ,many scholars have investigated extensively by applying this two results and obtained many important results[4-6].Let N be the class of functions p (z) analytic in the open unit disc U with p ( 0)=1.In the second part of this article,a property of p (z) at estremal points for argument on the boundary of the circle z = r<1 is derived.This main result is the generalization of one by Nunokawa and Owa etc.Some applications of our main result are also considered.Recently, many scholars have investigated some interesting integraloperators, for example, WhereΓdenotes the Gamma function and Some interesting subclasses of analytic functions, associated with theoperator Qβα,have been considered by Jung,Kin and Srivastava[7],Aoufetal[8],Lin[9],Owa and Liu[10] and others.Motivated by their work,we define a operator Qβα.Let A p be the class of functions analytic in the open unit disk U .A certain integral Ap→Ap is defined asMaking use of operator Qβαand differential subordination,certain subclasses Eαp ,β(η;A,B), Fpα,β(η;A,B), S *p ,α,β(η) and Kαp,β(η) are introduced. In the third part of this article, the inclusion relationships of these subclasses are obtained. Some other properties of operator Qβαare also given.