Some Applications Of Certain Convolution Operators And Differential Subordination

The studies of convolution operators and applications of differential subor-dination and superordination play a very important role in the analytic functiontheory. The main purpose of this thesis is to derive some properties of certainconvolution operators by using the methods of differential subordination and su-perordination. Furthermore, we discuss the Briot-Bouquet differential equation,subordination preserving integral operators, superordination preserving operators,the applications of subordination chain and some interesting properties of certainspecial classes analytic functions.In Chapter 2, by applying the Cho-Kwon-Srivastava operator and differentialsubordination, we deffne a new subclass of multivalent analytic functions. Thisclass of functions is a generalization of the familiar class of Bazileviˇc functions.We obtain some subordination and superordination properties,"sandwich"typedouble subordination properties, convolution properties, inclusion relationships,distortion theorems, coeffcient inequalities and suffcient conditions for multivalentstarlikeness. The results contain some new criteria for starlikeness.In Chapter 3, by using the Dziok-Srivastava operator and differential sub-ordination, we deffne some new subclasses of multivalent analytic functions withrespect to k-symmetric points. These classes of functions are generalizations ofstarlike and close-to-convex functions with respect to k-symmetric points. Wederive several inclusion relationships, integral representations, convolution prop-erties and integral-preserving properties for these classes of multivalent analyticfunctions.In Chapter 4, by applying the familiar Hurwitz-Lerch-Zeta function, we in-troduce the generalized Srivastava-Attiya operator. Some new subclasses of mul-tivalent analytic functions are deffned by this operator, these classes of analyticfunctions are generalizations of starlike, convex, close-to-convex and quasi-convexfunctions. Such results as inclusion relationships, integral-preserving propertiesand double subordination properties are proved.In Chapter 5, we derive certain new subclasses of meromorphic multivalentfunctions with respect to k-symmetric points, conjugate points and symmetricconjugate points, these classes of functions are deffned by the modiffed version ofthe Dziok-Srivastava operator in meromorphic function space. We obtain some in-clusion relationships, integral representations and convolution properties for theseclasses of functions. In Chapter 6, we introduce and investigate two new subclasses Mp(β) andNp(β) of meromorphic multivalent functions. We derive some subordination prop-erties, coeffcient estimates, integral representations, convolution properties, coef-ffcient suffcient conditions for meromorphic multivalent starlike and convex func-tions, criteria for meromorphic multivalent starlikeness and convexity. By deffninga new array associated with the power series, we prove the coeffcient bounds ofthese classes of functions in view of the method of mathematical induction.