Applications Of Basic(or Q-)series And Basic(or Q-)polynomials In Geometric Function Theory

Basic(or q-)series and basic(or q-)polynomials are particularity applicable in many areas of mathematics and other sciences.In this dissertation,by utilizing the idea of basic(or q-)series and basic(or q-)polynomials along with the technique of subordination we derive some new subfamilies of functions which are analytic and bi-univalent in U.For each of the classes of analytic and biunivalent functions,we obtain a number of new results,including for example,coefficients bounds and the Fekete-Szego type problem.Furthermore we study necessary and sufficient condition for a function whose coefficients are probabilities of q-Poisson distribution.Also,by using the higher-order q-derivatives,we define a number of new other subclasses of starlike functions.For the defined function classes of q-starlike functions,we find necessary and sufficient conditions,some inclusion results and Closure Theorems.Furthermore,by using the q-derivatives and the higher-order q-derivatives,we defined several classes with negative Taylor-Maclaurin coefficients and then for each of the defined functions classes with negative Taylor-Maclaurin coefficients,a number of interesting properties including(for example)distortion theorems,radius problems and the estimate on lower bounds for the ratio of analytic functions and their sequences of the partial sums.Most of the results in this dissertation are supported with a number of corollaries and consequences as well.Our results are demonstrated to be linked with diverse former works associated to the field of our present consideration.