Geometric properties of new subclasses of univalent and bi-univalent functions associated with subordination and generalised differential operator

This thesis is concerned with the geometric properties of the class of analytic univalent and bi-univalent functions in the complex plane. Thus, there are three major objectives to be achieved and the results are presented in three different chapters. The first objective is to apply the subordination method on the class of analytic univalent and bi-univalent functions involving radius and coefficient problems. The radii of starlikeness for concave univalent functions related to certain rational functions, lune, cardioid, and the exponential equation are obtained. Then, the Third Hankel Determinants |a3(a2 a4 - a3^2) + a4(a2 a3 - a4) + a5(a3 - a2^2)| and Generalised Zalcman Conjecture |an am - an+m-1| of bi-univalent functions are acquired.