Some properties associated with certain subclasses of univalent and multivalent functions / Wan Sabhi Salmi bt Wan Hassan - Wan Hassan, Wan Sabhi Salmi (2013)
This thesis investigates properties of certain analytic functions; in particular, functions which are univalent and multivalent in the unit disc U = {z 2 C : |z| < 1}. Let A denote the class of all normalised analytic functions of the form f(z) = z + X1 n=2 anzn. Interest is focused at several subclasses of A. Functions belonging to these subclasses are defined via some differential operator; namely the S˘al˘agean and Al- Oboudi operator. These classes formed are subclasses of S, the class of univalent functions. Let f 2 Bn() for > 0 and n = 0, 1, 2, . . . be defined by Re Dnf(z) z > . where Dn denote the S˘al˘agean operator. For functions f 2 Bn(), we obtain estimates for the second, third and fourth coefficients of the inverse functions. Further, we investigate similar coefficient problems for functions in the B n(), an extension of the above class defined via the Al-Oboudi operator. In addition, these are then applied to obtain the Fekete-Szeg¨o inequalities. Next, besides functions of the above normalised form, the thesis also looks at functions of the form f(z) = zp + X1 n=1 ap+nzp+n, where p a fixed positive integer. For functions of this form, we denote Ap as the class consisting of such functions. For such class we investigate sharp lower bounds on ii the real part of the quotients between the normalised functions and their sequence of partial sums for convex and starlike functions as well as their related classes, the uniformly convex and parabolic starlike functions which satisfy certain conditions. Finally, for function f(z) 2 Ap which are analytic in U, results on the preservation of two integral operators I p f(z) and J f(z) given by I p f(z) = (p + 1) zp