Geometric Properties of Partial Sums of Univalent Functions

The $n$th partial sum of an analytic function $f(z)=z+\sum_{k=2}^\infty a_k z^k$ is the polynomial $f_n(z):=z+\sum_{k=2}^n a_k z^k$. A survey of the univalence and other geometric properties of the $n$th partial sum of univalent functions as well as other related functions including those of starlike, convex and close-to-convex functions are presented

Bounds for the Second Hankel Determinant of Certain Univalent Functions

The estimates for the second Hankel determinant a_2a_4-a_3^2 of analytic function f(z)=z+a_2 z^2+a_3 z^3+...b for which either zf'(z)/f(z) or 1+zf"(z)/f'(z) is subordinate to certain analytic function are investigated. The estimates for the Hankel determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike, lemniscate starlike functions are obtained

Subordination And Convolution Of Multivalent Functions And Starlikeness Of Integral Transforms

This thesis deals with analytic functions as well as multivalent functions de- �ned on the unit disk U. In most cases, these functions are assumed to be normalized, either of the form f(z) = z + 1X k=2 akzk; or f(z) = zp + 1X k=1 ak+pzk+p; p a �xed positive integer. Let A be the class of functions f with the �rst normalization, while Ap consists of functions f with the latter normalization. Five research problems are discussed in this work. First, let f(q) denote the q-th derivative of a function f 2 Ap. Using the theory of di�erential subordination, su�cient conditions are obtained for the following di�erential chain to hold: f(q)(z) �(p; q)z