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

Certain subclasses of starlike functions of complex order involving the Hurwitz-Lerch Zeta function

Making use of the Hurwitz-Lerch Zeta function, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients of complex order denoted by $TS^\mu_b(\alpha, \beta, \gamma)$ and obtain coefficient estimates, extreme points, the radii of close to convexity, starlikeness and convexity and neighbourhood results for the class $TS^\mu_b(\alpha, \beta, \gamma)$. In particular, we obtain integral means inequalities for the function $f(z)$ belongs to the class $TS^\mu_b(\alpha, \beta, \gamma)$ in the unit disc