On moment conditions for the Girsanov Theorem

In this dissertation, the well-known Girsanov Theorem will be proved under a set of moment conditions on exponential processes. Our conditions are motivated by the desire to avoid using the local martingale theory in the proof of the Girsanov Theorem. Namely, we will only use the martingale theory to prove the Girsanov Theorem. Many sufficient conditions for the validity of the Girsanov Theorem have been found since the publication of the result by Girsanov in 1960. We will compare our conditions with some of these conditions. As an application of the Girsanov Theorem, we will show the nonexistence of an arbitrage in a market and will also explain a simplified version of Black-Scholes model

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

Coefficient Conditions for Starlikeness of Nonnegative Order

Sufficient conditions on a sequence {ak} of nonnegative numbers are obtained that ensures f(z)= E°°k=1 akzk is starlike of nonnegative order in the unit disk. A result of Vietoris on trigonometric sums is extended in this pursuit. Conditions for close to convexity and convexity in the direction of the imaginary axis are also established. These results are applied to investigate the starlikeness of functions involving the Gaussian hypergeometric functions

A Third-Order Differential Equation and Starlikeness of a Double Integral Operator

Functions f(z)= z+E°2 anzn that are analytic in the unit disk and satisfy the differential equation f'(z) + azf"(z)+yz2f"(z) = g(z) are considered, where g is subordinated to a normalized convex univalent function h. These functions f are given by a double integral operator of the form f(z) = (10(10G(ztμsν�t−μs−νds dt with G" subordinated to h. The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function h