Power series coefficients of some classical functions

In this thesis we consider several problems pertaining to extremal conditions arising in Geometric Function Theory. First, we extend the known extremal conditions of a particular function space to a slightly more general space and determine the condition of equality. Next, we discuss the current status of the Krzyz conjecture with a new observation. In the following chapter, we develop a mechanism to translate an extremal function of one space into another space and apply it to a special case of the Krzyz conjecture. The last chapter of the paper is devoted to discussing the current status of Brannan's conjecture with some observations that might lead to its proof.