If g(z) is regular in the open unit disk E, normalized by g(0) = 0 and g′(0)= 1, and there is a complex number ε, |ε| = 1, such that Re [εzg′(z)/g(z)] > 0 for z in E; then g(z) is said to be a spiral function. The spiral functions are used to define a new class of regular functions: f(z) is in H if and only if f(z) is regular in E, satisfies f(0) - 0 and f′(0) = 1, there exists a spiral function g(z) and a complex number ξ such that Re [ξzf′(z)/g(z)] > 0, z in E. The class H contains the spiral functions and the close-to-convex functions; it also contains functions which are not univalent.

The dissertation has four chapters. The first chapter is concerned with preliminary definitions and the decomposition of H into subclasses.

In chapter two the methods of interior variation are used to solve two general extremal problems over H.

Let f(z) = z + a₂z² + ... belong to H. Chapter 3 deals with bounds on arg [f′(z)], |f′(z)| and |a_n|. Upper bounds are also determined for | | a_n+1| - |a_n|| and |a₃ - μa₂²|, μ any complex number.

Let C denote the class of functions f(z) such that zf′(z) is a spiral function. In chapter four the radii of convexity and close-to-convexity of H is 2 - √3. Also an example of a univalent subclass of H is given by appealing to the Schwarzian derivative