Dziubinski defined the class of quasi-starlike functions. Due to a recent result by Brickman, we may generalize Dziubinski's definition to obtain a larger class of functions, which we will call quasi-spirallike functions. We consider also the class B of normalized functions which are analytic, univalent, and bounded on the unit disk and we show that the class of quasi-spirallike functions is a proper subclass of the class B.

A lower bound on the radius of starlikeness for the class of quasi-starlike functions is derived. Distortion theorems are obtained for all of the classes under consideration. The mapping properties of the extremal functions are discussed in detail.

For arbitrary complex β, sharp bounds are obtained for the maximum of the functional |b₃ - βb₂²| over the class of quasi-spirallike functions with a given normalization. A basic inequality is obtained by comparing these bounds with the sharp bounds on the maximum of the functional |b₃ - βb₂²| over the class B with the same normalization. Finally, max |b₂| and max |b₃| are computed for the various classes.