VOLTERRA-TYPE OPERATORS ON THE SUBCLASSES OF UNIVALENT FUNCTIONS

We examine necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order b (b not equal 0) and spirallike functions of type lambda (-pi/2 < lambda < pi/2) with complex order b (b not equal 0). We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions that belong to the indicated classes. In the main part of the paper, we obtain necessary and sufficient conditions of boundedness for the image of the open unit disk D = {z is an element of C : vertical bar z vertical bar < 1} under the action of a Volterra-type operator and the product of a composition operator and a Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate for the Schwartzian norm of the above operators in these spaces.