On the convolution and subordination of convex functions

In this work we present some new results on convolution and subordination in geometric function theory. We prove that the class of convex functions of order αα is closed under convolution with a prestarlike function of the same order. Using this, we prove that subordination under the convex function order αα is preserved under convolution with a prestarlike function of the same order. Moreover, we find a subordinating factor sequence for the class of convex functions. The work deals with several ideas and techniques used in geometric function theory, contained in the book Convolutions in Geometric Function Theory by Ruscheweyh (1982).