Family Of Analytic Functions Research Articles

Distortion Theorems in the Theory of Schlicht Functions

Let be the family of analytic functions F(z) which are regular and schlicht in the circle E: |z| < 1 and normalized at the origin such that F(0) = 0 and F′(0) = 1. Let be the subfamily of consisting of all functions, each of which possesses, as boundary of its image-domain., a closed Jordan curve which is supposed to be regular analytic. Then is everywhere dense in the original family . Hence, by an approximation theorem of Carathéodory on the kernel of sequence of domains, we may restrict ourselves within , so far as we concern the estimation of continuous functionals on .