Analytic functions close to mappings convex in one direction

Abstract:

By analogy to the class of close-to-convex functions we define a class of analytic functions which are close to a family $\Sigma$ of mappings onto domains convex in one direction. In contrast to the close-to-convex class the close-to-$\Sigma$ functions are not necessarily univalent. However, we determine the radius of convexity for $\Sigma$, and this gives a lower bound for the radius of univalence of close-to-$\Sigma$ functions. We next derive the coefficient estimate $|{A_n}| \leqq n|{A_1}|$ for close-to-$\Sigma$ functions and conclude with an elementary distortion theorem.