A theorem on the regularity of decrease in linearly invariant families of functions

Abstract

The fact that Uα = ∅ for α < 1 and U1 coincides with the class of convex functions, i. e., with the class of analytic functions which univalently map∆ onto convex domains, is proved in [1]. The interest towards l. i. f.’s is connected with the fact that many well-known classes of conformal mappings represent l. i. f.’s. Thus, one can study the properties of many classes of locally univalent in∆ functions in general. The most well-known example of an l. i. f. is the class S of univalent in ∆ functions (ordS = 2). In this class the theorem on the regularity of growth of functions and their first derivatives is established. For a continuous in∆ function φ denote M(r, φ) = max |z|=r |φ(z)|, m(r, φ) = min |z|=r |φ(z)|.