Coefficients of univalent functions which assume no pair of values W and ?W

Abstract

In this paper we study the behavior of the coefficients of functions\(\varphi \left( z \right) = 1 + \sum\nolimits_{\kappa = 1}^\infty {b_{\kappa ^z } ^\kappa } \) univalent in the disk ¦z¦< 1 and assuming there are no pair of values Wand −W. In particular, we establish the asymptotic behavior of bn (n→∞); for the coefficients we obtain the estimate ¦bn¦ < 2.34 exp {l/4n} (n = 2,3, ...) and for each function of the class indicated we prove, subject to a certain condition, the relationship ∥bn+1¦−¦bn∥=O(n−1/2).