The second coefficient of univalent Bieberbach-Eilenberg functions near the identity

Abstract

An asymptotic expansion is found for the maximum | b 2 | |{b_2}| , in terms of β = 1 − | b 1 | \beta = 1 - |{b_1}| , for functions F ( z ) = b 1 z + b 2 z 2 + ⋯ F(z) = {b_1}z + {b_2}{z^2} + \cdots which are in the class of univalent Bieberbach-Eilenberg functions and which are near the identity ( b 1 {b_1} near 1). The first two terms of this expansion are the same as the expansion of | b 2 | |{b_2}| in terms of 1 − b 1 1 - {b_1} for the functions which map the unit disc onto the interior of circles passing through ± 1 \pm 1 .