UNIVALENT FUNCTIONS with QUASICONFORMAL EXTENSIONS: BECKER'S CLASS and ESTIMATES of the THIRD COEFFICIENT

We investigate univalent functions f(z) = z+a2z2 +a3z3 +. . . in the unit disk D extendible to k-q.c.(=quasiconformal) automorphisms of C. In particular, we answer a question on estimation of |a3| raised by Kühnau and Niske [Math. Nachr. 78 (1977), pp. 185-192]. This is one of the results we obtain studying univalent functions that admit q.c.-extensions via a construction, based on Loewner's parametric representation method, due to Becker [J. Reine Angew. Math. 255 (1972), pp. 23-43]. Another problem we consider is to find the maximal k* ∈ (0, 1] such that every univalent function f in D having a k-q.c. extension to C with k < k* admits also a Becker q.c.-extension, possibly with a larger upper bound for the dilatation. We prove that k* > 1/6. Moreover, we show that in some cases, Becker's extension turns out to be the optimal one. Namely, given any k ∈ (0, 1), to each finite Blaschke product there corresponds a univalent function f in D that admits a Becker k-q.c. extension but no k_-q.c. extensions to C with k_< k.© 2020 American Mathematical Society.