2019/05/21 by Gumenyuk, Pavel, Hotta, Ikkei
#30C50 #30C75 #30D05 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30C62 #Secondary 30C35
paper · doi:10.48550/arxiv.1905.08666
We investigate univalent functions f(z)=z+a2z2+a3z3+… in the unit disk \mathbb D extendible to k-q.c.(=quasiconformal) automorphisms of \mathbb C. In particular, we answer a question on estimation of |a3| raised by Kühnau and Niske [Math. Nachr. 78 (1977) 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) 23-43]. Another problem we consider is to find the maximal k_*∈(0,1] such that every univalent function f in \mathbb D having a k-q.c. extension to \mathbb C with k\leqslant 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 \mathbb D that admits a Becker k-q.c. extension but no k'-q.c. extensions to \mathbb C with k'