Grunsky inequalities and quasiconformal extension

2006 ◽  
Vol 152 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Samuel Krushkal ◽  
Reiner Kühnau

1976 ◽  
Vol 51 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jan G. Krzyż


1977 ◽  
Vol 29 (3) ◽  
pp. 587-605
Author(s):  
Duane W. De Temple ◽  
David B. Oulton

The Grunsky inequalities [6] and their generalizations (e.g., [5; 14; 17]) have become an increasingly important tool for the study of the coefficients of normalized univalent functions defined on the unit disc. In particular, proofs based upon the Grunsky inequalities have now settled the Bieberbach conjecture for the fifth [15] and sixth [13] coefficients. For bounded univalent functions the situation is similar, although the Grunsky inequalities go over to those of Nehari [11].



2012 ◽  
Vol 141 (1-2) ◽  
pp. 195-209 ◽  
Author(s):  
Hidetaka Hamada ◽  
Gabriela Kohr


2001 ◽  
Vol 8 (4) ◽  
pp. 815-822
Author(s):  
S. L. Krushkal

Abstract Given a quasisymmetric map , let 𝑓0 be an extremal quasiconformal extension of ℎ onto the upper half-plane 𝘜 = {𝑧 ∈ ℂ : 𝔍𝑧 > 0} whose dilatation 𝑘(𝑓0) = inf{𝑘(𝑓) : 𝑓|∂𝘜 = ℎ0} ≕ 𝑘(ℎ). Let 𝑘𝑛 be the minimal dilatation of polygonal quasiconformal maps 𝑓 : 𝘜 → 𝘜 satisfying 𝑓(𝑥𝑗) = ℎ(𝑥𝑗), 𝑗 = 1, 2, . . . , 𝑛, for any 𝑛 points of (vertices of 𝑛-gons). Already a long time ago, the question was posed whether 𝑘(ℎ) = sup 𝑘𝑛, where the supremum is taken over all possible 𝑛-gons of such kind. The answer is obtained (in the negative) for the case of quadrilaterals (𝑛 = 4). We show that in the case of pentagons (𝑛 = 5) the answer is also negative, i.e., there are quasisymmetric ℎ with 𝑘(ℎ) > sup 𝑘5.



2004 ◽  
Vol 56 (6) ◽  
pp. 873-881
Author(s):  
V. V. Aseev ◽  
A. V. Sychev ◽  
A. V. Tetenov


2013 ◽  
Vol 38 ◽  
pp. 617-630 ◽  
Author(s):  
Rodrigo Hernández ◽  
María J. Martín






1986 ◽  
Vol 83 (2) ◽  
pp. 167-200
Author(s):  
Jacob Burbea
Keyword(s):  


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