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A Schwarz Lemma for the Symmetrized Bidisc
Author(s) -
Agler J.,
Young N. J.
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/33.2.175
Subject(s) - mathematics , lemma (botany) , function (biology) , pure mathematics , inverse , tangent , inequality , mathematical analysis , combinatorics , geometry , ecology , poaceae , evolutionary biology , biology
Let φ be an analytic function from D to the symmetrized bidiscΓ = def{ ( λ 1 + λ 2 , λ 1 λ 2 ) : ∣ λ 1 ∣ ⩽ 1 , ∣ λ 2 ⩽ 1 ∣ } .We show that if φ(0) = (0,0) and φ(λ) = ( s , p ) in the interior of Γ, then2 ∣ s − p s ¯ ∣ + ∣ s 2 − 4 p ∣4 −∣ s ∣ 2⩽ ∣ λ ∣ .Moreover, the inequality is sharp: we give an explicit formula for a suitable φ in the event that the inequality holds with equality. We show further that the inverse hyperbolic tangent of the left‐hand side of the inequality is equal to both the Caratheodory distance and the Kobayashi distance from (0,0) to ( s , p ) in int Γ