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The Sharpness of a Criterion of Maclane for the Class\s A
Author(s) -
Drasin David,
Wu Jang-Mei
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610702004015
Subject(s) - holomorphic function , mathematics , combinatorics , bounded function , class (philosophy) , function (biology) , point (geometry) , unit (ring theory) , mathematical analysis , geometry , computer science , artificial intelligence , mathematics education , evolutionary biology , biology
A holomorphic function defined in the unit disk Δ = { z : | z | < 1} belongs to the MacLane class A if each point ζ of a dense subset of ∂ Δ is the endpoint of a curve γ ζ (with γζ\ζ⊂ Δ) such that f ( z ) tends to a limit (perhaps ∞) as z → ζ on γ ζ . The classical Fatou theorem ensures that f ∈ A when f is bounded. G. R. MacLane introduced\s A in [ 5 ], where he proved that f ∈ A if there is a set E dense in ∂Δ with∫ 0 1( 1 − r )log + | f ( r e i θ)| d r < ∞( θ ∈ E )(1.1) For example, if f is the modular function and M(r) = max | z |=r | f ( z )| its maximum modulus, thenlog M ( r ) ⩽ log 1 1 − r + O ( 1 ) ,so that (1.1) applies. An ample discussion of A is in [ 4 , Chapter 10].