The Sharpness of a Criterion of Maclane for the Class\s A

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].