Bounded Analytic Functions by John Garnett

By John Garnett

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Since v(z) ≤ vr (z), u(z) is a harmonic majorant of v(z). 3, U (z) ≥ vr (z) for each r. Consequently, supr vr (0) < ∞, and again u(z) = limr vr (z) is finite and harmonic. Since vr (z) ≤ U (z), we have u(z) ≤ U (z), and so u(z) is the least harmonic majorant. Since by continuity u(z) = limr →1 u(r z), the least harmonic majorant of v(z) can also be written u(z) = lim r →1 Pz (θ)v(r eiθ ) dθ/2π. In particular, if v(z) ≥ 0 and if v(z) has a harmonic majorant, then its least harmonic majorant is the Poisson integral of the weak-star limit of the bounded positive measures v(r eiθ ) dθ/2π .

Hence E is dense in (z 0 , r ). Because E is closed this means (z 0 , r ) ⊂ E, and E is open. Since W was assumed to be connected, we have a contradiction and we conclude that a ≤ 0. Conversely, let z 0 ∈ and let (z 0 , r ) ⊂ . Since v is upper semicontinuous there are continuous functions u n (z) decreasing to v(z) on ∂ (z 0 , r ) as n → ∞. Let Un (z) be the harmonic function on (z 0 , r ) with boundary values u n (z). After a suitable change of scale, Un is obtained from u n by the Poisson integral formula for the unit disc.

Letting n → ∞ now yields 1 2π |B(eiθ )|dθ = 1. so that |B(eiθ )| = 1 almost everywhere. The purpose of the convergence factors −¯z n /|z n | is to make arg bn (z) converge. To remember the convergence factors, note that they are chosen so that bn (0) > 0. 2, the analytic function f (z) has a factorization f (z) = B(z)g(z), z ∈ D, where B(z) is a Blaschke product and where g(z) has no zeros on D, if and only if the subharmonic function log | f (z)| has a harmonic majorant. 3) yn < ∞, 1 + |z n |2 z n = xn + i yn , and the Blaschke product with zeros {z n } is B(z) = z−i z+i m z n =i |z n2 + 1| z − z n .

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