By Peter L. Duren
ISBN-10: 0486411842
ISBN-13: 9780486411842
A mix of classical and sleek suggestions and viewpoints, this article examines harmonic and subharmonic services, the fundamental constitution of Hp services, purposes, Taylor coefficients, Hp as a linear house, interpolation concept, the corona theorem, and extra. applicable for researchers or 2d- or 3rd-year graduate scholars. 1970 edition.
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M~(r,f) is = M(r) + e EXERCISES 51 Now let e ~ 0, integrate from 0 to r, and apply Lemma 1 : M/(r,f) ~ p [ [M(r)]P dr o r ~p JrP- (1- r)1 1 2P dr < oo 0 if 0 < p < ! This concludes the proof. 8). The Blaschke product B(z) obviously has at most one factor. Less obvious is the fact thatfcan have no singular part. 17. lf/(z) is analytic and schlicht in lzl factor S(z) =1. < 1, then its singular PROOF. Iff does not vanish in the open disk, then 1/fis analytic and schlicht, so 1/f E HP for all p < -t.
0 There is a similar factorization for/EN. A function F(z) of the form (6), where 1/J(t) ;;::: 0 and log 1/J(t) E L1 , will be called an outer function for the class N. (Note that the condition 1/1 E I! 9. Every functionf(z) oj. f(ei1)1). Conversely, every function of the form (7) belongs to N. PROOF. Letf(z) =eiYB(z)g(z), whereg E N,g(z) of. Oin lzl < 1, andg(O) > 0. Since loglg(z)l E h 1 , it has a representation as a Poisson-Stieltjes integral J2" 1 loglg(z)l = P(r, 8- t) dv(t) 2n v with respect to a function v(t) of bounded variation.
10, which he attributed to "Madame P. Kotchine". More about the class N+ may be found in Privalov's book [4]. The condition (9) actually characterizes Blaschke products among all analytic functions with 1/(z)l :::;;; l (see Exercise 6). This theorem is due to M. Riesz, but was first published in Frostman [1]. (See also Zygmund [4], Vol. I, p. 12 was mentioned in passing by Smirnov ([3], p. 341). There is a large literature on the boundary behavior of Blaschke products; see, for example, Frostman [2], Cargo [1, 2], and Somadasa [1].