By John Garnett

The e-book, which covers quite a lot of attractive issues in research, is intensely good prepared and good written, with based, unique proofs. The publication has proficient a complete new release of mathematicians with backgrounds in complicated research and serve as algebras. It has had a very good effect at the early careers of many prime analysts and has been extensively followed as a textbook for graduate classes and studying seminars in either the USA and abroad.

- From the quotation for the 2003 Leroy P. Steele Prize for Exposition

The writer has now not tried to provide a compendium. quite, he has chosen a number of themes in a many-faceted idea and, inside of that diversity, penetrated to massive depth...the writer has succeeded in bringing out the great thing about a conception which, regardless of its quite complicated age---now drawing close eighty years---continues to shock and to please its practitioners. the writer has left his mark at the subject.

- Donald Sarason, Mathematical Reviews

Garnett's ** Bounded Analytic Functions** is to operate thought as Zygmund's

**is to Fourier research.**

*Trigonometric Series***is extensively considered as a vintage textbook used world wide to teach present day practioners within the box, and is the first resource for the specialists. it really is fantastically written, yet deliberately can't be learn as a singular. really it provides simply the precise point of aspect in order that the encouraged pupil develops the considered necessary talents of the exchange within the technique of researching the great thing about the combo of genuine, advanced and sensible analysis.**

*Bounded Analytic Functions*- Donald E. Marshall, collage of Washington

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**Extra info for Bounded Analytic Functions**

**Sample text**

For 0 < p ≤ ∞, H p is complete. Sect. 2 51 blaschke products Proof. We can assume p < ∞. We give the proof in the upper half plane; the reasoning for the disc is very similar. 2) of Chapter I. It shows that any H p Cauchy sequence { f n } converges pointwise on H to an analytic function f (z). Fatou’s lemma then shows | f m (x + i y) − f n (x + i y)| p d x | f (x + i y) − f n (x + i y)| p d x ≤ lim m→∞ ≤ lim m→∞ Hence f − p fn H p ≤ limm→∞ f m − f n fm − fn p Hp, p Hp. and H p is complete. 2. Blaschke Products We show that the zeros {z n } of a nonzero H p function on the disc satisfy Blaschke’s condition (1 − |z n |) < ∞.

2) holds at every point of the Lebesgue set of f , which is independent of ϕ. (c) Formulate and prove a similar result about nontangential convergence and nontangential maximal functions. 12. If f (x) has support a bounded interval I, then only if I | f | log+ | f | d x < ∞ (Stein [1969]). I M f d x < ∞ if and 13. Let μ be a positive Borel measure on ޒ, finite on compact sets, and define 1 | f | dμ. Mμ f (x) = sup I x μ(I ) I Show μ({Mμ f (x) > λ}) ≤ C | f | dμ λ and |Mμ f | p dμ ≤ C p | f | p dμ, 1 < p < ∞.

2. (a) (b) and z 2 Suppose f (z) ∈ B . If f (z) has two distinct fixed points in D, then f (z) = z. Let ε > 0, and suppose ρ( f (z 1 ), z 1 ) < ε, and ρ( f (z 2 ), z 2 ) < ε for z 1 distinct points of D. If z lies on the hyperbolic geodesic arc joining z 1 40 Chap. I preliminaries to z 2 , then ρ( f (z), z) ≤ Cε 1/2 with C an absolute constant. (Hint: We can assume that z = 0, z 1 = −r < 0, z 2 = s > 0, and that f (z 1 ) = z 1 . ) (c) Let eiθ and eiϕ be distinct points of ∂ D. Suppose {z n } and {wn } are sequences in D such that z n → eiθ , wn → eiϕ and such that ρ( f (z n ), z n ) → 0, ρ( f (wn ), wn ) → 0.