Titlebook: Topics in Hardy Classes and Univalent Functions; Marvin Rosenblum,James Rovnyak Textbook 1994 Springer Basel AG 1994 Algebra.Blaschke prod

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Subharmonic Functions,tion that relaxes the smoothness assumption and permits . to take the value —∞ is given in §2.3. Examples of subharmonic functions include log ∣.∣, log. ∣.∣ = max (log ∣.∣,0), and ∣.∣. (0 < . < ∞), where . is any analytic function on Ω (this is a special case of Theorem 2.12).

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Coefficient Inequalities,ch such coefficients satisfy. Historically, three interrelated problems have played a central role in this subject. These are now solved thanks to a long effort by many people culminating in a complete resolution in de Branges [1985].

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Harmonic Functions,A complex-valued function . on an open subset Ω of the complex plane C is called . on Ω if . ∈ .(Ω) and . on . Here . is the Laplacian of . We often assume that Ω is a region (that is, an open and connected set) even when connectivity is not needed, and we are mainly interested in the case in which Ω is a disk or half-plane.

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Function Theory on a Half-Plane,The purpose of this chapter is to present the ideas of Chapters 1–4 in a half-plane setting. Often this is done by mapping the upper half-plane II = {.: Im . > 0} to the disk . by a linear fractional transformation. In some cases (such as the Stieltjes inversion formula 5.4), it is simplest to give a direct proof for the half-plane.

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