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Unit disk
About: Unit disk is a research topic. Over the lifetime, 4221 publications have been published within this topic receiving 42637 citations. The topic is also known as: open unit disk.
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TL;DR: In this article, the authors extend the duality between HI and BMO in terms of boundedness on L 2 of the commutator of a singular integral operator with a multiplication operator and show a close relationship between BMO functions and certain linear operators on various LI and H2 spaces.
Abstract: The purpose of this paper is to extend to Hardy spaces in several variables certain well known factorization theorems on the unit disk. The extensions will be carried out for various spaces of holomorphic functions on the unit ball of C" as well as for Hardy spaces defined by the Riesz systems on R". These results together with their proofs yield new characterizations of the space BMO (Bounded Mean Oscillation) and show a close relationship between BMO functions and certain linear operators on various LI and H2 spaces. The main tools are the result of Fefferman and Stein [8] on the duality between HI and BMO and a new characterization of BMO in terms of boundedness on L2 of the commutator of a singular integral operator with a multiplication operator. We begin by illustrating these ideas in the one dimensional case: Let F be holomorphic in {I z I < 1} and satisfy sup, 5 F(rete) I dO ? 1 (i.e., F is in H'(dO)). It is well known that F = GG2 with G1, G2 holomorphic and sup, I G,(rel0) 1' ! 1 (i.e., G, e H2(dO)). Write F = f + if, G, = gj + ig withf, g1, g, real. Thenf = Im(GG2) = sg1 1 + gi. Thusafunction f is an imaginary (or real) part of an HI function if and only if it can be represented as glg2 + g192 for L2 functions g, and g2. Furthermore,
1,304 citations
TL;DR: In this paper, the Nevanlinna characteristic function of a regular function is defined as a convex increasing function with bounded characteristic in the sense that it always exists as a finite or infinite limit.
Abstract: n(r, F) as the number of poles in ]z] d r and = f r n(t, F) dt N(r, F) J0 $ Then T(r, F) = re(r, F) + N(r, F) is called the Nevanlinna characteristic function of F(z). The function T(r, F ) i s convex increasing function of log r, so tha t T(1, F) = lim T(r, F) r--~l always exists as a finite or infinite limit. I f T(1, F) is finite we say tha t F(z) has bounded characteristic in ]z] < 1. Examples show tha t F(z) m a y have bounded characteristic in ]z] < 1, even i f / (z) does not.(1) We may take for instance /(z) to be a regular function
614 citations
TL;DR: Two interesting subclasses of normalized analytic and univalent functions in the open unit disk whose inverse has univalently analytic continuation to U is introduced and investigated.
532 citations
01 Dec 1992
TL;DR: In this article, the coefficients of the univalent functions of the Nevanlinna classes N1 and N2 were defined by conditions on the circle, and a certain class of generalized hypergeometric functions associated with the Hardy space of analytic functions.
Abstract: Univalent logharmonic extensions onto the unit disk or onto an annulus, Z Abdulhadi and W Hengartner hypergeometric functions and elliptic integrals, GD Anderson et al a certain class of caratheodory functions defined by conditions on the circle, J Fuka and ZJ Jakubowski recent advances in the theory of zero sets of the Bergman spaces, EA LeBlanc a coefficient functional for meromorphic univalent functions, L Liu spherical linear invariance and uniform local spherical convexity, W Ma and D Minda a special differential subordination and its application to univalency conditions, SS Miller and PT Mocanu on the Bernardi integral functions, S Owa analytic solutions of a class of Briot-Bouquet differential equations, S Owa and HM Srivastava a certain class of generalized hypergeometric functions associated with the Hardy space of analytic functions, HM Srivastava on the coefficients of the univalent functions of the Nevanlinna classes N1 and N2, PG Todorov
434 citations
TL;DR: In this paper, the problem of determining properties of functions p that satisfy the differential superordination Ω ⊂ {ψ(p(z), z 2 p"(z);z)|z ∈ U}.
Abstract: Let Ω be any set in the complex plane ℂ, let p be analytic in the unit disk U and let ψ (r, s, t; z). In this article we consider the problem of determining properties of functions p that satisfy the differential superordination Ω ⊂ {ψ(p(z), z 2 p"(z);z)|z ∈ U}.
380 citations