Equivalent norms on lipschitz-type spaces of holomorphic functions

Dyakonov, Konstantin M.

doi:10.1007/bf02392692

Cited by 119 publications

(113 citation statements)

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“…First, it is clear that our proof also works for the holomorphic Lipschitz space since norms corresponding to (1) and Lemma 2.1 are available in this situation. This will give us a fairly simple proof of Theorem 2 in [6]. In fact, Theorem 1.1 here can be considered a Besov space version of the mentioned theorem (see also [9] for a different simple proof).…”

Section: A Norm On the Holomorphic Besov Space 237mentioning

confidence: 79%

A norm on the holomorphic Besov space

Bøe

2002

Proc. Amer. Math. Soc.

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Section: A Norm On the Holomorphic Besov Space 237mentioning

confidence: 79%

A norm on the holomorphic Besov space

Bøe

2002

Proc. Amer. Math. Soc.

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“…Roughly speaking, the first and second version of Koebe theorem for analytic functions state that holomorphic functions have the same dilatation in all directions and it indicates similar behavior of holomorphic function and its modulus in certain sense and leads, via crucial estimate (2.3), to what we call geometric visual proof of Dyakonov's theorem [Dyk1] The original proof of Theorem A was complicated; for simple proofs see [P], [MM1], and [MM3].…”

Section: Applicationsmentioning

confidence: 96%

Versions of Koebe 1/4 theorem for analytic and quasiregular harmonic functions and applications

Mateljević

2008

Publ Inst Math (Belgr)

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Abstract. In this paper we mainly survey results obtained in [MM3]. For example, we give an elementary proof of two versions of Koebe 1/4 theorem for analytic functions (see Theorem 1.2 and Theorem 1.4 below). We also show a version of the Koebe theorem for quasiregular harmonic functions. As an application, we show that holomorphic functions (more generally quasiregular harmonic functions) and their modulus have similar behavior in a certain sense. Two versions of Koebe 1/4 theorem for analytic functionsThis paper can be considered as the review of some results presented in [MM3], but it also contains new results and proofs. We will use the following notation. If r > 0 and a is a complex number B(a; r) = {z ∈ C : |z − a| < r} is the open circular disc with center at a and radius r. Also we use notation ∆ r = B(0, r) and ∆ = ∆ 1 . First, we introduce a particulary interesting class of conformal mappings of the disc, the class S. We denote by S the class of holomorphic functions f in ∆ which are injective and satisfy the normalization conditions f (0) = 0 and f (0) = 1.The proof of Koebe's One-Quarter Theorem is based on the extremal property of the Grötzsch annulus, which we first need to discuss. Grötzsch Theorem and Koebe's One-Quarter Theorem.If Ω is a double connected domain, by M (Ω) we denote the modulus of Ω; and for a given family of curves Γ by M (Γ) modulus of the family of curves Γ. If Ω is a double connected domain and E 1 and E 2 the components of ∂Ω the extremal distance d Ω (E 1 , E 2 ) between E 1 and E 2 is the modulus of Ω. Note that if Γ is the family of curves which joins the components E 1 and E 2 , then d Ω (E 1 , E 2 ) = 1/M (Γ) and if Γ is the family of curves which separates the components E 1 and E 2 then M (Ω) = M (Γ ).Let 0 < r < 1 and c be any continum that contains {0, r}; and let Γ, Γ 0 and Γ c be the families of closed curves in the unit disk that separate {0, r}, s = [0, r] 2000 Mathematics Subject Classification: Primary 30C62, 30C80, 30C55; Secondary 30H05.

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“…There are more general weighted Lipschitz spaces which are extensively studied in ( [2], [3], [4]), and so on. Especially, in [3] they proved the same results in weighted Lipschitz spaces like as ours.…”

Section: Characterization Of Bm O or Lipschitz Functions 93mentioning

confidence: 99%