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

Abstract

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.

Distortion of Quasiregular Mappings and Equivalent Norms on Lipschitz-Type Spaces

“…Applying (a.3) to the disk , ∈ , we get (41). It is clear that (41) implies (42). For a planar hyperbolic domain in C, we denote by = and hyp = hyp; the hyperbolic density and metric of , respectively.…”

“…Lemma 36 (see [42], second version of Koebe theorem for analytic functions). Let = ( ; ); let be holomorphic function on , = ( ), ( ) = , and let the unbounded component ∞ of be not empty, and…”

On planar harmonic Lipschitz and planar harmonic Hardy classes

“…Again, for any z 1 , z 2 ∈ G, by (12) and (15), there exists a positive constant C 1 such that z 2 ) and the proof of this theorem is completed.…”

“…Later in [16, Theorems A], Pavlović came up with a relatively simple proof of the results of Dyakonov. Recently, many authors considered this topic and generalized Dyakonov's results to pseudo-holomorphic functions and real harmonic functions of several variables for some special majorant ω(t) = t α , where α > 0 (see [9,11,12,13,14]). In this paper, we first extend [16, Theorems A and B] to planar K-quasiregular harmonic mappings as follows, where K ≥ 1.…”

Lipschitz-type spaces of pluriharmonic mappings

“…Recently, many authors considered this topic and generalized the work of Dyakonov to the cases of holomorphic functions, quasiconformal mappings, pseudo-holomorphic functions and real harmonic functions with several variables, see [1, 3-6, 8, 10, 14]. Using version of Koebe theorem for analytic functions or Bloch theorem, a simple proof and generalization of Dyakonov are given in [15][16][17][18]. By using the Garsia-type norm in B n , the authors in [6] got some characterizations for holomorphic functions to be in Λ ω (B n ).…”