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In this paper we mainly survey results obtained in (MM3). For example, we give an elementary proof of two versions of Koebe 1/ 4t heorem 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. 1. Two versions of Koebe 1/4 theorem for analytic functions This 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> 0a nda 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 )a nd ∆=∆ 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 Grotzsch annulus, which we first need to discuss. 1.1. Grotzsch Theorem and Koebe's One-Quarter Theorem. If Ω is a

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