Ahlfors-Type Theorem for Hausdorff Measures

Suppose that Δ ⊂ C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}}$$\end{document} is a domain, f is an analytic function in Δ, D = f(Δ) is considered as a Riemann surface. Put lR = {z ∈ Δ : |f(z)| = R}. Let E ⊂ Δ be a closed set. Put hα,β(r) = rα| ln r|β, 0 < α < 1, 0 < β < 1. Let Λα,β(·), Λα+1,β(·) be the Hausdorff measures with respect to the functions hα,β, hα+1,β. Assume that Λα+1,β(E) < ∞. We introduce the sets lR,ε = {z ∈ lR : dist(z, ∂Δ) ≥ ε, |z| ≤ 1ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{\varepsilon }$$\end{document}} and TR,ε = f(lR,ε ∩ E), TR,ε ⊂ D. Put GεR=0ifΛα,βTR,ε=0orΛα,βTR,ε=∞,Λα,β1+ααE∩lR,εΛα,β1αTR,εif0<Λα,βTR,ε<∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G}_{\varepsilon }\left(R\right)=\left\{\begin{array}{l}0\ \ \ if\, {\Lambda }_{\alpha ,\beta }\left({T}_{R,\varepsilon }\right)=0\ or\ {\Lambda }_{\alpha ,\beta }\left({T}_{R,\varepsilon }\right)=\infty ,\\ \frac{{\Lambda }_{\alpha ,\beta }^{\frac{1+\alpha }{\alpha }}\left(E\cap {l}_{R,\varepsilon }\right)}{{\Lambda }_{\alpha ,\beta }^{\frac{1}{\alpha }}\left({T}_{R,\varepsilon }\right)}\ \ if\, 0<{\Lambda }_{\alpha ,\beta }\left({T}_{R,\varepsilon }\right)<\infty .\end{array}\right.$$\end{document}