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 Hardy-Littlewood Property and $\alpha$-Quasihyperbolic Metric Commun. Korean Math. Soc.Published online September 2, 2019 Ki Won Kim and Jeong Seog Ryu Silla University, Hongik University Abstract : Hardy and Littlewood found a relation between the smoothness of the radial limit of an analytic function on unit disk $D \subset {\mathbb C}$ and the growth of its derivative. It is reasonable to expect an analytic function to be smooth on the boundary if its derivative grows slowly, and conversely. Gehring and Martio showed this principle for uniform domains in ${\mathbb R}^{2}$. Astala and Gehring proved quasiconformal analogue of this principle for uniform domains in ${\mathbb R}^n$. We consider \textit{$\alpha$-quasihyperbolic metric}, $k_D^{\alpha}$ and we extend it to proper domains in ${\mathbb R}^n$. Keywords : Hardy-Littlewood Property, Quasiconformal Mapping, Quasihyperbolic Metric MSC numbers : 30C65 Full-Text :