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Let 
${{B}_{p1,p2}}\,=\,\left\{ z\,\in \,{{\mathbb{C}}^{n}}\,:\,{{\left| {{z}_{1}} \right|}^{{{p}_{1}}}}\,+\,{{\left| {{z}_{2}} \right|}^{{{p}_{2}}}}\,+\,\cdots \,+\,{{\left| {{z}_{n}} \right|}^{{{p}_{2}}}}\,<\,1 \right\}$ be an egg domain in 
${{\mathbb{C}}^{n}}$. In this paper, we first characterize the Kobayashi metric on 
${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge 1,\,{{p}_{2}}\,>\,1 \right)$ and then establish a new type of classical boundary Schwarz lemma at 
${{z}_{0}}\in \partial {{B}_{{{p}_{1}},{{p}_{2}}}}$ for holomorphic self-mappings of 
${{B}_{{{p}_{1}},{{p}_{2}}}}\,\left( {{p}_{1}}\,\ge \,1,\,{{p}_{2}}\,>\,1 \right)$), where 
${{z}_{0}}={{\left( {{e}^{i\theta }},\,0,\ldots ,0 \right)}^{\prime }}$ and 
$\theta \,\in \,\mathbb{R}$.
