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
$\theta \,\in \,\mathbb{R}$
.