主函数的变分公式（II）在调和跨度变分中的应用
Variation formulas for principal functions (II) Applications to variation for harmonic spans
摘要：
对于$\mathbb{C}_z$中具有光滑边界的域$D$和$a,b\in D, a\ne b$，我们在$D$上有圆形（径向）狭缝映射$P(z)(Q(z))$，使得$P(z)- \frac{1}{z-a}\ (Q(z)- \frac{1}{z-a})$在$a$和$P(b)(Q(b))=0$处是正则的，我们称$p(z)=\log |P(z)|\ (q(z)=\log|Q(z)|)$为$L_1$ -（$L_0$ -）主函数;对于$D$，\ $α=\log|P'(b)|$ $(β=\log|Q'(b)|)$为$L_1$ -（$L_0$ -）常数，\ $s=α- β$为调和跨度。S.\，Hamano在\cite{hamano-2}中给出了$L_1$ -常数$α(t)$的二阶变分公式，对于$\mathbb{C}_z$中的移动畴$D(t)$，用$t \in B:=\{t\in \mathbb{C}: |t|<ρ\}$ .给出了$D(t)$的$L_0$ -常数$β(t)$的相应公式，并结合这些公式得到：如果全空间${\mathcal D}=\cup_{t\in B}(t, D(t)) $在$ B \times \mathbb{C}_z$中是伪凸的，则$s(t)$在$B$上是次调和的.由于给出了$s(t)$的几何意义，这一事实给出了每个纤维$D(t), t\in B$上的共形映射与${\mathcal D}$的伪凸性之间的关系之一。作为一个简单的应用，我们获得了$\log \cosh d(t)$在$B$上的亚谐性，其中$d(t)$是$a$和$b$之间的庞加莱距离。
英文摘要：
For a domain $D$ in $\mathbb{C}_z$ with smooth boundary and for $a,b\in D, a\ne b$, we have the circular (radial) slit mapping $P(z)(Q(z))$ on $D$ such that $P(z)- \frac{1}{z-a}\ (Q(z)- \frac{1}{z-a})$ is regular at $a$ and $P(b)(Q(b))=0$, and we call $p(z)=\log |P(z)|\ (q(z)=\log|Q(z)|)$ the $L_1$-($L_0$-)principal function; \ $α=\log|P'(b)|$ $(β=\log|Q'(b)|)$ the $L_1$-($L_0$-)constant, and \ $s=α- β$ the harmonic span, for $D$. S.\,Hamano in \cite{hamano-2} showed the variation formula of the second order for the $L_1$-const. $α(t)$ for the moving domain $D(t)$ in $\mathbb{C}_z$ with $t \in B:=\{t\in \mathbb{C}: |t|<ρ\}$. We show the corresponding formula for the $L_0$-const. $β(t)$ for $D(t)$, and combine these formulas to obtain, if the total space ${\mathcal D}=\cup_{t\in B}(t, D(t)) $ is pseudoconvex in $ B \times \mathbb{C}_z$, then $s(t)$ is subharmonic on $B$. Since the geometric meaning of $s(t)$ is showed, this fact gives one of the relations between the conformal mappings on each fiber $D(t), t\in B$ and the pseudoconvexity of ${\mathcal D}$. As a simple application we obtain the subharmonicity of $\log \cosh d(t)$ on $B$, where $d(t)$ is the Poincaré distance between $a$ and $b$.
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