Variation formulas for principal functions, II: Applications to variation for harmonic spans

Abstract

A domain $D\subset {\mathbb{C}}_z$ admits the circular slit mapping $P\left(z\right)$ for $a,b\in D$ such that $P\left(z\right)-1/(z-a)$ is regular at $a$ and $P\left(b\right)=0$. We call $p\left(z\right)=log\left|P\right(z\left)\right|$ the $L_1$-principal function and $\alpha =log\left|P\text{'}\right(b\left)\right|$ the $L_1$-constant, and similarly, the radial slit mapping $Q\left(z\right)$ implies the $L_0$-principal function $q\left(z\right)$ and the $L_0$-constant $\beta$. We call $s=\alpha -\beta$ the harmonic span for $(D,a,b)$. We show the geometric meaning of $s$. Hamano showed the variation formula for the $L_1$-constant $\alpha \left(t\right)$ for the moving domain $D\left(t\right)$ in ${\mathbb{C}}_z$ with $t\in B:=\{t\in \mathbb{C}:|t|<\rho \}$. We show the corresponding formula for the $L_0$-constant $\beta \left(t\right)$ for $D\left(t\right)$ and combine these to prove that, if the total space $\mathcal{D}={\bigcup }_{t\in B}(t,D(t\left)\right)$ is pseudoconvex in $B\times {\mathbb{C}}_z$, then $s\left(t\right)$ is subharmonic on $B$. As a direct application, we have the subharmonicity of $logcoshd\left(t\right)$ on $B$, where $d\left(t\right)$ is the Poincaré distance between $a$ and $b$ on $D\left(t\right)$.