Variation formulas for principal functions (II) Applications to variation for harmonic spans

For a domain $D$ in ${\mathbb{C}}_z$ with smooth boundary and for $a,b\in D, a\ne b$,wehavethecircular(radial)slitmapping$P(z)(Q(z))$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 $p(z)=\log \mathrm{\mid }P(z)\mathrm{\mid }(q(z)=\log \mathrm{\mid }Q(z)\mathrm{\mid })$ the $L_1$-($L_0$-)principal function; \ $\alpha =\log \mathrm{\mid }{P}^{\mathrm{\prime }}(b)\mathrm{\mid }$ $(\beta =\log|Q’(b)|)$the$L_1$-($L_0-)constant, and \s=\alpha - \beta $ the harmonic span, for $D$. S.,Hamano in \cite{hamano-2} showed the variation formula of the second order for the $L_1$-const. $\alpha (t)$ for the moving domain $D(t)$ in ${\mathbb{C}}_z$ with $t\in B:=\{t\in \mathbb{C}:\mathrm{\mid }t\mathrm{\mid }<\rho \}$. We show the corresponding formula for the $L_0$-const. $\beta (t)$ for $D(t)$, and combine these formulas to obtain, if the total space ${\mathcal D}=\cup_{t\in B}(t, D(t)) $ispseudoconvexin$ B \times $\mathbb{C}$_z$,then$s(t)$issubharmonicon$B$\mathrm{.}Sincethegeometricmeaningof$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 Poincar'e distance between $a$ and $b$.