Variation formulas for principal functions (II) Applications to variation for harmonic spans
S. Hamano, F. Maitani, H. Yamaguchi · arXiv (Cornell University) · 2010
For a domain $D$ in $\mathbb{C}_z$ with smooth boundary and for $a,b\in D, a e b$, 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|