English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 $D$ suchthat $P(z)- \frac{1}{z-a}\ (Q(z)- \frac{1}{z-a})$ 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; \ $\alpha =\log|P'(b)|$ $(\beta=\log|Q'(b)|)$ the $L_1$-($L_0$-)constant, and \ $s=\alpha - \beta $ theharmonic span, for $D$. S.\,Hamano in \cite{hamano-2} showed the variationformula of the second order for the $L_1$-const. $\alpha (t)$ for the movingdomain $D(t)$ in $\mathbb{C}_z$ with $t \in B:=\{t\in \mathbb{C}: |t|<\rho\}$.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 ${\mathcalD}=\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 eachfiber $D(t), t\in B$ and the pseudoconvexity of ${\mathcal D}$. As a simpleapplication we obtain the subharmonicity of $\log \cosh d(t)$ on $B$, where$d(t)$ is the Poincar\'e distance between $a$ and $b$.

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