A domain admits the circular slit mapping for such that is regular at and . We call the -principal function and the -constant, and similarly, the radial slit mapping implies the -principal function and the -constant . We call the harmonic span for . We show the geometric meaning of . Hamano showed the variation formula for the -constant for the moving domain in with . We show the corresponding formula for the -constant for and combine these to prove that, if the total space is pseudoconvex in , then is subharmonic on . As a direct application, we have the subharmonicity of on , where is the Poincaré distance between and on .
"Variation formulas for principal functions, II: Applications to variation for harmonic spans." Nagoya Math. J. 204 19 - 56, December 2011. https://doi.org/10.1215/00277630-1431822