Interpolation by Rational ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-Inner Function and Rational ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-Inner Function

We study several function-theoretic aspects of the domains $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$. We introduce and investigate the notions of ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-inner functions and ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-inner functions, and establish their relationships with ${\mathrm{\Gamma }}_{E(2;2;1,1)}$-inner functions and ${\mathrm{\Gamma }}_{E(2;1;2)}$-inner functions. We provide explicit constructions and representations of rational ${\mathrm{\Gamma }}_{E(3;3;1,1,1)}$-inner functions and rational ${\mathrm{\Gamma }}_{E(3;2;1,2)}$-inner functions. As an application of these results, we obtain partial solutions to the associated interpolation problems and construct explicit rational inner interpolants for interpolation data lying in suitable subsets of $G_{E(3;3;1,1,1)}$ and $G_{E(3;2;1,2)}$, respectively.