On the subrings of the ring of analytic functions and conformally equivalance

Abstract

We consider the discrete sets $D_isubset G_1$. of the regions $G_1$ in the complex plane e and study the subrings of the rings of analytic functions $A(G_1)$ corresponding to these discrete sets $D_i (i=12)$. Furthermore, we prove that the sets of zeros of the functions which map onto each other under the e-isomorphism $Phi: A(G_1)rightarrow A(G_2)$, are also mapped onto each other by a conformal mapping $Phi: G_2rightarrow G_1$, where $Phi(f)=fophi$.