On additive automorphic and rotation automorphic functions (Q792486)

A function \(W(z)\) meromorphic in the unit disk D is said to be additive automorphic relative to the Fuchsian group \(\Gamma\) if for each transformation \(T\in \Gamma\) there exists a constant \(A_ T\) such that \(W(T(z))=W(z)+A_ T\) for each \(z\in D\). A function \(f(z)\) meromorphic in D is said to be a normal function if there exists a constant \(N_ f\) such that \((1-| z|^ 2)| f'(z)| /(1+| f(z)|^ 2)\leq N_ f\) for each \(z\in D\). The main theorem is the following: There exists an additive automorphic function W(z) relative to a Fuchsian group \(\Gamma\) such that \(W(z)\) is not a normal function, W(z) has only imaginary periods, and \(\iint_{F}| W'(z)|^ 2dxdy<\infty,\) where F denotes the fundamental region of \(\Gamma\). We say that the harmonic function \(u(z)\) defined in D is a normal function if \(\sup \{(1- | z|^ 2)| \text{grad} u(z)| /(1+| u(z)|^ 2):\quad z\in D\}<\infty.\) As a corollary from the main theorem we obtain: There exists an additive automorphic function \(W(z)\) such that \(\iint_{F}| W'(z)|^ 2dxdy<\infty\) and \(u(z)=re(W(z))\) is an automorphic harmonic function which is not a normal function but \(\iint_{F}(u^ 2_ x(z)+u^ 2_ y(z))dxdy<\infty.\) We say that the meromorphic function \(G(z)\) in D is a rotation automorphic function relative to a Fuchsian group \(\Gamma\) if for each \(T\in \Gamma\) there exists a linear fractional transformation \(S_ T\), where \(S_ T\) is a rotation of the Riemann sphere, such that \(G(T(z))=S_ T(G(z))\) for each \(z\in D\). We prove the following: There exists a rotation automorphic function \(G(z)\) such that \(G(z)\) is not a normal function and \(\iint_{F}| G'(z)|^ 2dxdy<\infty.\)