On the functional equation $f^{2}=e^{2\phi _{1}}+e^{2\phi _{2}}+e^{2\phi _{3}}\$ and a new Picard theorem

Abstract:

By analogy with E. Borel’s reduction of the classical Picard theorem to an analytic statement about linear relations among exponentials of entire functions, a new Picard theorem is proved by considering the functional relation ${f^2} = {e^{2{\phi _1}}} + {e^{2{\phi _2}}} + {e^{2{\phi _3}}}$ for entire functions. The analytic techniques used are those of Nevanlinna theory.