Asymptotic Functions of Entire Functions

If f is an entire function and a complex number, said to be asymptotic value of if there exists path $$\gamma $$ from 0 infinity such that $$f(z) - a$$ tends as z along . The Denjoy–Carleman–Ahlfors Theorem asserts has n distinct values, then the rate growth at least order n/2, mean type. A long-standing problem asks whether this conclusion holds for functions having (entire) functions, each most 1/2, minimal In paper conditions on associated paths are obtained sufficient guarantee satisfies Theorem. addition, positive integer n, example given distinct, prescribed less than 1/2.